Download Quantum Mechanics

Quantum Mechanics
Uwe-Jens Wiese
Institute for Theoretical Physics
Albert Einstein Center for Fundamental Physics
Bern University
November 30, 2015
2
Preface
Learning quantum mechanics is probably the most exciting and, at the same time, the
most disturbing experience in the education of a physicist. The development of quantum
physics is one of the greatest intellectual achievements of science in the twentieth century.
It has opened the door to aspects of reality that are far beyond what we can access with the
concepts of classical physics. At atomic scales Nature is governed by the rules of quantum
physics. In the twenty-first century, about one hundred years after the beginnings of quantum mechanics, it remains the basis for research at the forefront of fundamental physics.
Particle and nuclear physics, atomic and molecular physics, chemistry, condensed matter
and mesoscopic physics, quantum optics, as well as astrophysics and modern cosmology
are all unthinkable without quantum physics. Understanding quantum mechanics is thus
an indispensable prerequisite for contributing to these exciting fields of current research.
In contrast to classical physics which emerges from quantum physics at macroscopic
scales, quantum physics is — as far as we know today — truly fundamental. While
classical physics fails at microscopic scales because it is just not applicable there, no
experiment has ever indicated a failure of quantum physics. On the contrary, numerous
theoretical predictions of quantum mechanics, which seem paradoxical from a classical
physics point of view, have been verified experimentally in great detail. In contrast to
classical physics, quantum physics is not directly accessible to our everyday experience.
Consequently, its theoretical description is more abstract than the one of classical physics.
While classical physics is deterministic, quantum physics is probabilistic. As a result, some
classical concepts like the path of a particle or the distinguishability of elementary objects
break down at microscopic scales. After a century of research at the quantum level, even
the interpretation and meaning of quantum mechanics remain partly unsettled issues. It
is a big effort and a great intellectual challenge for a physics student to go beyond the
intuitive concepts of macroscopic classical physics and to understand the abstract quantum
reality at microscopic scales. The rewards are plenty because understanding quantum
mechanics opens the door to a whole world of exciting phenomena ranging from the physics
of single elementary particles to atoms, molecules, laser light, Bose-Einstein condensates,
strongly correlated electron systems including high-temperature superconductors, to the
dense matter at the core of neutron stars, and the hot gas of electrons, photons, quarks,
and other elementary particles that filled the early Universe.
These lectures address the curious student learning quantum mechanics at an early
3
4
stage, willing to enrich his or her mathematical tool box, eager to move on to the questions
that drive current research. As an additional motivation, the lectures connect quantum
mechanics to some big questions that we may hope to answer during this century. The
author wants to encourage the student to face these problems, and think about them at a
deep level. This should serve as a strong motivation to penetrate the subject of quantum
mechanics in a profound manner. Although quantum mechanics is already about onehundred years old, it is one of the most promising tools that will allow us to push the
frontiers of present knowledge further into the unknown.
At the forefront of current research in fundamental physics a number of advanced
concepts are needed. These concepts include, for example, group theory, Abelian and
non-Abelian gauge fields, or topological considerations, which are usually not emphasized much in the teaching of elementary quantum mechanics. In order to facilitate a
smooth progression to quantum field theory and other more advanced topics, these lectures introduce these concepts already within quantum mechanics. For example, we will
encounter non-Abelian gauge fields when studying the adiabatic approximation and the
Pauli equation, we will learn about the groups O(4) and SU (3) when investigating accidental symmetries of the hydrogen atom and the harmonic oscillator, and we will use the
permutation group SN when discussing the statistics of identical particles. The current
text attempts to present a modern view of quantum physics. While still covering the
classical topics that can already be found in old books on the subject, we will focus on
topics of current interest including neutrino oscillations, the cosmic background radiation,
the quark content of protons and neutrons, the Aharonov-Bohm effect, the quantum Hall
effect, quantum spin systems and quantum computation, as well as on the Berry phase.
Modern physics is a rich and very diverse field. It covers all length scales from single
elementary particles to the entire cosmos, all time scales from the shortest laser pulse to
the age of the Universe, and all energy scales from the coldest Bose-Einstein condensate to
the extremely hot early Universe at the Planck scale. Necessarily, physicists specialize in
one of the many exciting subfields, and it is sometimes difficult to see physics as a whole.
Quantum mechanics is one of the uniting themes that allows us to address physics in its
entirety. These lectures want to underscore that physics is one entity, containing countless
exciting facets, all held together by mathematics, the universal language that Nature has
chosen to express herself in. The fascination that results from this fact has driven the
author in all of his work and also in writing these lecture notes.
Quantum mechanics is such an important subject that the student needs a sufficient
amount of time to understand the material at a deep level. At Bern University and at
MIT undergraduate quantum mechanics is taught in three semesters. This reflects a motto
(well known at MIT) which the author has tried to follow in his teaching:
Victor Weisskopf: “It is better to uncover a little than to cover a lot.”
In physics as well as in mathematics a given theoretical framework can be understood
completely. Achieving complete command of a complex subject such as quantum mechanics needs time, but leaves us with a sense of empowerment and an urge to progress to more
5
advanced topics. Empowering the curious student and encouraging him or her to think
about Nature’s biggest puzzles at a deep level is a major goal of these lectures.
6
Contents
1 Introduction
13
1.1
Motivation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2
The Cube of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3
What is Quantum Mechanics? . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Beyond Classical Physics
21
2.1
Photons — the Quanta of Light . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2
The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3
The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4
Some Big Questions and the Cosmic Background Radiation . . . . . . . . . 24
2.5
Planck’s Formula for Black Body Radiation . . . . . . . . . . . . . . . . . . 28
2.6
Quantum States of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.7
The Double-Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.8
Estimating Simple Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 De Broglie Waves
43
3.1
Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2
Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3
Coordinate and Momentum Space Representation of Operators . . . . . . . 46
3.4
Heisenberg’s Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . 47
3.5
Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7
8
CONTENTS
3.6
Spreading of a Gaussian Wave Packet . . . . . . . . . . . . . . . . . . . . . 51
4 The Schrödinger Equation
53
4.1
From Wave Packets to the Schrödinger Equation . . . . . . . . . . . . . . . 53
4.2
Conservation of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3
The Time-Independent Schrödinger Equation . . . . . . . . . . . . . . . . . 56
5 Square-Well Potentials and Tunneling Effect
59
5.1
Continuity Equation in One Dimension
. . . . . . . . . . . . . . . . . . . . 59
5.2
A Particle in a Box with Dirichlet Boundary Conditions . . . . . . . . . . . 60
5.3
The Tunneling Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4
Semi-Classical Instanton Formula . . . . . . . . . . . . . . . . . . . . . . . . 66
5.5
Reflection at a Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.6
Quantum Height Anxiety Paradox . . . . . . . . . . . . . . . . . . . . . . . 68
5.7
Free Particle on a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.8
Free Particle on a Discretized Circle . . . . . . . . . . . . . . . . . . . . . . 71
6 Spin, Precession, and the Stern-Gerlach Experiment
73
6.1
Quantum Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2
Neutron Spin Precession in a Magnetic Field . . . . . . . . . . . . . . . . . 77
6.3
The Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 79
7 The Formal Structure of Quantum Mechanics
81
7.1
Wave Functions as Vectors in a Hilbert Space . . . . . . . . . . . . . . . . . 81
7.2
Observables as Hermitean Operators . . . . . . . . . . . . . . . . . . . . . . 82
7.3
Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.4
Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.5
Ideal Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.6
Simultaneous Measurability and Commutators . . . . . . . . . . . . . . . . 87
CONTENTS
9
7.7
Commutation Relations of Coordinates, Momenta, and Angular Momenta . 89
7.8
Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8 Contact Interactions in One Dimension
93
8.1
Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.2
A Simple Toy Model for “Atoms” and “Molecules” . . . . . . . . . . . . . . 94
8.3
Shift Symmetry and Periodic Potentials . . . . . . . . . . . . . . . . . . . . 98
8.4
A Simple Toy Model for an Electron in a Crystal . . . . . . . . . . . . . . . 99
9 The Harmonic Oscillator
103
9.1
Solution of the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . 103
9.2
Operator Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.3
Coherent States and the Classical Limit . . . . . . . . . . . . . . . . . . . . 107
9.4
The Harmonic Oscillator in Two Dimensions . . . . . . . . . . . . . . . . . 111
10 The Hydrogen Atom
115
10.1 Separation of the Center of Mass Motion . . . . . . . . . . . . . . . . . . . . 115
10.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.3 Solution of the Radial Equation . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.4 Relativistic Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
11 EPR Paradox, Bell’s Inequality, and Schrödinger’s Cat
123
11.1 The Einstein-Podolsky-Rosen Paradox . . . . . . . . . . . . . . . . . . . . . 123
11.2 The Quantum Mechanics of Spin Correlations . . . . . . . . . . . . . . . . . 125
11.3 A Simple Hidden Variable Model . . . . . . . . . . . . . . . . . . . . . . . . 125
11.4 Bell’s Inequality
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
11.5 Schrödinger’s Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
12 Abstract Formulation of Quantum Mechanics
131
10
CONTENTS
12.1 Dirac’s Bracket Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.2 Unitary Time-Evolution Operator
. . . . . . . . . . . . . . . . . . . . . . . 136
12.3 Schrödinger versus Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . 136
12.4 Time-dependent Hamilton Operators . . . . . . . . . . . . . . . . . . . . . . 137
12.5 Dirac’s Interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
13 Quantum Mechanical Approximation Methods
141
13.1 The Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
13.2 Non-Degenerate Perturbation Theory to Low Orders . . . . . . . . . . . . . 143
13.3 Degenerate Perturbation Theory to First Order . . . . . . . . . . . . . . . . 146
13.4 The Hydrogen Atom in a Weak Electric Field . . . . . . . . . . . . . . . . . 147
13.5 Non-Degenerate Perturbation Theory to All Orders . . . . . . . . . . . . . . 149
13.6 Degenerate Perturbation Theory to All Orders . . . . . . . . . . . . . . . . 151
14 Charged Particle in an Electromagnetic Field
155
14.1 The Classical Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . 155
14.2 Classical Particle in an Electromagnetic Field . . . . . . . . . . . . . . . . . 157
14.3 Gauge Invariant Form of the Schrödinger Equation . . . . . . . . . . . . . . 159
14.4 Magnetic Flux Tubes and the Aharonov-Bohm Effect . . . . . . . . . . . . . 161
14.5 Flux Quantization for Monopoles and Superconductors . . . . . . . . . . . . 163
14.6 Charged Particle in a Constant Magnetic Field . . . . . . . . . . . . . . . . 165
14.7 The Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
14.8 The “Normal” Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 169
15 Coupling of Angular Momenta
171
15.1 Quantum Mechanical Angular Momentum . . . . . . . . . . . . . . . . . . . 171
15.2 Coupling of Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
15.3 Coupling of Orbital Angular Momentum and Spin . . . . . . . . . . . . . . 176
CONTENTS
11
16 Systems of Identical Particles
177
16.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
16.2 The Pauli Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
16.3 The Helium Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
16.4 Perturbation Theory for the Helium Atom . . . . . . . . . . . . . . . . . . . 184
17 Quantum Mechanics of Elastic Scattering
187
17.1 Differential Cross Section and Scattering Amplitude . . . . . . . . . . . . . 187
17.2 Green Function for the Schrödinger Equation . . . . . . . . . . . . . . . . . 189
17.3 The Lippmann-Schwinger Equation . . . . . . . . . . . . . . . . . . . . . . . 190
17.4 Abstract Form of the Lippmann-Schwinger Equation . . . . . . . . . . . . . 190
18 The Adiabatic Berry Phase
18.1 Abelian Berry Phase of a Spin
193
1
2
in a Magnetic Field . . . . . . . . . . . . . 193
18.2 Non-Abelian Berry Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
18.3 SO(3) Gauge Fields in Falling Cats . . . . . . . . . . . . . . . . . . . . . . . 197
18.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A Physical Units and Fundamental Constants
201
A.1 Units of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.2 Units of Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A.3 Units of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
A.4 Units of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.5 Natural Planck Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
A.6 The Strength of Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 204
A.7 The Feebleness of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
A.8 Units of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A.9 Various Units in the MKS System . . . . . . . . . . . . . . . . . . . . . . . 206
12
CONTENTS
B Scalars, Vectors, and Tensors
209
B.1 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
B.2 Tensors and Einstein’s Summation Convention . . . . . . . . . . . . . . . . 210
B.3 Vector Cross Product and Levi-Civita Symbol . . . . . . . . . . . . . . . . . 211
C Vector Analysis and Integration Theorems
213
C.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
C.2 Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
C.3 Integration Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
D Differential Operators in Different Coordinates
217
D.1 Differential Operators in Spherical Coordinates . . . . . . . . . . . . . . . . 217
D.2 Differential Operators in Cylindrical Coordinates . . . . . . . . . . . . . . . 218
E Fourier Transform and Dirac δ-Function
219
E.1 From Fourier Series to Fourier Transform . . . . . . . . . . . . . . . . . . . 219
E.2 Properties of the δ-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Chapter 1
Introduction
In the following, we will discuss how quantum mechanics is related to the rest of physics
and we will get a first impression what quantum mechanics might be.
1.1
Motivation
In our everyday life we experience a world which seems to be governed by the laws of
classical physics. The mechanics of massive bodies is determined by Newton’s laws and
the dynamics of the electromagnetic field follows Maxwell’s equations. As we know, these
two theories — non-relativistic classical mechanics and classical electrodynamics — are
not consistent with one another. While Maxwell’s equations are invariant against Lorentz
transformations, Newton’s laws are only Galilei invariant. As first realized by Albert
Einstein (1879 - 1955), this means that non-relativistic classical mechanics is incomplete.
Once it is replaced by relativistic classical mechanics (special relativity) it becomes consistent with electromagnetism, which — as a theory for light — had relativity built in from
the start. The fact that special relativity replaced non-relativistic classical mechanics does
not mean that Sir Isaac Newton (1643 - 1727) was wrong. In fact, his theory emerges from
Einstein’s in the limit c → ∞, i.e. if light would travel with infinite speed. As far as our
everyday experience is concerned this is practically the case, and hence for these purposes
Newton’s theory is sufficient. There is no need to convince a mechanical engineer to use
Einstein’s theory because her airplanes are not designed to reach speeds anywhere near
the speed of light
c = 2.99792456 × 108 m sec−1 .
(1.1.1)
Still, as physicists we know that non-relativistic classical mechanics is not the whole story
and relativity (both special and general) has totally altered our way of thinking about
space and time. It is certainly a great challenge for a physics student to abstract from our
everyday experience of flat space and absolute time and to deal with Lorentz contraction,
time dilation, or even curved space-time. At the end, it is Nature that forces us to do
so, because experiments with objects moving almost with the speed of light are correctly
13
14
CHAPTER 1. INTRODUCTION
described by relativity, but not by Newton’s theory. The classical theories of special and
general relativity and of electromagnetism are perfectly consistent with one another.
At the beginning of the twentieth century some people thought that physics was a
closed subject. Around that time a young student — Werner Heisenberg (1901 - 1976,
Nobel Prize in 1932) — was therefore advised to study mathematics instead of physics.
He ignored this advice and got involved in the greatest scientific revolution that has ever
taken place. Fortunately, he knew enough mathematics to formulate the rules of the new
theory — quantum mechanics.
In fact, it was obvious already at the end of the nineteenth century that classical physics
was headed for a crisis. The spectra of all kinds of substances consisted of discrete lines,
indicating that atoms emit electromagnetic radiation at characteristic discrete frequencies.
Classical physics offers no clue to why the frequency of the radiation is quantized. In
fact, classically an electron orbiting around the atomic nucleus experiences a constant
centripetal acceleration and would hence constantly emit radiation. This should soon
use up all its energy and should lead to the electron collapsing into the atomic nucleus:
classically atoms would be unstable.
The thermodynamics of light also was in bad shape. The classical Rayleigh-Jeans law
predicts that the total energy radiated by a black body should be infinite. It required the
genius of Max Planck (1858 - 1947, Nobel prize for the year 1918) to solve this complicated
problem by postulating that matter emits light of frequency ν only in discrete energy units
E = hν. Interestingly, it took a young man like Einstein to take this idea really seriously,
and conclude that light actually consists of quanta — so-called photons. Not for relativity
but for the explanation of the photoelectric effect he received the Nobel prize in 1921.
Planck’s constant
h = 6.6205 × 10−34 kg m2 sec−1
(1.1.2)
is a fundamental constant of Nature. It plays a similar role in quantum physics as c
plays in relativity theory. In particular, it sets the scale at which the predictions of
quantum mechanics differ significantly from those of classical physics. Again, the existence
of quantum mechanics does not mean that classical mechanics is wrong. It is, however,
incomplete and should not be applied to the microscopic quantum world. In fact, classical
mechanics is entirely contained within quantum mechanics as the limit h → 0, just like it
is the c → ∞ limit of special relativity.
Similarly, classical electrodynamics and special relativity can be extended to quantum
electrodynamics — a so-called quantum field theory — by “switching on” h. Other interactions like the weak and strong nuclear forces are also described by quantum field theories
which form the so-called standard model of particle physics. Hence, one might think that
at the beginning of the twenty-first century, physics finally is complete. However, the force
of gravity that is classically described by general relativity has thus far resisted quantization. In other words, again our theories don’t quite fit together when we apply them to
extreme conditions under which quantum effects of gravity become important. Ironically,
1.2. THE CUBE OF PHYSICS
15
today everything fits together only if we put Newton’s gravitational constant
G = 6.6720 × 10−11 kg−1 m3 sec−2
(1.1.3)
to zero. The “simplest” interaction — gravity — which got physics started, is now causing
trouble. This is good news because it gives us something to think about for this century.
In fact, there are promising ideas how to quantize gravity by using, for example, string
theories which might be a consistent way of letting G 6= 0, c 6= ∞ and h 6= 0 all at the
same time.
1.2
The Cube of Physics
In order to orient ourselves further in the space of physics theories let us consider what one
might call the “cube of physics”. Quantum mechanics — the subject of this text — has
its well-deserved place in the space of all theories. Theory space can be spanned by three
axes labeled with the three most fundamental constants of Nature: Newton’s gravitational
constant G, the velocity of light c (which would deserve the name Einstein’s constant),
and Planck’s quantum h. Actually, it is most convenient to label the three axes by G,
1/c, and h. For a long time it was not known that light travels at a finite speed or that
there is quantum mechanics. In particular, Newton’s classical mechanics corresponds to
c = ∞ ⇒ 1/c = 0 and h = 0, i.e. it is non-relativistic and non-quantum, and it thus takes
place along the G-axis. If we also ignore gravity and put G = 0 we are at the origin of
theory space doing Newton’s classical mechanics but only with non-gravitational forces. Of
course, Newton realized that in Nature G 6= 0, but he couldn’t take into account 1/c 6= 0 or
h 6= 0. James Clerk Maxwell (1831 - 1879) and the other pioneers of electrodynamics had
c built into their theory as a fundamental constant, and Einstein realized that Newton’s
classical mechanics needs to be modified to the theory of special relativity in order to take
into account that 1/c 6= 0. This opened a new dimension in theory space which extends
Newton’s line of classical mechanics to the plane of relativistic theories. When we take
the limit c → ∞ ⇒ 1/c → 0, special relativity reduces to Newton’s non-relativistic
classical mechanics. Once special relativity was discovered, it became clear that there
must be a theory that takes into account 1/c 6= 0 and G 6= 0 at the same time. After
years of hard work, Einstein was able to construct this theory — general relativity —
which is a relativistic theory of gravity. The G-1/c-plane contains classical (i.e. nonquantum) relativistic and non-relativistic physics. Classical electrodynamics fits together
naturally with general relativity, and thus relativistic classical physics forms a completely
consistent system. Still, this is not sufficient because it does not describe Nature correctly
at microscopic scales.
A third dimension in theory space was discovered by Planck who started quantum
mechanics and introduced the fundamental action quantum h. When we put h = 0, quantum physics reduces to classical physics. Again, the existence of quantum mechanics does
not mean that classical mechanics is wrong. It is, however, incomplete and should not
be applied to the microscopic quantum world. Classical mechanics is contained within
16
CHAPTER 1. INTRODUCTION
quantum mechanics as the limit h → 0. Quantum mechanics was first constructed nonrelativistically (i.e. by putting 1/c = 0). When we allow h 6= 0 as well as 1/c 6= 0 (but
put G = 0) we are doing relativistic quantum physics. This is where the quantum version
of classical electrodynamics — quantum electrodynamics (QED) — is located in theory
space. Also the entire standard model of elementary particle physics which includes QED
as well as its analog for the strong force — quantum chromodynamics (QCD) — is located
there. Today we know that there must be a consistent physical theory that allows G 6= 0,
1/c 6= 0, and h 6= 0 all at the same time. Although string theory is a promising candidate,
a generally accepted theory of relativistic quantum gravity has not yet been found. Indeed,
this is one of the big questions for this century:
What is the correct theory of quantum gravity?
Obviously, this question directly involves quantum physics. If we want to be able to
think about it, we must understand quantum mechanics.
1.3
What is Quantum Mechanics?
Before one can get involved in an enterprise such as quantum gravity, we must go back
about one hundred years and understand what got people like Planck and Heisenberg
excited in those days. Even more than understanding relativity, this is a tremendous
challenge for a physics student because it requires to leave our everyday experience behind,
and be ready to think in completely new categories. Like for Heisenberg, this will only be
possible if we equip ourselves with some mathematical tools (like matrices, eigenvalues,
eigenvectors, and partial differential equations). The rewards are plenty because this will
open the door to the quantum world of atoms, molecules, atomic nuclei, and elementary
particles, with phenomena that are totally different from anything in our macroscopic
everyday experience.
Learning quantum mechanics can be a truly disturbing — but exciting — experience
because the concept of quantum reality differs from the one of classical physics. Classical
physics is deterministic: given some initial conditions for position and velocity of a particle,
classical mechanics makes definite predictions for the trajectory of the particle. In practice
the predictive power of classical mechanics may be limited because we never know the
initial conditions with absolute precision. This is especially important for chaotic systems.
Still, the classical concept of reality implies that the initial conditions could, at least in
principle, be determined exactly. In the microscopic quantum world this is not possible,
even in principle. In fact, Heisenberg’s uncertainty principle,
∆x∆p ≥
h
,
4π
(1.3.1)
states that position x and momentum p of a particle cannot be determined simultaneously
with unlimited precision. If the position is measured absolutely precisely (∆x = 0) the
momentum will be completely uncertain (∆p = ∞) and vice versa. At best we can
1.3. WHAT IS QUANTUM MECHANICS?
17
achieve ∆x∆p = h/4π. In quantum reality position and momentum cannot be determined
simultaneously. This immediately implies that the concept of a classical particle trajectory
does not make sense at the quantum level. After all, the trajectory just specifies x and p
simultaneously as functions of time.
As Heisenberg’s uncertainty principle indicates, quantum mechanics is not deterministic. In fact, measurements on identically prepared systems generally lead to different
results. For example, a sample of identical uranium atoms shows different behavior of
individual atoms. Some decay right in front of us, others may wait for years to decay, and
we have no way of predicting when a particular atom will dissolve. Still, statistically we
can predict how many atoms will decay on average during a certain period of time. In fact,
quantum mechanics only makes predictions of statistical nature and does not say what an
individual particle “will do next”. It only predicts the probability for a particular result
of a measurement. Hence, only when we prepare identical experiments and repeat them
over and over again, we can determine the statistical distribution of the results and thus
test the predictions of quantum mechanics. The quantum concept of reality implies that
a particle has no definite position or momentum before we measure it. During the measurement process the particle “throws the dice” and “chooses” its position from a given
statistical distribution. According to the standard Copenhagen interpretation of quantum
mechanics a definite position of a particle is not part of quantum reality. Hence, it is not
a failure of quantum mechanics that it does not predict it.
Starting with Einstein, some physicists have not been happy with the quantum concept of reality. (Einstein: “God does not play dice.”) Since the quantum concept of
reality contradicts our classical everyday experience one can construct various kinds of
“paradoxes” involving half-dead animals like Schrödinger’s cat. At the end they are all
due to our difficulties to abstract from our usual concept of classical reality, and they get
resolved when we accept the different concept of quantum reality. Some people including
Einstein, who had a hard time to accept this, have suggested alternative theories, e.g.
involving so-called hidden variables. This is equivalent to Nature telling every uranium
atom deterministically when to decay, but such that it is indistinguishable from a statistical distribution. Remarkably, any experiment that has been performed until today is
correctly described by quantum mechanics and the Copenhagen interpretation, even if it
may contradict our classical concept of reality.
While it is amusing to think about Schrödinger’s cat, one can do so in a meaningful
manner only when one understands the basics of quantum mechanics. This requires some
mathematical formalism. In fact, a beginner will lack intuition for the phenomena in
the quantum world and it is the formalism that will help us to reach correct conclusions.
Fortunately, we do not need a whole new mathematics to do quantum physics. An essential
feature of quantum mechanics is that it only predicts probabilities. For example, instead
of predicting where a particle is (not an element of quantum reality), it predicts the
probability density with which it can be found at a particular position. In effect, the
particle’s position is uncertain and in some sense the particle is therefore smeared out. In
this respect, it behaves like a wave (which also exists simultaneously in different regions of
space). In fact, in quantum mechanics the probability density to find a particle at position
18
CHAPTER 1. INTRODUCTION
x at time t is given by the absolute square of its wave function Ψ(x, t) which is in general
complex-valued.
The simplest formulation of non-relativistic quantum mechanics is the wave mechanics
which Erwin Schrödinger (1887 - 1961, Nobel prize in 1933) developed in 1926. It is based
on the Schrödinger equation
i~
∂Ψ(x, t)
~2 ∂ 2 Ψ(x, t)
=−
+ V (x)Ψ(x, t).
∂t
2M
∂x2
(1.3.2)
Here ~ = h/2π = 1.055 × 10−27 erg sec and M is the mass of a particle moving under the
influence of a potential V (x). The Schrödinger equation is a partial differential equation
(just like Maxwell’s equations or other wave equations) that describes the time evolution
of the wave function. In fact, it is the quantum analog of Newton’s equation. Since
the Schrödinger equation is a differential equation, to a large extent the mathematics of
quantum mechanics is that of differential equations.
A more abstract formulation of quantum mechanics is due to Paul Adrien Maurice
Dirac (1902 - 1984, Nobel prize shared with Schrödinger in 1933). He would write the
Schrödinger equation as
∂|Ψ(t)i
i~
= H|Ψ(t)i.
(1.3.3)
∂t
Here |Ψ(t)i is a state vector (which is equivalent to Schrödinger’s wave function) in an
abstract Hilbert space (a vector space with infinitely many dimensions). The object
H=
p2
+V
2M
(1.3.4)
is the so-called Hamilton operator describing kinetic and potential energy of the particle.
The Hamilton operator can be viewed as a matrix (with infinitely many rows and columns)
in the Hilbert space. Also the momentum p is an operator which can be represented as
p=
~ ∂
,
i ∂x
(1.3.5)
and can again be viewed as a matrix in Hilbert space. It was Heisenberg who first formulated quantum mechanics as matrix mechanics. Then the various observables (like position
x and momentum p) are represented by matrices which not necessarily commute with each
other. For example, in quantum mechanics one finds
px − xp =
~
.
i
(1.3.6)
This is the origin of Heisenberg’s uncertainty relation. Two observables cannot be measured simultaneously with arbitrary precision when the corresponding matrices (operators) do not commute. In the classical limit ~ → 0 we recover px = xp, such that we
can then determine position and momentum simultaneously with arbitrary precision. For
time-independent problems the Schrödinger equation reduces to
H|Ψi = E|Ψi,
(1.3.7)
1.3. WHAT IS QUANTUM MECHANICS?
19
which can be viewed as the eigenvalue equation of the Hamilton operator matrix H. The
(sometimes quantized) energy E is a (sometimes discrete) eigenvalue and the state vector
|Ψi is the corresponding eigenvector. In the early days of quantum physics there were a lot
of arguments whether Heisenberg’s or Schödinger’s formulation was correct. These ended
when Schrödinger showed that both formulations are equivalent. Yet another equivalent
formulation of quantum mechanics was developed later by Richard Feynman (1918 - 1988,
Nobel prize in 1965). His path integral method is applicable to non-relativistic quantum
mechanics, but it shows its real strength only for relativistic quantum field theories.
We may now have a vague idea what quantum mechanics might be, but it will take
a while before we fully understand the Schrödinger equation. However, it is time now
to switch to a more quantitative approach to the problem. Hopefully you are ready to
explore the exciting territory of the microscopic quantum world. Please, leave behind
your concept of classical reality and grab your mathematical tool kit. It will help you to
understand an environment in which reality is probabilistic and where cats can be half
alive and half dead.
20
CHAPTER 1. INTRODUCTION
Chapter 2
Beyond Classical Physics
In this chapter we will familiarize ourselves with some phenomena of the quantum world
that played a crucial role in the historical development of quantum mechanics. Just like
the early pioneers of quantum mechanics, we do not yet have the entire formalism of
quantum mechanics at our disposal. Hence, some of our arguments will necessarily still
be semi-classical. Only in chapter 3 we will begin to develop the formalism of quantum
mechanics in a systematic manner. A reader less interested in the historical beginnings of
quantum mechanics, who wants to immediately progress to the development of the modern
formalism, may skip this chapter and move on directly to chapter 3.
2.1
Photons — the Quanta of Light
Historically the first theoretical formula of quantum physics was Planck’s law for black
body radiation. For example, the cosmic background radiation — a remnant of the big
bang — is of the black body type. Planck was able to explain the energy distribution of
black body radiation by assuming that light of frequency ν and thus of angular frequency
ω = 2πν is absorbed and emitted by matter only in discrete amounts of energy
E = hν = ~ω.
(2.1.1)
Nowadays, we would associate Planck’s radiation formula with a rather advanced field of
quantum physics. Strictly speaking, it belongs to the quantum statistical mechanics of
the electromagnetic field, i.e. it is part of quantum field theory — clearly a subject for
graduate courses. It is amazing that theoretical quantum physics started with such an
advanced topic. Still, we will try to understand Planck’s radiation formula later in this
chapter.
Einstein took Planck’s idea more seriously than Planck himself and concluded that
light indeed consists of quanta of energy E = ~ω which he called photons. In this way he
was able to explain the photoelectric effect, which will be discussed below. According to
21
22
CHAPTER 2. BEYOND CLASSICAL PHYSICS
Maxwell’s equations an electromagnetic wave of wave vector ~k has angular frequency
ω = |~k|c.
(2.1.2)
Recall that the wave length is given by λ = 2π/|~k| such that λ = c/ν. From special
relativity we know that a massless particle such as a photon with momentum p~ has energy
E = |~
p|c.
(2.1.3)
Combining eqs.(2.1.1), (2.1.2), and (2.1.3) suggests that the momentum of a photon in an
electromagnetic wave of wave vector ~k is given by
p~ = ~~k.
(2.1.4)
The Compton effect results from an elastic collision of electron and photon in which both
energy and momentum are conserved. The Compton effect, to be discussed below, is correctly described when one uses the above quantum mechanical formula for the momentum
of a photon.
It is common to refer to photons as particles, e.g. because they can collide with electrons
almost like billiard balls. Then one may get confused because light can behave both like
waves or like particles. Books about quantum physics are full of remarks about particlewave duality and about how one picture may be more appropriate than the other depending
on the phenomenon in question. The resulting confusion is entirely due to the fact that
our common language is often inappropriate to describe the quantum world. At the end
the appropriate language is mathematics whose answers are always unique. For example,
when we refer to the photon as a particle, we may think of it like a classical billiard ball.
This is appropriate only as far as its energy and momentum are concerned. A classical
billiard ball also has a definite position in space. Due to Heisenberg’s uncertainty relation
this is not the case for a photon of definite momentum. When we say that the photon is a
particle we should hence not think of a classical billiard ball. Then what do we mean by a
particle as opposed to a wave? Our common language distinguishes between the two, but
in the mathematical language of quantum physics both are the same thing. It is a fact
that quantum theory works with wave equations and objects never have simultaneously a
well defined position and momentum. In this sense we are always dealing with waves. In
our common language we sometimes like to call these objects particles as well. If you find
this confusing, just don’t worry about it. It has nothing to do with the real issue. Let
mathematics be our guide in the quantum world: in quantum reality there is no distinction
between particles and waves.
2.2
The Compton Effect
In the Compton effect we consider the elastic collision of a photon and an electron. In
1922 Arthur Holly Compton (1892 — 1962, Nobel prize in 1927) performed his original
experiment with electrons bound in the atoms of a metallic foil. The foil was experimentally essential but it complicates the theoretical issue and we thus ignore it in our
2.3. THE PHOTOELECTRIC EFFECT
23
discussion. The main lesson to be learned here is that a photon in an electromagnetic
wave of angular frequency ω and wave vector ~k carries energy E = ~ω and momentum
p~ = ~~k, and can exchange energy and momentum by colliding with other particles like
electrons. When the energy of the photon changes in the collision, the frequency of the
corresponding electromagnetic wave changes accordingly.
Let us consider an electron at rest. Then its energy is
Ee = M c2 ,
(2.2.1)
where M = 8.29 × 10−7 erg/c2 is its rest mass and its momentum is p~e = 0. Now we collide
the electron with a photon of energy Ep = ~ω and momentum p~p = ~~k. Of course, we
then have ω = |~k|c. After the collision the electron will no longer be at rest, but will pick
up a momentum p~e 0 . Then its energy will be given by
p
Ee0 = (M c2 )2 + |~
pe 0 |2 c2 .
(2.2.2)
Also energy and momentum of the photon will change during the collision. The energy
after the collision will be Ep0 = ~ω 0 and the momentum will be p~p 0 = ~~k 0 . The scattering
angle θ of the photon is hence given by
cos θ =
~k · ~k 0
.
|~k||~k 0 |
(2.2.3)
Using energy and momentum conservation one can show that
1
1
~
− =
(1 − cos θ).
0
ω
ω
M c2
(2.2.4)
This means that the frequency of the scattered photon changes depending on the scattering
angle. This is clearly a quantum effect, which disappears in the classical limit ~ → 0.
Let us consider two extreme cases for the photon scattering angle: θ = 0 and θ = π.
For θ = 0 the photon travels forward undisturbed, also leaving the electron unaffected.
Then cos θ = 1 and hence ω 0 = ω. In this case the photon does not change its frequency.
For θ = π the photon reverses its direction and changes its frequency to
ω0 =
ω
< ω.
1 + 2~ω/M c2
(2.2.5)
The photon’s frequency — and hence its energy — is smaller than before the collision
because it has transferred energy to the electron.
2.3
The Photoelectric Effect
In 1886 Heinrich Hertz (1857 — 1894) discovered that light shining on a metal surface
may knock out electrons from the metal. The crucial observation is that light of too small
24
CHAPTER 2. BEYOND CLASSICAL PHYSICS
frequency cannot knock out any electrons even if it is arbitrarily intense. Once a minimal
frequency is exceeded, the number of electrons knocked out of the metal is proportional to
the intensity of the light. The kinetic energy of the knocked out electrons was measured to
rise linearly with the difference between the applied frequency and the minimal frequency.
All these observations have a natural explanation in quantum physics, but cannot be
understood using classical concepts.
Einstein explained the photoelectric effect as follows. Electrons are bound within
the metal and it requires a minimal work W to knock out an electron from the metal’s
surface. Any excess energy will show up as kinetic energy of the knocked out electron. The
energy necessary to knock out an electron is provided by a single photon in the incoming
electromagnetic wave. Photons with frequencies below the minimum value simply do not
have enough energy. Also it is extremely unlikely that many photons of small energy can
“collaborate” and combine their energies to knock out an electron. Instead, an electron
is usually hit by one photon at a time and none of them is sufficiently energetic to knock
out an electron. As we have learned before, such a photon has energy E = ~ω where ω is
the angular frequency of the light wave. Only when ω exceeds a minimal frequency
ω0 =
W
,
~
(2.3.1)
an electron can actually be knocked out. The excess energy shows up as kinetic energy
T = ~ω − W = ~(ω − ω0 )
(2.3.2)
of the knocked out electron. This is exactly what was observed experimentally.
2.4
Some Big Questions and the Cosmic Background Radiation
In this section we address some big questions of current research that arise in the context of
the cosmic microwave background radiation. A reader only interested in a straightforward
introduction to quantum mechanics may skip this section and move on directly to section
2.5.
A system of photons in thermal equilibrium has been with us from the beginning of
our Universe. Immediately after the big bang the energy density — and hence the temperature — was extremely high and all kinds of elementary particles (among them photons,
electrons, and their anti-particles — positrons — as well as neutrinos and anti-neutrinos)
have existed as an extremely hot gas filling all of space. These particles interacted with
each other, e.g., via Compton scattering. As the Universe expanded, the temperature
decreased and electrons and positrons annihilated into a vast number of photons. A very
small fraction of the electrons (actually all the ones in the Universe today) exceeded the
number of positrons and thus survived annihilation. At this time — a few seconds after the big bang — no atom had ever been formed. As a consequence, there were no
2.4. SOME BIG QUESTIONS AND THE COSMIC BACKGROUND RADIATION 25
characteristic colors of selected spectral lines. This is what we mean when we talk about
the cosmic photons as black body radiation. About 380’000 years after the big bang the
Universe had expanded and cooled so much that electrons and atomic nuclei could settle
down to form neutral atoms. At that time the Universe became transparent. The photons
that emerged from the mass extinction of electrons and positrons were left alone and are
still floating through our Universe today. However, during the past 13.7 × 109 years the
Universe has expanded further and the cosmic photon gas has cooled down accordingly.
Today the temperature of the cosmic background radiation is 2.735 K (degrees Kelvin
above absolute zero temperature).
The investigation of the cosmic microwave background radiation is related to several
big questions which can be addressed only if we master quantum mechanics. First of all,
one may ask why some particles (like protons, neutrons, and electrons) survived the mass
extinction of matter and anti-matter that gave rise to the cosmic background radiation.
The tiny surplus of matter is essential for our own existence, because it provides all ordinary matter (consisting of atoms and thus of protons, neutrons, and electrons) that exists
in the Universe today. Protons and neutrons belong to a class of elementary particles
known as baryons, while anti-protons and anti-neutrons are anti-baryons. Without an
asymmetry between baryons and anti-baryons we would have no matter in the Universe
today, just cosmic background radiation. Hence, one of the big questions that we may
hope to answer in the future is:
What is the origin of the baryon asymmetry?
There are already some very interesting ideas concerning this question related to cosmic phase transitions, neutrino physics, or grand unified theories (GUT) that extend the
standard model of elementary particle physics to very high energies almost at the Planck
scale. Whatever the solution of the baryon puzzle may be, it will certainly involve quantum
physics.
There are other big questions related to the cosmic background radiation. Already in
1933, applying the virial theorem to galaxy clusters, the astronomer Fritz Zwicky (1898
— 1974) predicted that galaxies must contain large amounts of non-luminous, so-called
dark matter. The presence of dark matter can also be inferred from measurements of
the rotation speed of stars around the center of distant galaxies. One candidate for dark
matter are the so-called MACHOS — massive compact halo objects consisting of ordinary
matter made of atoms. For example, the huge planet Jupiter, which is nevertheless not
sufficiently massive to ignite nuclear fusion in its core, and which is thus non-luminous,
falls in this category. Numerical simulations of the formation of large scale structures in
the Universe suggest that besides MACHOS there must also be other more exotic forms of
dark matter. A detailed study of the tiny angle-dependent temperature variations in the
cosmic background radiation, measured by the Wilkinson Microwave Anisotropy Probe
(WMAP) satellite, indicates that about 20 percent of the energy of the Universe indeed
resides in an exotic form of dark matter, while only about 5 percent is in the form of
ordinary matter consisting of atoms. Naturally, one of the big questions today is:
26
CHAPTER 2. BEYOND CLASSICAL PHYSICS
What is the nature of the exotic dark matter?
At present, the most prominent candidates are so-called WIMPs — weakly interacting
massive particles. For example, the minimal supersymmetric (SUSY) extension of the
standard model of particle physics (the so-called MSSM) postulates the existence of heavy
partners for all of the existing elementary particles. The lightest of these SUSY partners
— the lightest supersymmetric particle (also known as the LSP) — is a promising WIMP
candidate. The large hadron collider (LHC) at CERN in Geneva is the first particle accelerator that can reach energies high enough to perhaps produce WIMPs or other SUSY
particles. At LHC one already found the so-called Higgs particle which is predicted by
the standard model but had not been produced in earlier experiments at lower energies.
It goes without saying that the standard model as well as its extensions all rely on quantum physics. If we want to understand the outcome of the exciting LHC experiments, we
certainly need a solid background in quantum mechanics.
Since 1923, when Edwin Hubble (1889 - 1953) observed that the light from distant
galaxies is red-shifted, we know that the Universe is expanding. Until the end of the
twentieth century, it was believed that the expansion is decelerating due to the gravitational attraction between the galaxies. In such a scenario, the Universe might re-collapse
and finally end in a big crunch. In 1998 two international collaborations, the supernova
cosmology project and the high-z supernova search team, have reported observations of
supernova explosions in very distant galaxies which revealed that the expansion of the
Universe is actually accelerating. What is counteracting the gravitational pull that should
slow down the expansion? The same detailed analysis of the temperature variations in
the cosmic background radiation has also answered that question: about 75 percent of
the energy of the Universe is not of material origin, but is uniformly spread out through
“empty” space. This vacuum energy is also known as dark energy, a cosmological constant,
or in a more dynamical variant as so-called quintessence. The many names just reflect our
ignorance about this major fraction of the energy of our Universe. Indeed, another big
question is:
What is the nature of the vacuum energy?
Originally, the cosmological constant had been introduced by Einstein as an additional
parameter of general relativity besides Newton’s constant. By tuning the cosmological
constant to a specific value, Einstein was able to describe a static Universe. When he
learned about Hubble’s discovery of the expanding Universe, he supposedly called the
introduction of the cosmological constant his “biggest blunder”. Today we know that
an extremely tiny, but non-zero, cosmological constant is needed to explain the observed
accelerated expansion of the Universe. This leads us to yet another big question — the
cosmological constant problem:
Why is the cosmological constant so incredibly small, but still non-zero?
2.4. SOME BIG QUESTIONS AND THE COSMIC BACKGROUND RADIATION 27
This may be one of the hardest problems in physics today. Perhaps it can be answered
only after the correct theory of quantum gravity has been found. A currently popular attempt of an explanation uses the anthropic principle. If the cosmological constant would
be much bigger, the Universe would accelerate very strongly. Then everything would be
pulled apart so quickly that no complex structures such as galaxies, solar systems, or intelligent life-forms could ever develop. In other words, life is possible only in a Universe with
a tiny value of the vacuum energy. If one accepts the anthropic principle, but does not
want to subscribe to “theories” of intelligent design, one may assume that our Universe
is part of a much bigger Multiverse. Most regions in the Multiverse would have a large
value of the vacuum energy and would thus not be inhabitable. The author would prefer
not to subscribe to this way of thinking. Instead, he hopes that perhaps some very clever
reader of this text will eventually find a natural explanation for the cosmological constant
problem in the one Universe we will always be confined to.
It is amazing that the temperature of the cosmic background radiation is to a very
high degree of precision the same, no matter what corner of the Universe the cosmic
photons come from. How can very distant regions that seem causally disconnected have
thermalized at the same temperature? This puzzle was first explained by Alan Guth from
MIT using the idea of the inflationary Universe: an exponential expansion of the very
early Universe, let us say at about 10−30 seconds after the big bang, would imply that
our entire observable Universe had been in causal contact very early on. The idea of an
early epoch of inflationary exponential expansion would also solve at least three other big
questions. While dimensional analysis suggests that the
p natural life-time of a randomly
created Universe is about the Planck time, tP lanck = G~/c5 = 5.3904 × 10−44 sec, our
Universe has reached the respectable age of about 13.7 billion years. Hence the question
arises:
Why is the Universe so old?
Again, one could invoke the anthropic principle and argue that we can only live in a
Universe that exists sufficiently long to allow intelligent life-forms to develop. Guth has
another explanation: inflation would naturally lead to a Universe that can become very
old. General relativity allows space to be curved. However, observation indicates that our
Universe is flat. Remarkably, the next big question that the idea of inflation can answer is:
Why is space flat?
And yet another big question potentially solved by inflation is:
Why are there no magnetic monopoles?
The grand unified extensions of the standard model of elementary particles (GUT theories) predict the production of magnetically charged particles at about 10−34 seconds
after the big bang. However, despite numerous intensive searches, no magnetic monopole
has ever been found. If there was an epoch of inflation after monopoles were produced,
28
CHAPTER 2. BEYOND CLASSICAL PHYSICS
they would get extremely diluted by the exponential inflationary expansion of the Universe. This would explain why no monopoles are to be found today. Apparently, inflation
is a very powerful idea which may solve several big questions about our Universe. However,
it still remains to be seen whether Nature has actually made use of this idea. Obviously,
we must ask:
Was there an early inflationary epoch?
Remarkably, the tiny temperature variations in the cosmic background radiation seem
to indicate that this was indeed the case. If inflation is indeed established as a fact, it
leads to the next big question:
What could be the cause of inflation?
Obviously, there is a lot to think about in this century. We see over and over again
that we can address some of the biggest questions that drive current research only with
a solid basis in quantum physics. This should be motivation enough to once again step
about one hundred years back, and understand in detail what quantum mechanics is all
about.
2.5
Planck’s Formula for Black Body Radiation
Now that we have familiarized ourselves with the properties of photons, we may turn to
their thermodynamics. Thermodynamics deals with systems of many particles that are
in thermal equilibrium with each other and with a thermal bath. The energies of the
individual states are statistically distributed, following a Boltzmann distribution for a
given temperature T . The thermal statistical fluctuations are of a different nature than
those related to quantum uncertainty. Thermal fluctuations are present also at the classical
level. Following the classical concept of reality, it is possible, at least in principle, to treat
a system with a large number of particles deterministically. In practice, it is, however,
much more appropriate to use a classical statistical description. In the thermodynamics
of photons, i.e. in quantum statistical mechanics, we deal with thermal and quantum
fluctuations at the same time.
How does one measure the temperature of a system of photons? The temperature
is defined via the Boltzmann distribution, in our case by the intensity of radiation with
a certain frequency. Hence, by measuring the photon spectrum one can determine the
temperature. This is exactly what the Wilkinson Microwave Anisotropy Probe (WMAP)
satellite has been doing when it measured the temperature distribution of the cosmic
microwave background radiation, which was generated immediately after the big bang by
the annihilation of electrons and their anti-particles — the positrons. Equipped with the
idea of Planck, let us now derive this spectrum theoretically. For simplicity we replace
the Universe by a large box of spatial size L × L × L with periodic boundary conditions.
2.5. PLANCK’S FORMULA FOR BLACK BODY RADIATION
29
This is just a technical trick that will allow us to simplify the calculation. At the end
we will let L → ∞. The calculation proceeds in three steps. First, we work classically
and classify all possible modes of the electromagnetic field in the box. Then we switch to
quantum physics and populate these modes with light quanta (photons). Finally, we apply
quantum statistical mechanics by summing over all quantum states using the Boltzmann
distribution.
What are the modes of the electromagnetic field in an L3 periodic box? First of all,
we can classify them by their wave vector ~k which is now restricted to discrete values
~k = 2π m,
~ mi ∈ Z.
L
The frequency of this mode is given by
ω = |~k|c.
(2.5.1)
(2.5.2)
Each of the modes can exist in two polarization states.
Now we turn to quantum mechanics and populate the classical modes with photons.
As we have learned, a mode of frequency ω can host photons of energy
E(~k) = ~ω = ~|~k|c
(2.5.3)
only. Photons are so-called bosons. This means that an arbitrary number of them can
occupy a single mode of the electromagnetic field. Electrons and neutrinos, for example,
behave very differently. They are so-called fermions, i.e. at most one of them can occupy
a single mode. All elementary particles we know are either fermions or bosons. We can
completely classify a quantum state of the electromagnetic field by specifying the number
of photons n(~k) ∈ {0, 1, 2, ...} occupying each mode (characterized by wave vector ~k and
polarization, which we suppress in our notation). It is important to note that it does not
matter “which photon” occupies which mode. Individual photons are indistinguishable
from each other, they are like perfect twins. Hence specifying their number per mode
determines their state completely.
Now that we have classified all quantum states of the electromagnetic field by specifying the photon occupation numbers for each mode, we can turn to quantum statistical
mechanics. We must then evaluate the so-called partition function (a normalization factor
for the probability) by summing over all states. Since the different modes are completely
independent of one another, the total partition function
Y
Z=
Z(~k)2
(2.5.4)
~k
factorizes into partition functions Z(~k) for each individual mode. The square of Z(~k) arises
because each wave vector ~k is associated with two polarization states. Let us consider the
single-mode partition function
Z(~k) =
∞
X
n(~k)=0
∞
X
exp −n(~k)E(~k)/kB T =
exp −βn(~k)~|~k|c .
n(~k)=0
(2.5.5)
30
CHAPTER 2. BEYOND CLASSICAL PHYSICS
The state of each mode is weighed by its Boltzmann factor exp(−n(~k)E(~k)/kB T ) which
is determined by its total energy of photons n(~k)E(~k) occupying the mode and by the
temperature T . The Boltzmann constant kB = 1.38 × 10−16 erg/K arises purely for dimensional reasons. Had we decided to measure temperature in energy units, there would be
no need for Boltzmann’s constant. We have also introduced
β = 1/kB T.
(2.5.6)
Now we make use of the well-known summation formula for a geometric series
∞
X
n=0
xn =
1
.
1−x
(2.5.7)
Using x = exp(−β~|~k|c) we obtain the partition function corresponding to the so-called
Bose-Einstein statistics
1
.
(2.5.8)
Z(~k) =
1 − exp(−β~|~k|c)
We are interested in the statistical average of the energy in a particular mode, which is
given by
∞
X
1
n(~k)E(~k) exp −βn(~k)E(~k)
Z(~k)
hn(~k)E(~k)i =
n(~k)=0
= −
∂ log Z(~k)
~|~k|c
1 ∂Z(~k)
.
=−
=
∂β
Z(~k) ∂β
exp(β~|~k|c) − 1
Finally, we are interested in the average total energy as a sum over all modes
3 Z
X
L
~
~
hEi = 2
hn(k)E(k)i → 2
d3 k hn(~k)E(~k)i.
2π
(2.5.9)
(2.5.10)
~k
Here a factor 2 arises due to the two polarization states. In the last step we have performed
the infinite volume limit L → ∞. Then the sum over discrete modes turns into an integral.
It is no surprise that our result grows in proportion to the volume L3 . We should simply
consider the energy density u = hEi/L3 . We now perform the angular integration and
also replace |~k| = ω/c to obtain
Z ∞
1
~ω 3
u= 2 3
dω
.
(2.5.11)
π c 0
exp(β~ω) − 1
Before we do the integral we read off the energy density per frequency unit for modes of
a given angular frequency ω
du(ω)
1
~ω 3
= 2 3
.
dω
π c exp(~ω/kB T ) − 1
(2.5.12)
This is Planck’s formula that was at the origin of quantum mechanics. If you have followed
the above arguments: congratulations, you have just mastered a calculation in quantum
field theory!
2.6. QUANTUM STATES OF MATTER
31
Since we have worked hard to produce this important result, let us discuss it in some
detail. Let us first consider the classical limit ~ → 0. Then we obtain the classical
Rayleigh-Jeans law
du(ω)
ω 2 kB T
=
.
(2.5.13)
dω
π 2 c3
Integrating this over all frequencies gives a divergent result from the high frequency end
of the spectrum. This is the so-called ultraviolet Jeans catastrophe. The classical thermodynamics of the electromagnetic field gives an unphysical result. Now we go back to
Planck’s quantum result and perform the integral over all frequencies ω. This gives the
Stefan-Boltzmann law
4 T4
π 2 kB
u=
,
(2.5.14)
15~3 c3
which is not only finite, but also agrees with experiment. Again, in the classical limit
the result would be divergent. It is interesting that for high temperatures and for low
frequencies, i.e. for kB T ~ω, Planck’s formula also reduces to the classical result.
Quantum effects become import only in the low-temperature or high-frequency regimes.
Now we can understand how WMAP measures the temperature of the cosmic background radiation. The energy density is measured for various frequencies and is then
compared with Planck’s formula which leads to a high precision determination of T . The
WMAP data tell us a lot about how our Universe began. In fact, the early history of
the Universe is encoded in the photons left over from the big bang. Sometimes one must
understand the very small before one can understand the very large.
2.6
Quantum States of Matter
As we have seen, at the quantum level the energy of a light wave is carried by photons
which sometimes behave like particles. Similarly, in the quantum world objects that
we like to think about as particles may also behave like waves. Typical properties of
classical electromagnetic waves are diffraction and interference. Similar experiments can
also be performed with electrons, neutrons, or atoms. Quantum mechanically they all
show typical wave behavior as, e.g., observed in the experiments by Davisson and Germer
in 1927. A famous experiment that illustrates the diffraction of matter waves is the doubleslit experiment, in analogy to Young’s experiment using light waves. De Broglie (1892 —
1987, Nobel prize in 1929) was first to suggest that not only photons but any particle of
momentum p~ has a wave vector ~k associated with it. Just as for photons, he postulated
that also for massive particles
p~ = ~~k.
(2.6.1)
The de Broglie wave length of a particle is hence given by
λ=
h
2π
=
.
|~
p|
|~k|
(2.6.2)
32
CHAPTER 2. BEYOND CLASSICAL PHYSICS
When one wants to detect the wave properties of matter experimentally, one must work
with structures of the size of the wave length. In the classical limit h → 0 the wave length
goes to zero. Then even on the smallest scales one would not detect any wave property.
However, in the real world h 6= 0 and wave properties are indeed observed.
Another manifestation of quantum mechanics in the material world are the discrete
spectra of atoms and molecules. They arise from transitions between quantized energy
levels of the electrons surrounding the atomic nucleus. The Schrödinger equation predicts
the existence of discrete energy values. For example, in the hydrogen atom (consisting of
an electron and a much heavier proton that acts as the atomic nucleus) the allowed energy
values in the discrete spectrum are given by
En = −
M e4
, n ∈ {1, 2, 3, ...},
2~2 n2
(2.6.3)
where M and e are mass and charge of the electron. When an atom is in an excited state
with n0 > 1 it can emit a photon and thus jump into a lower lying state with n < n0 . The
energy difference between the two states shows up as the energy (and hence the frequency)
of the emitted photon, i.e.
M e4 1
1
0
En − En =
= ~ω.
(2.6.4)
−
2~2 n2 n0 2
This simple observation led Niels Bohr (1885 — 1962, Nobel prize in 1922) to a basic
understanding of atomic spectra. His model of the atom was ultimately replaced by the
Schrödinger equation, but it played an important role in the early days of quantum physics.
2.7
The Double-Slit Experiment
Double-slit experiments where originally performed with electromagnetic waves. The interference patterns resulting from light passing through the slits in a screen reflect the wave
properties of electromagnetic radiation. They also confirm that electromagnetic waves can
be linearly superimposed. The fact that Maxwell’s equations are linear in the electromagnetic field are a mathematical manifestation of the superposition principle. The double slit
experiment has also been used to illustrate the wave properties of matter. Again, interference patterns emerge that point to a superposition principle for matter waves. Indeed,
Schrödinger’s wave equation is linear in the quantum mechanical wave function. Experimentally it is much harder to perform double-slit experiments with matter waves than
with light waves in the visible range of the spectrum. This is because the wave length of
matter waves is usually a lot shorter. Hence, the slits that cause the diffraction pattern
must be much more closely spaced than in experiments with light waves. Still, experiments
with electron beams, for example, have been performed and they clearly demonstrate the
wave properties of matter.
Let us first discuss the double-slit experiment for light waves that was originally performed by Thomas Young at the beginning of the nineteenth century. We consider a plane
2.7. THE DOUBLE-SLIT EXPERIMENT
33
electromagnetic wave propagating in the x-direction with wave number k and angular frequency ω = kc. The wave falls on an opaque planar screen oriented in the y-z-plane with
two holes located at x = z = 0 and y1 = 0, y2 = d. Geometrical optics would predict that
only two rays of light will make it through the slits, and the screen creates a sharp shadow
behind it. This prediction is correct only when the wave length of the light is very small
compared to the size of the holes in the screen. Here we are interested in the opposite limit
of very small holes compared to the wave length. Then the predictions of geometrical optics fail completely and only Maxwell’s equations predict the correct behavior. For matter
waves the analog of geometrical optics is classical physics. Once the wave properties of
matter become important, only the Schrödinger equation yields correct results. For light
waves each of the two holes in the screen acts like an antenna that radiates a circular
wave. Let us restrict ourselves to the x-y-plane. Then the electric field of an antenna is
given by
~ t) = A sin(ωt − kr)~ez ,
E(r,
(2.7.1)
r
where r is the distance from the antenna. Also we have assumed polarization in the zdirection. For an observer at large distance, the two antennae corresponding to the two
holes have, to a first approximation, the same distance r. However, this approximation
is not good enough for our purposes. Because we are interested in interference effects,
we must be able to detect differences in the lengths of the optical paths of the order of a
wavelength. Still, for an observer in the far zone we can assume that all incoming light
rays are practically parallel. The difference between the path lengths of the circular waves
emanating from the two holes is d sin ϕ where ϕ is the angle between the x-axis and the
direction of observation behind the screen. When d sin ϕ = λm is an integer multiple of the
wavelength the two waves arrive at the observer in phase and we obtain a large intensity.
To see this analytically, we add up the two contributions to the total electric field
~ ϕ, t) = E
~ 1 (r1 , t) + E
~ 2 (r2 , t)
E(r,
A
A
=
sin(ωt − kr1 )~ez + sin(ωt − kr2 )~ez
r1
r2
A
=
Im[exp i(ωt − kr) + exp i(ωt − kr + δ)]~ez .
r
(2.7.2)
Here we have used r = r1 ≈ r2 in the amplitude of the waves. In the phase we need to be
more precise. There we have used
r2 = r − d sin ϕ,
(2.7.3)
and we have introduced the phase shift
δ = kd sin ϕ.
(2.7.4)
The complex notation is not really necessary but it simplifies the calculation. We now
write
~ ϕ, t) =
E(r,
=
A
Im{exp[i(ωt − kr + δ/2)][exp(−iδ/2) + exp(iδ/2)]}~ez
r
A
sin(ωt − kr + δ/2) 2 cos(δ/2)~ez .
r
(2.7.5)
34
CHAPTER 2. BEYOND CLASSICAL PHYSICS
~ average
We are interested in the time averaged intensity of the field. Hence, we square E,
over t, and obtain the intensity
I(ϕ) = 4I0 cos2 (δ/2).
(2.7.6)
Here I0 is the ϕ-independent intensity in the presence of just a single hole. What is the
intensity observed at the angle ϕ = 0? Naively, one might expect 2I0 , but the value is
indeed 4I0 . Furthermore, what happens at the angles ϕ for which d sin ϕ = λm? In that
case
δ = kd sin ϕ = kλm = 2πm.
(2.7.7)
The intensity is 2π-periodic in δ and hence the intensity is again 4I0 . For these particular
angles both waves are in phase, thus yielding a large intensity. The minima of the intensity,
on the other hand, correspond to δ = π.
Let us now turn to the diffraction and interference of matter waves. We can think
of a beam of electrons incident on the same screen as before. We prepare an ensemble
of electrons with momentum px in the x-direction and with py = pz = 0. As we will
understand later, this state is described by a quantum mechanical wave function which is
again a plane wave. We will detect the diffracted electrons with a second detection screen
a distance l behind the screen with the two holes and parallel to it. We are interested
in the probability to find electrons in certain positions at the detection screen at x = l.
Again, we limit ourselves to z = 0 and ask about the probability as a function of y.
As for any problem in quantum mechanics, one can answer this question by solving
the Schrödinger equation. Since we have not yet discussed the Schrödinger equation in
sufficient detail, we will proceed differently. In fact, we will make a detour to so-called
path integrals — a very interesting formulation of quantum mechanics that was discovered
by Richard Feynman (1918 — 1988, Nobel prize in 1965). He noted that classical and
quantum physics are more closely related than one might think based on the Schrödinger
equation. In particular, Feynman realized that quantum physics is governed by the action
of the corresponding classical problem. Let us introduce the action in the context of a
particle of mass M moving under the influence of a potential V (x) in one dimension. The
action is a number associated with any possible path x(t) connecting a point x(0) at the
initial time t = 0 to a point x(T ) at a final time T . The path need not be the one that
a classical particle will actually follow according to Newton’s equation. Even a path that
cannot be realized classically has an action associated with it. The action is defined as
#
Z T " M dx(t) 2
S[x(t)] =
dt
− V (x(t)) ,
(2.7.8)
2
dt
0
i.e. one simply integrates the difference of kinetic and potential energy over time. The
action has a quite remarkable property. It is minimal (or at least extremal) for the classical
path — the one that the particle takes following Newton’s equation. All other (classically
not realized) paths have a larger action than the actual classical path. This can be
understood most easily when we discretize time into N small steps of size ε such that
2.7. THE DOUBLE-SLIT EXPERIMENT
35
N ε = T and tn = nε. At the end we will let N → ∞ and ε → 0 such that T remains
fixed. The discretized version of the action takes the form
#
" N
−1
X
M x(tn+1 ) − x(tn ) 2
S[x(t)] =
− V (x(tn )) .
(2.7.9)
ε
2
ε
n=0
Note that we have also replaced the derivative dx(t)/dt by its discretized variant — a
corresponding quotient of differences. Let us now find the path that minimizes the action
simply by varying S[x(t)] with respect to all x(tn ). We then obtain
dS[x(t)]
x(tn+1 ) − x(tn )
x(tn ) − x(tn−1 ) dV (x(tn ))
= ε −M
+M
−
= 0.
(2.7.10)
dx(tn )
ε2
ε2
dx(tn )
We know that the gradient of the potential determines the force, i.e.
−
dV (x)
= F (x),
dx
(2.7.11)
such that the above equation takes the form
M
x(tn+1 ) − 2x(tn ) + x(tn−1 )
= F (x(tn )).
ε2
(2.7.12)
The term on the left-hand side is a discretized second derivative. Taking the limit ε → 0
we obtain
d2 x(t)
M
= F (x(t)).
(2.7.13)
dt2
This is exactly Newton’s equation that determines the classical path. Hence, we have
shown that the classical path is a minimum (more generally an extremum) of the action.
In fact, in modern formulations of classical mechanics the action plays a central role.
Feynman discovered that the action also governs quantum mechanics. In fact, he found
a new path integral formulation of quantum mechanics that is equivalent to Heisenberg’s
and Schrödinger’s formulations, and that is often used in quantum field theories. In
quantum mechanics, on the other hand, solving the Schrödinger equation is usually easier
than doing the path integral. Therefore, we will soon concentrate on Schrödinger’s method.
Still, to describe the diffraction of matter waves, the path integral is very well suited.
Quantum mechanically we should not think of a particle tracing out a classical path.
This follows already from Heisenberg’s uncertainty relation which tells us that position
and momentum of a particle are not simultaneously measurable with arbitrary precision.
Feynman realized that the concept of a particle’s path can still be useful when one wants
to determine the probability for finding a particle (which started at time t = 0 at position
~x = 0) at a final time t = T in a new position ~x(T ). More precisely we are interested
in the transition probability amplitude Ψ(~x(T )) — a complex number closely related
to the quantum mechanical wave function — whose absolute value |Ψ(~x(T ))|2 defines a
probability density, i.e. |Ψ(~x(T ))|2 d3 x is the probability to find the particle at time T
in a volume element d3 x around the position ~x(T ). Recall that we have assumed that
the particle started initially at time t = 0 at position ~x = 0. Feynman’s idea was that
36
CHAPTER 2. BEYOND CLASSICAL PHYSICS
Ψ(~x(T )) receives contributions from all possible paths that connect ~x = 0 with ~x(T ),
not just from the classical path that minimizes the action. The Feynman path integral
represents Ψ(~x(T )) as an integral over all paths
Z
i
Ψ(~x(T )) = D~x(t) exp
S[~x(t)] .
(2.7.14)
~
The essential observation is that the action S[x(t)] of a path x(t) determines its contribution to the total path integral. Defining precisely what we mean by the above path
integral expression, and finally performing the path integral, goes beyond the scope of
this chapter. What we will need for the calculation of electron diffraction is just the path
integral for a free particle. In that case one finds
i
Ψ(~x(T )) = A exp
S[~xc (t)] .
(2.7.15)
~
Here ~xc (t) is the classical path — the one determined by Newton’s equation. Hence, for
a free particle the whole path integral is proportional to the contribution of the classical
path. The proportionality factor A is of no concern for us here.
What is the action of the classical path of a free particle moving from ~x = 0 at time
t = 0 to ~x(T ) at time t = T ? Newton’s equation tells us that a free particle moves along
a straight line, and hence the classical path is
~xc (t) =
~x(T )
t.
T
(2.7.16)
The derivative
d~xc (t)
~x(T )
=
,
dt
T
is just the constant velocity. The action of the classical path is thus given by
Z T
M |~x(T )|2
M |~x(T )|2
S[~xc (t)] =
dt
=
,
2 T2
2T
0
and Feynman’s transition probability amplitude takes the form
iM |~x(T )|2
Ψ(~x(T )) = A exp
.
2~T
(2.7.17)
(2.7.18)
(2.7.19)
The amplitude depends on the distance squared |~x(T )|2 that the particle traveled during
the time T .
Now we are ready to calculate the probability distribution of electrons at the detection
screen. Compared to the previous calculation for light waves, here it is more convenient
to shift the positions of the two holes in the screen from 0 and d to ±d/2. An electron
detected at a position y may have originated either from the hole at y1 = d/2 or from the
hole at y2 = −d/2, and we don’t know through which hole it came. In the first case it has
traveled a distance squared |~x(T )|2 = l2 + (y − d/2)2 , and in the second case the distance
2.7. THE DOUBLE-SLIT EXPERIMENT
37
squared is |~x(T )|2 = l2 + (y + d/2)2 . Feynman’s path integral method requires to add the
contributions of the two paths such that the total transition probability amplitude is
iM [l2 + (y + d/2)2 ]
iM [l2 + (y − d/2)2 ]
+ A exp
Ψ(y) = A exp
2~T
2~T
2
2
2
iM yd
iM yd
iM [l + y + d /4]
exp
+ exp −
= A exp
2~T
2~T
2~T
2
2
2
iM [l + y + d /4]
M yd
= A exp
2 cos
.
(2.7.20)
2~T
2~T
To obtain the probability density we take the absolute value squared
2 M yd
2
2
2 M yd
= 4ρ0 cos
.
ρ(y) = |Ψ(y)| = 4|A| cos
2~T
2~T
(2.7.21)
Here ρ0 is the y-independent probability density that one finds in an experiment with just
a single hole.
The above result is very similar to the one for diffraction of light waves. In particular,
again we find the cos2 (δ/2) distribution if we identify
δ=
M yd
.
~T
(2.7.22)
In the case of light waves we had
δ = kd sin ϕ.
(2.7.23)
In the present case we have y = l tan ϕ ≈ l sin ϕ for small ϕ, such that we can identify
k=
Ml
.
~T
(2.7.24)
Here l/T is the distance that the electron traveled in the x-direction divided by the time
T . Multiplying this velocity by the mass we obtain the momentum component in the
x-direction
Ml
px =
.
(2.7.25)
T
Hence, we finally identify
px = ~k.
(2.7.26)
In other words, the diffraction pattern of electrons of momentum px is the same as the
one of light waves with wave number k = px /~. Then it is natural to associate this wave
number also with the electrons. The corresponding wave length
λ=
2π
2π~
h
=
= ,
k
px
px
(2.7.27)
is the so-called de Broglie wave length of a particle of momentum px . It is remarkable that
we obtain the same relation as for photons. Generalizing to three dimensions we hence
associate with any particle of momentum p~ a wave with wave number vector ~k such that
p~ = ~~k.
(2.7.28)
38
CHAPTER 2. BEYOND CLASSICAL PHYSICS
It is important to note that the above double slit experiment can be performed with
one individual particle at a time. In particular, it should be stressed that the observed
interference patterns do not result from multi-particle interactions. Of course, a single
electron that passes through a double slit just leaves a single black spot somewhere at the
detection screen. Only after the experiment has been repeated with many individual electrons (one after the other) the collection of numerous black spots reveals the interference
pattern. In a double slit experiment, it is essential that we did not determine through
which slit the particle (a photon or an electron) actually went. In fact, we added contributions from both “alternatives”, and only that gave rise to the interference pattern. Once
we would figure out through which slit the particle went (for example, by putting a small
detector) the interference pattern would disappear. This is a consequence of Heisenberg’s
uncertainty relation. At the quantum level the influence that a measuring device has on
the observed phenomenon can in general not be made arbitrarily small. A small detector, for example, that determines through which slit the particle went would unavoidably
change the particle’s momentum, which then wipes out the interference pattern.
2.8
Estimating Simple Atoms
A series of experiments by Ernest Rutherford (1871 — 1937, Nobel prize in chemistry in
1908) and his collaborators in 1911 revealed that atoms consisting of negatively charged
electrons and a positively charged atomic nucleus have the positive charge concentrated
in a tiny region compared to the size of the atom. Consequently, as far as atomic physics
is concerned, the atomic nucleus can essentially be considered point-like. Of course, the
atomic nucleus itself consists of protons and neutrons, which in turn consist of quarks
and gluons, but this is physics at much smaller length scales. In contrast to the positive
charge, the negative charge of the electrons is distributed over the whole volume of the
atom, i.e. over a region with a radius of about 10−10 m. The electron itself has no internal
structure, at least as far as we know today. Hence, the distribution of negative charge over
the atom is entirely due to the fact that electrons move inside the atom. The forces that
govern the electron dynamics are of electromagnetic origin. The Coulomb force attracts
the negatively charged electron to the positively charged atomic nucleus. As opposed to
the charge, the mass of an atom is almost entirely concentrated in the nucleus, simply
because protons and neutrons have masses roughly a factor 2000 bigger than the electron
mass. The above picture of the atom that emerged from various experiments is inconsistent
with classical physics. In fact, an electron that surrounds an atomic nucleus is constantly
accelerated and should hence constantly emit electromagnetic radiation. This should lead
to a fast energy loss, causing the electron to spiral into the nucleus. After this collapse
the classical atom would not be bigger than the nucleus itself.
Based on the experimental observations of Rutherford, Bohr constructed a model of
the atom that eliminated the classical collapse problem by postulating that electrons can
move around the nucleus in certain discrete stable orbits without emitting electromagnetic
radiation. These discrete orbits were singled out by certain quantization conditions. The
2.8. ESTIMATING SIMPLE ATOMS
39
energies of the discrete orbits are then also quantized. Atoms can absorb or emit electromagnetic radiation by undergoing transitions between the various quantized energy levels.
The energy difference between the levels is then turned into the energy of a photon, and
hence manifests itself in the characteristic spectral lines that are observed experimentally.
In this way Bohr was able to describe the spectra of various atoms. Although in the early
days of quantum mechanics Bohr’s model of the atom played an important role, from a
modern perspective Bohr’s quantization method is considered to be semi-classical, or if
one wants semi-quantum. A full quantum description of the atom emerged only when one
was able to derive atomic spectra from the Schrödinger equation.
It will take us until the end of the second semester of this course to solve the Schrödinger
equation for the simplest atom — the hydrogen atom consisting of a single electron and a
proton that plays the role of the atomic nucleus. Hence, we will now use Bohr’s quantization method to get at least a first estimate of simple atoms. Let us consider hydrogen-like
atoms with just a single electron of charge −e, but with Z protons in the atomic nucleus
and hence with nuclear charge Ze. The Coulomb potential exerted on the electron is then
given by
Ze2
V (r) = −
,
(2.8.1)
r
where r is the distance of the electron from the atomic nucleus. Since the nucleus is so
much heavier than the electron we can assume that it remains at rest in the center of
mass frame of the whole atom. Further, let us assume that the electron moves around
the nucleus classically in a circular orbit of radius r with angular frequency ω. Then its
acceleration is given by a = ω 2 r, and thus Newton’s equation tells us that
M ω 2 r = M a = F (r) =
Ze2
.
r2
(2.8.2)
Let us now use Feynman’s path integral again. We have learned that a particle that
started at ~x = 0 at t = 0 has a transition probability amplitude Ψ(~x(T )) for arriving at a
later time T at the new position ~x(T ). This amplitude is given by a path integral, and for
a free particle the amplitude is proportional to the contribution of the classical path, i.e.
i
Ψ(~x(T )) = A exp
S[~xc (t)] .
(2.8.3)
~
An electron that is bound inside an atom is obviously not free. Therefore the above
equation can only be approximately correct. In fact, the assumption of the dominance
of the classical path implies a semi-classical approximation. The motion of the electron
on its circular orbit is periodic with period T = 2π/ω. Hence, Feynman’s transition
amplitude for a complete period, with the particle returning to its initial position, will be
approximately given by the action corresponding to the periodic path. As we will learn
later, the phase of the quantum mechanical wave function of a stationary state with energy
E changes by exp(−iET /~) during the time T . Consistency with Feynman’s transition
amplitude therefore requires
i
i
S[~xc (t)] = exp − ET ,
(2.8.4)
exp
~
~
40
CHAPTER 2. BEYOND CLASSICAL PHYSICS
and hence
S[~xc (t)] + ET = 2πn~.
(2.8.5)
Here n ∈ {1, 2, 3, ...}. This is the so-called Bohr-Sommerfeld quantization condition for
stationary states with energy E.
Let us use the Bohr-Sommerfeld quantization condition to estimate the energies of
stationary states of hydrogen-like atoms. First, we need to compute the action of the
periodic circular path of the electron. For this path the kinetic energy is given by 21 M ω 2 r2
and the potential energy is −Ze2 /r, which are both time-independent for a circular path.
The action for a complete period — as the time integral of the difference between kinetic
and potential energy — is then given by
Ze2
1
2 2
S[~xc (t)] = T
Mω r +
.
(2.8.6)
2
r
The total energy, on the other hand, is
Ze2
1
.
E = M ω2 r2 −
2
r
(2.8.7)
Hence, the quantization condition takes the form
S[~xc (t)]/T + E = M ω 2 r2 = 2πn~/T = n~ω,
(2.8.8)
Lz = M ωr2 = n~.
(2.8.9)
and therefore
This is nothing but the angular momentum, more precisely its component Lz perpendicular
to the plane of the circular path, which is hence quantized in integer multiples of ~. Using
this together with eq.(2.8.2) one obtains
r=
n2 ~2
n2 ~
=
.
Ze2 M
ZαM c
(2.8.10)
Here we have introduced the fine-structure constant
α=
1
e2
≈
,
~c
137.04
(2.8.11)
which characterizes the strength of the electromagnetic interaction. One notices that the
radius of the electronic orbit grows in proportion to the quantum number n squared. The
orbit of smallest radius corresponds to n = 1. In the hydrogen atom, i.e. for Z = 1, this
orbit has a radius
~
r=
≈ 5.3 × 10−9 cm.
(2.8.12)
αM c
This so-called Bohr radius sets the scale for the typical size of atoms. We also note that
the radius of the smallest orbit is proportional to 1/Z. Thus in heavy atoms electrons can
come closer to the atomic nucleus, simply because there is a stronger Coulomb force due
2.8. ESTIMATING SIMPLE ATOMS
41
to the larger number of protons in the nucleus. Using eq.(2.8.7) together with eq.(2.8.2)
and eq.(2.8.10) one obtains
E=−
M c2 (Zα)2
M Z 2 e4
=−
.
2
2
2n ~
2n2
(2.8.13)
These are the correct quantized energy values of the hydrogen spectrum. Although our
calculation was only semi-classical, we got the full quantum answer — the same that one
obtains from Schrödinger’s equation. This is a peculiarity of the hydrogen atom and will
in general not be the case. The state of lowest energy of the hydrogen atom — its so-called
ground state — corresponds to n = 1. If we want to ionize the hydrogen atom out of its
ground state, i.e. if we want to strip off the electron from the proton, we thus need to
invest the energy
M e4
M c2 α 2
E=
=
≈ 13.6eV.
(2.8.14)
2~2
2
Similarly, if we want to raise the energy of the hydrogen atom from its ground state with
n = 1 to its first excited state with n = 2 we must provide the energy difference
M c2 α 2
1
E = E2 − E1 =
1−
≈ 10.2eV.
(2.8.15)
2
4
A hydrogen atom in its first excited state will, after a while, emit a photon and return to
its ground state. The energy of the emitted photon is then given by the above expression,
and hence its angular frequency is
E
3M c2 α2
=
.
~
8~
(2.8.16)
2πc
16π~
=
≈ 1.2 × 10−5 cm,
ω
3α2 M c
(2.8.17)
ω=
The corresponding wave length then is
λ=
which is indeed observed in the ultraviolet part of the hydrogen spectrum.
42
CHAPTER 2. BEYOND CLASSICAL PHYSICS
Chapter 3
De Broglie Waves
In chapter 2 we have seen that — at the quantum level — light, which is classically described by Maxwell’s wave equations, also displays particle features. On the other hand,
matter, which is classically described as particles, shows wave behavior at the small scales
of the quantum world. In fact, in quantum physics we should think of particle and wave
as the same thing. In particular, the classical particle picture of an object with definite
position and momentum always breaks down at very short distances. Schrödinger’s formulation of quantum mechanics uses the wave function Ψ(x, t) to describe the probability
amplitude to find a particle at position x at time t. The Schrödinger equation is the
wave equation for Ψ(x, t). In this chapter we discuss the quantum description of free
non-relativistic particles by de Broglie waves.
3.1
Wave Packets
We have seen that quantum mechanically particles with momentum p = ~k behave like
waves with wave number k. The corresponding wave function given by
Ψ(x) = A exp(ikx),
(3.1.1)
describes a particle with definite momentum. When we form the probability density we
find
ρ(x) = |Ψ(x)|2 = |A|2 ,
(3.1.2)
which is independent of the position x. Hence, all positions are equally probable and it is
completely uncertain where the particle is. This is consistent with Heisenberg’s uncertainty
relation. To describe a particle that is localized in a particular region of space, one needs
to go beyond a simple plane wave. In general one can superimpose plane waves with
different k values to construct a wave packet
Z ∞
1
Ψ(x) =
dk Ψ(k) exp(ikx).
(3.1.3)
2π −∞
43
44
CHAPTER 3. DE BROGLIE WAVES
Here Ψ(k) is the amplitude of the plane wave with wave number k. The above equation
describes a Fourier transformation and Ψ(k) is called the Fourier transform of the wave
function Ψ(x). Note that Ψ(k) and Ψ(x) are two different functions. They are denoted by
the same letter Ψ just because they are closely related via the Fourier transform which is
further discussed in appendix E.
As we have learned, |Ψ(x)|2 is the probability density to find the particle at position
x. Consequently, |Ψ(x)|2 dx is the probability to find it in an interval dx around the point
x. The integral
Z
∞
dx |Ψ(x)|2 = 1
(3.1.4)
−∞
therefore is the probability to find the particle somewhere in space. This total probability
should be normalized to one. The normalization condition for Ψ(x) implies a normalization
condition for Ψ(k) which can be derived by inserting eq.(3.1.3) into eq.(3.1.4) such that
Z ∞
Z ∞
Z ∞
1
1
dx
dk Ψ(k)∗ exp(−ikx)
dk 0 Ψ(k 0 ) exp(ik 0 x) =
2π
2π
−∞
Z ∞
Z−∞
Z ∞ −∞
∞
1
1
dk
dk 0 Ψ(k)∗ Ψ(k 0 )
dx exp(i(k 0 − k)x) =
2π −∞
2π −∞
−∞
Z ∞
Z ∞
1
dk
dk 0 Ψ(k)∗ Ψ(k 0 )δ(k 0 − k) =
2π −∞
−∞
Z ∞
1
dk Ψ(k)∗ Ψ(k).
(3.1.5)
2π −∞
Here we have identified the Dirac δ-function
Z ∞
1
0
δ(k − k) =
dx exp(i(k 0 − k)x),
2π −∞
which is further discussed in appendix E, and we have used
Z ∞
dk 0 f (k 0 )δ(k 0 − k) = f (k).
(3.1.6)
(3.1.7)
−∞
For the normalization condition in momentum space we hence obtain
Z ∞
1
dk |Ψ(k)|2 = 1.
2π −∞
(3.1.8)
It is important that the wave function in momentum space is properly normalized to 1
only if the integral of |Ψ(k)|2 is divided by 2π. Hence,
1
ρ(~k) =
|Ψ(k)|2 ,
2π
(3.1.9)
is the probability density to find the particle at wave number k (or equivalently with
momentum p = ~k). It is important to realize that unlike Ψ(k) which is the Fourier
transform of Ψ(x), ρ(k) is not the Fourier transform of ρ(x).
3.2. EXPECTATION VALUES
3.2
45
Expectation Values
Since |Ψ(x)|2 dx is the probability to find the particle in an interval dx around the position
x we can compute the expectation value of the position, i.e. the average over a large
number of position measurements, as
Z ∞
hxi =
dx x|Ψ(x)|2 .
(3.2.1)
−∞
Similarly we obtain
hx2 i =
Z
∞
dx x2 |Ψ(x)|2 .
(3.2.2)
−∞
The variance ∆x of the position is given by
(∆x)2 = h(x − hxi)2 i = hx2 − 2xhxi + hxi2 i
= hx2 i − 2hxihxi + hxi2 = hx2 i − hxi2 .
Similarly, one defines expectation values of momenta (or k values) as
Z ∞
Z ∞
1
1
hki =
dk k|Ψ(k)|2 , hk 2 i =
dk k 2 |Ψ(k)|2 ,
2π −∞
2π −∞
and (up to a factor of ~) the variance of the momentum is given by
p
∆k = hk 2 i − hki2 .
(3.2.3)
(3.2.4)
(3.2.5)
One can also derive hki directly from Ψ(x). This follows from
Z ∞
dx Ψ(x)∗ (−i∂x )Ψ(x) =
Z−∞
Z ∞
Z ∞
∞
1
1
∗
dx
dk Ψ(k) exp(−ikx)(−i∂x )
dk 0 Ψ(k 0 ) exp(ik 0 x) =
2π
2π
−∞
Z−∞
Z−∞
Z ∞
∞
∞
1
1
∗
0
dx
dk Ψ(k) exp(−ikx)
dk Ψ(k 0 )k 0 exp(ik 0 x) =
2π
2π
−∞
Z ∞
Z−∞
Z ∞−∞
∞
1
1
dk
dk 0 Ψ(k)∗ Ψ(k 0 )k 0
dx exp(i(k 0 − k)x) =
2π −∞
2π −∞
−∞
Z ∞
Z ∞
1
dk
dk 0 Ψ(k)∗ Ψ(k 0 )k 0 δ(k 0 − k)
2π −∞
−∞
Z ∞
1
dk k|Ψ(k)|2 = hki.
(3.2.6)
2π −∞
Here ∂x is the derivative with respect to x. Similarly, one finds
Z ∞
dx Ψ(x)∗ (−i∂x )2 Ψ(x) = hk 2 i.
−∞
(3.2.7)
46
3.3
CHAPTER 3. DE BROGLIE WAVES
Coordinate and Momentum Space Representation of
Operators
In quantum mechanics observables like x and k are generally described by operators, and
−i∂x is the operator that represents k in coordinate space. Similarly, the operator i∂k
represents x in momentum space.
In three dimensions, the operator that represents the momentum vector acting on a
coordinate space wave function Ψ(~x) is given by
~
p~ = −i~(∂x , ∂y , ∂z ) = −i~∇.
(3.3.1)
In momentum space, on the other hand, the operator p~ = ~~k acts on a wave function Ψ(~k)
simply by multiplication, i.e.
p~ Ψ(~k) = ~~kΨ(~k).
(3.3.2)
Similarly, the operator that represents the position vector ~x of the particle, acts on a wave
function Ψ(~x) by a simple multiplication ~x Ψ(~x), while it acts on a momentum space wave
function Ψ(~k) by differentiation, i.e.
~ k,
~x = i(∂kx , ∂ky , ∂kz ) = i∇
(3.3.3)
~ k Ψ(~k).
~x Ψ(~k) = i∇
(3.3.4)
such that
The kinetic energy of a free non-relativistic particle of mass M is given by
T =
p~ 2
.
2M
(3.3.5)
The corresponding operator that acts on a position space wave function Ψ(~x) is hence
given by
2
2
p~ 2
1
~ 2 = − ~ (∂x2 + ∂y2 + ∂z2 ) = − ~ ∆,
T =
=
(−i~∇)
(3.3.6)
2M
2M
2M
2M
where ∆ is the Laplace operator, such that
T Ψ(~x) = −
~2
∆Ψ(~x).
2M
(3.3.7)
In momentum space, on the other hand, the kinetic energy operator acts again simply by
multiplication
p~ 2
~2~k 2 ~
T Ψ(~k) =
Ψ(~k) =
Ψ(k).
(3.3.8)
2M
2M
These results are summarized in table 3.3.
3.4. HEISENBERG’S UNCERTAINTY RELATION
Object
wave function in 1-d
particle coordinate in 1-d
momentum operator in 1-d
wave function in 3-d
particle coordinate in 3-d
momentum operator in 3-d
coordinate space
Ψ(x)
x
−i~∂x
Ψ(~x)
~x
~ = −i~(∂x , ∂y , ∂z )
−i~∇
kinetic energy operator in 3-d
~
~
− 2M
∆ = − 2M
(∂x2 + ∂y2 + ∂z2 )
2
2
47
momentum space
Ψ(k)
i∂k
p = ~k
Ψ(~k)
~ k = i(∂k , ∂k , ∂k )
i∇
x
y
z
p~ = ~~k
T =
~2~k 2
2M
Table 3.1: Representations of operators acting in coordinate and in momentum space.
3.4
Heisenberg’s Uncertainty Relation
We are now in a position to state the Heisenberg uncertainty relation. It takes the form
1
∆x∆k ≥ ,
2
(3.4.1)
or equivalently ∆x∆p ≥ ~2 . Gaussian wave packets have a minimal uncertainty product,
i.e. for them ∆x∆k = 12 . The corresponding wave function takes the form
x2
Ψ(x) = A exp − 2 ,
2a
(3.4.2)
p √
where A = 1/ a π follows from the normalization condition. For a Gaussian wave packet
2
one finds hxi = 0 and hx2 i = a2 such that ∆x = √a2 . Performing the Fourier transform to
momentum space one finds
2 2
a k
Ψ(k) = B exp −
,
(3.4.3)
2
p √
where B = 2a π again follows from the normalization condition. In momentum space
we find hki = 0 and hk 2 i = 2a12 such that ∆k = √12a . Hence, for a Gaussian wave packet
indeed ∆x∆k = 12 .
Let us now prove Heisenberg’s relation for general wave packets. For that purpose we
construct the function
Φ(x) = ∂x Ψ(x) + αxΨ(x) + βΨ(x).
(3.4.4)
Here we choose α to be real, while β may be complex. The idea of the proof is to evaluate
the integral
Z ∞
I=
dx |Φ|2 ≥ 0,
(3.4.5)
−∞
which is positive by construction, and then vary α and β to find the minimum value of this
48
CHAPTER 3. DE BROGLIE WAVES
integral. The resulting relation will turn out to be Heisenberg’s inequality. One obtains
Z ∞
dx (∂x Ψ∗ + αxΨ∗ + β ∗ Ψ∗ )(∂x Ψ + αxΨ + βΨ)
I =
−∞
Z ∞
dx (∂x Ψ∗ ∂x Ψ + ∂x Ψ∗ αxΨ + ∂x Ψ∗ βΨ + αxΨ∗ ∂x Ψ
=
−∞
+ αxΨ∗ αxΨ + αxΨ∗ βΨ + β ∗ Ψ∗ ∂x Ψ + β ∗ Ψ∗ αxΨ + β ∗ Ψ∗ βΨ)
Z ∞
=
dx (−Ψ∗ ∂x2 Ψ − α|Ψ|2 − Ψ∗ αx∂x Ψ − Ψ∗ β∂x Ψ + αxΨ∗ ∂x Ψ
−∞
2 2
+ α x |Ψ|2 + αβx|Ψ|2 + β ∗ Ψ∗ ∂x Ψ + αβ ∗ x|Ψ|2 + |β|2 |Ψ|2 )
= hk 2 i − α + 2βi hki + α2 hx2 i + 2αβr hxi + βr2 + βi2 .
(3.4.6)
Here β = βr + iβi . Now we search for the minimum of the integral
dI
= −1 + 2αhx2 i + 2βr hxi = 0,
dα
dI
= 2αhxi + 2βr = 0,
dβr
dI
= 2hki + 2βi = 0.
dβi
(3.4.7)
These equations imply
α=
1
hxi
, βr = −
, βi = −hki.
2(∆x)2
2(∆x)2
(3.4.8)
Inserting this back into eq.(3.4.6) yields
I = (∆k)2 −
1
≥ 0,
4(∆x)2
(3.4.9)
and hence
1
∆x∆k ≥ .
2
(3.4.10)
From the above derivation it follows that Heisenberg’s inequality can be saturated, i.e.
∆x∆k = 12 can be reached, only if Φ = 0. For the Gaussian wave packet
x2
Ψ(x) = A exp − 2 ,
2a
we had hxi = 0, hki = 0 and ∆x =
indeed
√a .
2
Hence, in that case α =
Φ(x) = ∂x Ψ(x) + αxΨ(x) + βΨ(x) = −
(3.4.11)
1
a2
and βr = βi = 0 and
x
x
Ψ(x) + 2 Ψ(x) = 0.
2
a
a
(3.4.12)
3.5. DISPERSION RELATIONS
3.5
49
Dispersion Relations
Classically, the electromagnetic field is described by Maxwell’s wave equations which are
consistent with special relativity. As a consequence, an electromagnetic plane wave in
vacuum with space-time dependence exp(i(~k · ~x − ωt)) obeys the dispersion relation
ω = |~k|c.
(3.5.1)
E = ~ω, p~ = ~~k,
(3.5.2)
Using the quantum physics relations
for photons, this is indeed equivalent to the relativistic energy-momentum relation
E = |~
p|c
(3.5.3)
for massless particles. For massive particles with rest mass M the corresponding relation
takes the form
p
EM = (M c2 )2 + (~
pc)2 .
(3.5.4)
Here we will restrict ourselves to the non-relativistic limit of small momenta or large
masses, such that
p~ 2
EM = M c2 +
.
(3.5.5)
2M
The rest energy M c2 represents a constant (momentum-independent) energy shift, which
in non-relativistic physics is usually absorbed in a redefinition of the energy, and hence
E = EM − M c2 =
p~ 2
.
2M
(3.5.6)
De Broglie was first to postulate the general validity of eq.(3.5.2) not just for photons
but for any free massive particle as well. This immediately implies that for a free nonrelativistic particle the dispersion relation reads
ω=
~~k 2
p~ 2
=
.
2M ~
2M
(3.5.7)
Indeed, in Schrödinger’s wave mechanics a free particle with momentum p~ is described by
the wave function
!!
~~k 2
~
Ψ(~x, t) = A exp i k · ~x −
t
.
(3.5.8)
2M
The time evolution of a 3-dimensional wave packet is then determined by
!!
Z
~k 2
1
~
3
Ψ(~x, t) =
d k Ψ(~k) exp i ~k · ~x −
t
(2π)3
2M
Z
1
=
d3 k Ψ(~k, t) exp(i~k · ~x).
(2π)3
(3.5.9)
50
CHAPTER 3. DE BROGLIE WAVES
The time-dependent momentum space wave function takes the form
!
~k 2
~
Ψ(~k, t) = Ψ(~k) exp −i
t .
2M
(3.5.10)
In particular, the probability density in momentum space
ρ(~k, t) =
1
1
|Ψ(~k, t)|2 =
|Ψ(~k)|2 ,
3
(2π)
(2π)3
(3.5.11)
is time-independent. This is typical for the propagation of a free particle. Its momentum
is a conserved quantity, and therefore the probability distribution of momenta also does
not change with time. The position of the particle, on the other hand, is obviously
not conserved, since it is moving with a certain velocity. Let us now calculate the time
evolution of a 1-dimensional wave function in coordinate space, first approximately for a
wave packet that has a narrow distribution Ψ(k) in momentum space centered around a
wave number k0 . In that case we can approximate the dispersion relation by
ω(k) = ω(k0 ) +
dω
(k0 )(k − k0 ) + O((k − k0 )2 ).
dk
(3.5.12)
Neglecting quadratic and higher order terms, we find
Z ∞
1
Ψ(x, t) =
dk Ψ(k) exp(i(kx − ω(k)t))
2π −∞
Z ∞
1
dω
=
dk Ψ(k) exp i kx − ω(k0 ) +
(k0 )(k − k0 ) t
2π −∞
dk
Z ∞
dω
1
dk Ψ(k) exp ik x −
(k0 )t
=
2π −∞
dk
dω
× exp −i ω(k0 ) −
(k0 )k0 t
dk
dω
dω
(k0 )t, 0) exp −i ω(k0 ) −
(k0 )k0 t .
(3.5.13)
= Ψ(x −
dk
dk
Up to a phase factor, the wave function is just the one at time t = 0 shifted in space by
dω
dk (k0 )t, i.e. the wave packet is moving with the so-called group velocity
vg =
dω
~k0
(k0 ) =
,
dk
M
(3.5.14)
which indeed is the velocity of a non-relativistic particle with momentum ~k0 . Still, it
is important to note that the above calculation is only approximate because we have
neglected quadratic terms in the dispersion relation. As a consequence, the wave packet
traveled undistorted like an electromagnetic wave in vacuum. Once the quadratic terms are
included, the wave packet will also change its shape because the different wave number
components travel with different velocities. This is similar to an electromagnetic wave
traveling in a medium. Unfortunately, the spreading of a quantum mechanical wave packet
is difficult to work out for a general wave packet. Therefore we restrict ourselves to
Gaussian wave packets in the following.
3.6. SPREADING OF A GAUSSIAN WAVE PACKET
3.6
51
Spreading of a Gaussian Wave Packet
Let us consider a properly normalized 1-dimensional Gaussian wave packet
2
q
√
a (k − k0 )2
,
Ψ(k) = 2a π exp −
2
(3.6.1)
centered around k0 in momentum space. The corresponding wave function in coordinate
space is given by
2
Z ∞
q
√
1
~k 2
a (k − k0 )2
Ψ(x, t) =
dk 2a π exp −
exp i kx −
t
2π −∞
2
2M
2 2
Z ∞
q
√
a k
1
dk 2a π exp −
=
2π −∞
2
~(k + k0 )2
t
× exp i (k + k0 )x −
2M
Z ∞
q
√
~t k 2
~k0
1
dk 2a π exp − a2 + i
exp ik x −
t
=
2π −∞
M 2
M
2
~k
× exp i k0 x − 0 t
.
(3.6.2)
2M
At this point we use a result obtained before
2 2
Z ∞
q
√
1
a k
1
x2
dk 2a π exp −
exp(ikx) = p √ exp − 2 .
2π −∞
2
2a
a π
(3.6.3)
We can therefore read off
r
Ψ(x, t) =
a
~k02
x02
√ exp − 02 exp i k0 x −
t
,
2a
2M
a02 π
(3.6.4)
where x0 = x − ~k0 t/M , i.e. the particle moves with velocity v = ~k0 /M and
a02 = a2 + i
~t
1
a2 − i~t/M
⇒ 02 = 4
.
M
a
a + (~t/M )2
(3.6.5)
Inserting this in the previous result and squaring the wave function we obtain the probability density
a
(x − ~k0 t/M )2 a2
2
ρ(x, t) = |Ψ(x, t)| = p
.
(3.6.6)
√ exp − 4
a + (~t/M )2
a4 + (~t/M )2 π
The width of the wave packet in coordinate space increases with time according to
s
~t 2
2
a(t) = a +
.
(3.6.7)
Ma
52
CHAPTER 3. DE BROGLIE WAVES
At t = 0 we have a(t) = a and at very large times we have a(t) = ~t/M a. The speed
of spreading of the wave packet is ~/M a, i.e. a packet that was very localized initially
(had a small a) will spread very quickly. This is understandable because a spatially
1
localized wave packet (with a small ∆x = √a2 ) necessarily has a large spread ∆k = a√
in
2
momentum space. Hence, the various wave number contributions to the packet travel with
very different speeds and consequently the wave packet quickly spreads out in coordinate
space. One should not think of the spread ∆x as the size of the quantum mechanical
particle. Indeed, we have assumed the particle to be point-like. The increasing spread
√ simply tells us that with time we know less and less about where the particle
∆x(t) = a(t)
2
is. This is simply because it has traveled with an uncertain velocity that is picked from
the assumed Gaussian probability distribution.
Chapter 4
The Schrödinger Equation
In this chapter we introduce the Schrödinger equation as the fundamental dynamical
equation of quantum mechanics. While we will motivate the Schrödinger equation based
on the dispersion relation of de Broglie waves, it can presently not be derived from a
more fundamental physical theory. Even our most fundamental theory — the standard
model of particle physics — which is a relativistic quantum field theory is governed by a
corresponding (functional) Schrödinger equation.
4.1
From Wave Packets to the Schrödinger Equation
In chapter 3 we have learned how to deal with wave functions describing free non-relativistic
particles. In particular, those are entirely determined by an initial distribution Ψ(~k) in
momentum space. The time evolution of the wave function in 3-dimensional coordinate
space is then given by
1
Ψ(~x, t) =
(2π)3
Z
~|~k|2
d k Ψ(~k) exp i ~k · ~x −
t
2M
3
!!
.
(4.1.1)
Writing
1
Ψ(~x, t) =
(2π)3
Z
d3 k Ψ(~k, t) exp(i~k · ~x),
(4.1.2)
we can thus identify
!
~k|2
~|
Ψ(~k, t) = Ψ(~k) exp −i
t ,
2M
53
(4.1.3)
54
CHAPTER 4. THE SCHRÖDINGER EQUATION
such that |Ψ(~k, t)|2 = |Ψ(~k)|2 is time-independent. Let us investigate the space- and
time-dependence of Ψ(~x, t) by taking the corresponding derivatives
!!
Z
2 |~
2
~k|2
1
~
k|
~|
3
i~∂t Ψ(~x, t) =
d k Ψ(~k)
exp i ~k · ~x −
t
,
(2π)3
2M
2M
!!
Z
~k|2
~|
1
d3 k Ψ(~k)(i~k)2 exp i ~k · ~x −
t
.
∆Ψ(~x, t) =
(2π)3
2M
(4.1.4)
Here ∆ = ∂x2 + ∂y2 + ∂z2 is the Laplacian. One thus obtains
i~∂t Ψ(~x, t) = −
~2
∆Ψ(~x, t),
2M
(4.1.5)
which is indeed the Schrödinger equation for a free particle. By our construction it is clear
that the Schrödinger equation resulted just from the non-relativistic dispersion relation
E = |~
p|2 /2M . Indeed
~ 2 = p~ 2 ,
−~2 ∆ = (−i~∇)
(4.1.6)
is nothing but the square of the momentum operator
~ = −i~(∂x , ∂y , ∂z ).
p~ = −i~∇
(4.1.7)
It is now straightforward to include interactions of the particle with an external potential
V (~x). The total energy of the classical problem then receives contributions from the kinetic
and the potential energy
|~
p|2
E=
+ V (~x).
(4.1.8)
2M
Hence, we should include V (~x) also in the Schrödinger equation
i~∂t Ψ(~x, t) = −
~2
∆Ψ(~x, t) + V (~x)Ψ(~x, t).
2M
(4.1.9)
In fact, this is the equation that Schrödinger wrote down originally. In quantum mechanics
the total energy is represented by the operator
H=
p~ 2
~2
+ V (~x) = −
∆ + V (~x),
2M
2M
(4.1.10)
which is known as the Hamilton operator or Hamiltonian. The kinetic part of the Hamiltonian acts on the wave function by taking its second derivative. The potential part, on
the other hand, just multiplies the wave function with V (~x). The Schrödinger equation is
the wave equation for the quantum mechanical wave function, i.e. it determines the time
evolution of Ψ(~x, t) once we specify an initial distribution Ψ(~x, 0). Hence, Schrödinger’s
equation plays the same role in quantum mechanics that Maxwell’s wave equations play
in electrodynamics.
4.2. CONSERVATION OF PROBABILITY
4.2
55
Conservation of Probability
We have learned that ρ(~x, t) = |Ψ(~x, t)|2 is the probability density to find the particle at
position ~x at time t. For this interpretation of the wave function to be consistent we must
demand that the wave function is normalized, i.e.
Z
Z
3
d x ρ(~x, t) = d3 x |Ψ(~x, t)|2 = 1,
(4.2.1)
simply because it should always be possible to find the particle somewhere in space. In
particular, this should be so at any moment in time. In other words, the total probability
should be conserved and should be equal to 1 at all times. Let us calculate the change of
the probability density with time
i~∂t ρ(~x, t) = i~∂t |Ψ(~x, t)|2 = i~Ψ(~x, t)∗ ∂t Ψ(~x, t) + i~Ψ(~x, t)∂t Ψ(~x, t)∗
~2
~2
∗
= Ψ(~x, t) −
∆ + V (~x) Ψ(~x, t) − Ψ(~x, t) −
∆ + V (~x) Ψ(~x, t)∗ +
2M
2M
~2
~ · ~j(~x, t).
[Ψ(~x, t)∗ ∆Ψ(~x, t) − Ψ(~x, t)∆Ψ(~x, t)∗ ] = −i~∇
(4.2.2)
= −
2M
Here we have introduced the probability current density
h
i
~ x, t) − Ψ(~x, t)∇Ψ(~
~ x, t)∗ .
~j(~x, t) = ~ Ψ(~x, t)∗ ∇Ψ(~
2M i
(4.2.3)
Indeed, taking the divergence of the current, we obtain
i
h
2
~ · Ψ(~x, t)∗ ∇Ψ(~
~ x, t) − Ψ(~x, t)∇Ψ(~
~ x, t)∗
~ · ~j(~x, t) = − ~ ∇
−i~∇
2M
~2 h ~
~ x, t) + Ψ(~x, t)∗ ∆Ψ(~x, t)
∇Ψ(~x, t)∗ · ∇Ψ(~
= −
2M
i
~ x, t) · ∇Ψ(~
~ x, t)∗ − Ψ(~x, t)∆Ψ(~x, t)∗
− ∇Ψ(~
= −
~2
[Ψ(~x, t)∗ ∆Ψ(~x, t) − Ψ(~x, t)∆Ψ(~x, t)∗ ] .
2M
(4.2.4)
Combining these results we thus obtain
~ · ~j(~x, t) = 0.
∂t ρ(~x, t) + ∇
(4.2.5)
This is a continuity equation that guarantees probability conservation. It relates the probability density ρ(~x, t) to the probability current density ~j(~x, t) in the same way in which
electric charge and current density are related in electrodynamics. In that case, charge
conservation follows from Maxwell’s equations. Here probability conservation follows from
the Schrödinger equation.
Let us consider the probability to find the particle in a region Ω of space. Its rate of
change with time is given by
Z
Z
Z
3
2
3
∂t
d x |Ψ(~x, t)| =
d x ∂t ρ(~x, t) = −
d2~s · ~j(~x, t).
(4.2.6)
Ω
Ω
∂Ω
56
CHAPTER 4. THE SCHRÖDINGER EQUATION
Here we have used the Gauss law. The continuity equation tells us that the probability
to find the particle in a region Ω can change only if there is a probability current flowing
through the boundary ∂Ω of that region. Assuming that the probability current density
~j(~x, t) vanishes at spatial infinity immediately implies that the total probability remains
constant.
4.3
The Time-Independent Schrödinger Equation
We have seen that the Schrödinger equation governs the time evolution of the quantum
mechanical wave function. In many cases, however, one is interested in stationary solutions
of the Schrödinger equation, for example, in the stationary states of an atom. Also it turns
out that once one has found all stationary solutions of the Schrödinger equation, one can
use them to construct the general time-dependent solution. In a stationary state the
probability density ρ(~x, t) = |Ψ(~x, t)|2 = |Ψ(~x)|2 is time-independent. This does not mean
that Ψ(~x, t) is time-independent. In fact, it may still have a time-dependent phase. Hence
we make the ansatz
Ψ(~x, t) = Ψ(~x) exp(iϕ(t)).
(4.3.1)
Inserting this in the Schrödinger equation implies
~2
−~∂t ϕ(t)Ψ(~x) exp(iϕ(t)) = −
∆ + V (~x) Ψ(~x) exp(iϕ(t)).
2M
(4.3.2)
Dividing this equation by exp(iϕ(t)) one finds that the right-hand side becomes timeindependent. Hence, the left-hand side should also be time-independent and thus
−~∂t ϕ(t) = E,
(4.3.3)
where E is a constant. Integrating this equation one obtains
1
ϕ(t) = − Et,
~
and thus
i
Ψ(~x, t) = Ψ(~x) exp − Et .
~
(4.3.4)
(4.3.5)
The remaining equation for the time-independent wave function Ψ(~x) takes the form
~2
−
∆ + V (~x) Ψ(~x) = EΨ(~x).
(4.3.6)
2M
This is the so-called time-independent Schrödinger equation. Identifying the Hamilton
operator
~2
H=−
∆ + V (~x),
(4.3.7)
2M
we can also write it as
HΨ(~x) = EΨ(~x).
(4.3.8)
4.3. THE TIME-INDEPENDENT SCHRÖDINGER EQUATION
57
As we will soon understand, this can be viewed as an eigenvalue problem for the operator
H with the eigenvalue E and the eigenvector Ψ(~x). We can now calculate the expectation
value of the Hamiltonian (the operator that describes the total energy). Using the timeindependent Schrödinger equation we find
Z
Z
Z
3
∗
3
∗
d x Ψ(~x, t) HΨ(~x, t) = d x Ψ(~x) HΨ(~x) = d3 x E|Ψ(~x)|2 = E.
(4.3.9)
This clarifies the meaning of the constant E. It is nothing but the total energy.
58
CHAPTER 4. THE SCHRÖDINGER EQUATION
Chapter 5
Square-Well Potentials and
Tunneling Effect
We now know enough quantum mechanics to go ahead and start applying it to physical
problems. Doing quantum mechanics essentially means solving the Schrödinger equation. As we will see in chapter 7, for time-independent Hamilton operators the general solution of the time-dependent Schrödinger equation follows from those of the timeindependent Schrödinger equation. Hence, in this chapter we will concentrate on the
time-independent equation. For a general potential it may be difficult to solve the timeindependent Schrödinger equation. In fact, in general one will not be able to find an
analytic solution, and hence one must then use some approximate numerical method.
Still, there are a number of potentials for which the Schrödinger equation can be solved
analytically, among them several examples in one dimension. Obviously, it is simpler to
work in one than in three dimensions. This is the main reason for studying 1-dimensional
problems in this chapter. We will learn techniques to solve the Schrödinger equation that
will be useful also in two and three spatial dimensions. In this chapter we will solve
the 1-dimensional time-independent Schrödinger equation with square-well potentials. In
particular, we will investigate a genuine quantum phenomenon — the so-called tunneling
effect — which is classically forbidden, but allowed at the quantum level. We will also
encounter a paradoxical quantum reflection problem, which we denote as the “quantum
height anxiety paradox”, and finally we will study a free particle moving on a circle.
5.1
Continuity Equation in One Dimension
As we have seen, quantum mechanical probability conservation is encoded in the continuity
equation (4.2.5). In one spatial dimension the continuity equation takes the form
∂t ρ(x, t) + ∂x j(x, t) = 0,
59
(5.1.1)
60
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
with the probability density ρ(x, t) = |Ψ(x, t)|2 and the probability current density
j(x, t) =
~
[Ψ(x, t)∗ ∂x Ψ(x, t) − Ψ(x, t)∂x Ψ(x, t)∗ ] .
2M i
(5.1.2)
As we learned from eq.(4.3.5), stationary solutions of the time-dependent Schrödinger
equation with energy E take the factorized form
i
Ψ(x, t) = Ψ(x) exp − Et ,
(5.1.3)
~
such that
ρ(x, t) = |Ψ(x)|2 = ρ(x) ⇒ ∂t ρ(x, t) = ∂t ρ(x) = 0.
(5.1.4)
Furthermore,
j(x, t) =
~
[Ψ(x)∗ ∂x Ψ(x) − Ψ(x)∂x Ψ(x)∗ ] = j(x),
2M i
(5.1.5)
such that the continuity equation (5.1.1) reduces to
∂t ρ(x, t) + ∂x j(x, t) = ∂x j(x) = 0 ⇒ j(x) = j.
(5.1.6)
Hence, the probability current is, in fact, a constant j independent of x. States that are
localized and thus do not extend to spatial infinity have Ψ(±∞) = 0, such that for them
j = 0.
5.2
A Particle in a Box with Dirichlet Boundary Conditions
One of the simplest problems in quantum mechanics concerns a particle in a 1-dimensional
box with perfectly reflecting walls. We can realize this situation by switching on a potential
that is infinite everywhere outside an interval [0, a], and zero inside this interval. The
infinite potential energy implies that the particle cannot exist outside the box, and hence
Ψ(x) = 0 for x ∈
/ [0, a]. Consequently, the constant probability current density j = 0 also
vanishes everywhere. This is consistent with the fact that the particle is indeed localized
inside the finite interval [0, a].
The most general boundary conditions that are consistent with the vanishing probability current density are the so-called mixed or Robin boundary conditions
γ(0)Ψ(0) + ∂x Ψ(0) = 0, γ(a)Ψ(a) + ∂x Ψ(a) = 0,
(5.2.1)
with γ(0), γ(a) ∈ R. Indeed, the probability current density at x = 0 then takes the form
j(0) =
=
~
[Ψ(0)∗ ∂x Ψ(0) − Ψ(0)∂x Ψ(0)∗ ]
2M i
~
[−Ψ(0)∗ γΨ(0) + Ψ(0)γ(0)∗ Ψ(0)∗ ] = 0.
2M i
(5.2.2)
5.2. A PARTICLE IN A BOX WITH DIRICHLET BOUNDARY CONDITIONS
61
Similarly, for the boundary at x = a we obtain j(a) = 0. The problem of the particle in
the box is characterized by a 2-parameter family of boundary conditions parametrized by
the two real-valued parameters γ(0) and γ(a). These parameters are often ignored in the
textbook literature, which usually restricts itself to the Dirichlet boundary conditions
Ψ(0) = 0, Ψ(a) = 0.
(5.2.3)
This is the specific choice, corresponding to γ(0) = γ(a) = ∞, for which the wave function
is continuous at x = 0 and at x = a. As we will see later, discontinuous wave functions (which arise for general Robin boundary conditions) are perfectly acceptable both
physically and mathematically. In this chapter, for simplicity we restrict ourselves to the
standard textbook procedure and thus adopt Dirichlet boundary conditions.
The time-independent Schrödinger equation inside the box takes the form
−
~2 d2 Ψ(x)
= EΨ(x),
2M dx2
(5.2.4)
which we consider with the Dirichlet boundary condition Ψ(0) = Ψ(a) = 0. We make the
ansatz
Ψ(x) = A sin(kx).
(5.2.5)
Then the boundary condition implies
πn
,
a
(5.2.6)
~2 k 2
~2 π 2 n2
=
.
2M
2M a2
(5.2.7)
k=
and the Schrödinger equation yields
E=
As a consequence of the infinite potential on both sides of the interval the energy spectrum
is entirely discrete. Also, the ground state energy
E=
~2 π 2
2M a2
(5.2.8)
— i.e. the lowest possible energy — is non-zero. This means that the particle in the box
cannot be at rest. It always has a non-zero kinetic energy. This is a genuine quantum
effect. Classically, there would be nothing wrong with a particle being at rest in a box,
and indeed in the classical limit ~ → 0 the ground state energy goes to zero.
The fact that the ground state energy is non-zero already follows from the Heisenberg
uncertainty relation
~
∆x∆p ≥ .
(5.2.9)
2
p
For the particle in the box hpi = 0, and hence ∆p = hp2 i. Also the spread of the wave
function in x can at most be a/2 such that ∆x ≤ a/2. Therefore the kinetic energy of the
particle is restricted by
~2
p2
h
i≥
,
(5.2.10)
2M
2M a2
62
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
which is consistent with the actual ground state energy.
Let us also use the semi-classical Bohr-Sommerfeld quantization for this problem. The
quantization condition has the form
S[xc (t)] + ET = 2πn~,
(5.2.11)
where S[xc (t)] is the action of a periodic classical path with period T . The action is the
time integral of the difference between kinetic and potential energy. Since the potential
vanishes inside the box, the action is simply the integrated kinetic energy, which here
corresponds to the conserved (i.e. time-independent) total energy. Therefore, in this case
S[xc (t)] = ET,
(5.2.12)
ET = πn~.
(5.2.13)
and thus
Inside the box the particle moves with constant speed v bouncing back and forth between
the two walls. The period of a corresponding classical path is
2a
,
v
(5.2.14)
M v2
.
2
(5.2.15)
T =
and the energy is
E=
The quantization condition thus implies
ET = M av = πn~ ⇒ v =
πn~
,
Ma
(5.2.16)
and hence
M v2
~2 π 2 n2
=
,
(5.2.17)
2
2M a2
which is indeed the exact spectrum that we also obtained from the Schrödinger equation.
As we will see, this is not always the case.
E=
Let us now modify the above problem and make it more interesting. We still consider
an infinite potential for all negative x, and a vanishing potential in the interval [0, a], but
we set the potential to a finite value V0 > 0 for x > a. Then the semi-classical analysis
already gets modified. In fact, a classical path with an energy larger than V0 will no longer
be confined to the interior of the box, but will simply go to infinity. Such a path is not
periodic, and can therefore not be used in the Bohr-Sommerfeld quantization procedure.
Consequently, the predicted discrete energy spectrum still is
E=
~2 π 2 n2
,
2M a2
(5.2.18)
but with the additional restriction E < V0 . This means that the allowed values of n are
limited to
√
2M a2 V0
n<
,
(5.2.19)
~π
5.2. A PARTICLE IN A BOX WITH DIRICHLET BOUNDARY CONDITIONS
63
and only a finite set of discrete energy values exists.
Let us see what the Schrödinger equation implies for the modified problem. First, let us
again assume that E < V0 . Again using Dirichlet boundary conditions, the wave function
must still vanish at x = 0. However, it can now extend into the classically forbidden region
x > a. In that region the Schrödinger equation takes the form
−
such that
~2 d2 Ψ(x)
+ V0 Ψ(x) = EΨ(x),
2M dx2
(5.2.20)
2M (V0 − E)
d2 Ψ(x)
=
Ψ(x).
2
dx
~2
(5.2.21)
Ψ(x) = B exp(−κx),
(5.2.22)
We make the ansatz
and obtain
p
2M (V0 − E)
κ=±
.
(5.2.23)
~
Only the positive solution is physical because it corresponds to an exponentially decaying
normalizable wave function. The other solution is exponentially rising, and thus not
normalizable. Inside the box the solution of the Schrödinger equation still takes the form
Ψ(x) = A sin(kx),
(5.2.24)
but now the wave function need not vanish at x = a. Later we will discuss the most
general boundary condition that can be imposed in case of a finite potential step. In this
chapter, we again restrict ourselves to the standard textbook procedure by demanding
that both the wave function and its first derivative are continuous at x = a. This implies
A sin(ka) = B exp(−κa), Ak cos(ka) = −Bκ exp(−κa),
(5.2.25)
κ = −k cot(ka).
(5.2.26)
and thus
We can also use the relations for k and κ and obtain
2M E 2M (V0 − E)
2M V0
+
=
.
(5.2.27)
2
2
~
~
~2
√
This equation describes a circle of radius 2M V0 /~ in the k-κ-plane, which suggests a
graphical method to solve for the energy spectrum. In fact, plotting κ = −k cot(ka) as
well as the circle yields the quantized energies as the intersections of the two curves. The
condition for at least n intersections to exist is
√
2M V0
π
1
>
n−
.
(5.2.28)
~
a
2
k 2 + κ2 =
For n = 1 this implies
V0 >
~2 π 2
.
8M a2
(5.2.29)
64
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
The potential must be larger than this critical value in order to support at least one discrete
energy level. A particle in that state is localized essentially inside the box, although it has
a non-zero probability to be at x > a. Therefore, such a state is called a bound state. The
above condition for the existence of at least one bound state is not the same as the one
that follows from the semi-classical method. In that case, we found at least one bound
state if
~2 π 2
V0 >
.
(5.2.30)
2M a2
This shows that Bohr-Sommerfeld quantization does not always provide the exact answer.
Still, it gives the right answer in the limit of large quantum numbers n. For example,
semi-classical quantization predicts that in order to support n bound states the strength
of the potential must exceed
~2 π 2 n2
V0 >
,
(5.2.31)
2M a2
which is consistent with the exact result of eq.(5.2.28) obtained from the Schrödinger
equation in the limit of large n.
5.3
The Tunneling Effect
Tunneling is a genuine quantum phenomenon that allows a particle to penetrate a classically forbidden region of large potential energy. This effect plays a role in many branches
of physics. For example, the radioactive α-particle decay of an atomic nucleus can be
described as a tunneling phenomenon. Also the tunneling of electrons between separate
metal surfaces has important applications in condensed matter physics, for example in the
tunneling electron microscope.
Here we consider tunneling through a square well potential barrier without putting
the particle in a box. Then the potential is V (x) = V0 > 0 for 0 ≤ x ≤ a, and zero
otherwise. Again, we are interested in energies E = ~2 k 2 /2M < V0 , i.e. classically the
particle would not have enough energy to go over the barrier. We prepare a state in which
the particle has momentum k at x → −∞. Then the incoming probability wave will be
partially reflected back to −∞, and partly it will tunnel through the barrier and then go
to +∞. The corresponding ansatz for the wave function is
Ψ(x) = exp(ikx) + R exp(−ikx),
(5.3.1)
Ψ(x) = T exp(ikx),
(5.3.2)
for x ≤ 0, and
for x ≥ a. Here T is the tunneling amplitude and |T |2 is the tunneling rate. Inside the
barrier the wave function takes the form
Ψ(x) = A exp(κx) + B exp(−κx).
(5.3.3)
Continuity of the wave function at x = 0 implies
1 + R = A + B,
(5.3.4)
5.3. THE TUNNELING EFFECT
65
and continuity of its derivative yields
ik(1 − R) = κ(A − B),
(5.3.5)
ik
ik
(1 − R), 2B = 1 + R − (1 − R).
κ
κ
(5.3.6)
which implies
2A = 1 + R +
Similarly, at x = a we obtain
T exp(ika) = A exp(κa) + B exp(−κa),
(5.3.7)
ikT exp(ika) = Aκ exp(κa) − Bκ exp(−κa),
(5.3.8)
as well as
and thus
ik
ik
ik
T exp(ika) 1 +
= 2A exp(κa) = 1 +
+R 1−
exp(κa),
κ
κ
κ
ik
ik
ik
= 2B exp(−κa) = 1 −
+R 1+
exp(−κa).
T exp(ika) 1 −
κ
κ
κ
Eliminating R from these equations we obtain
"
# ik 2
ik 2
ik 2
ik 2
T exp(ika) 1 +
exp(−κa) − 1 −
exp(κa) = 1 +
− 1−
,
κ
κ
κ
κ
(5.3.9)
which results in
T = exp(−ika)
(k 2
−
2ikκ
.
+ 2ikκ cosh(κa)
κ2 ) sinh(κa)
(5.3.10)
The tunneling rate then takes the form
|T |2 =
4k 2 κ2
.
(k 2 + κ2 )2 sinh2 (κa) + 4k 2 κ2
(5.3.11)
In the limit of a large barrier (κa → ∞) the tunneling rate is exponentially suppressed
|T |2 =
16k 2 κ2
exp(−2κa).
(k 2 + κ2 )2
(5.3.12)
The most important term is the exponential suppression factor. The prefactor represents
a small corrections to this leading effect, and hence we can write approximately
!
r
2M
2
|T | ≈ exp(−2κa) = exp −2a
[V0 − E] .
(5.3.13)
~2
66
5.4
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
Semi-Classical Instanton Formula
In practical applications tunneling barriers in general do not have square well shapes.
Therefore, let us now turn to tunneling through an arbitrary potential barrier. Then
we will, in general, not be able to solve the Schrödinger equation exactly. Still, we can
perform a semi-classical approximation (similar to Bohr-Sommerfeld quantization) and obtain a closed expression for the tunneling rate. Of course, in contrast to Bohr-Sommerfeld
quantization, for a tunneling process no corresponding classical path exists, simply because
tunneling is forbidden at the classical level. Still, as we will see, classical mechanics at
purely imaginary “time” will provide us with a tunneling path. The concept of imaginary
or so-called Euclidean time plays an important role in the modern treatment of quantum
field theories, which simplifies the mathematics of path integrals. Also here imaginary
time is just a mathematical trick to derive an expression for the tunneling rate in a simple
manner. We should keep in mind that in real time there is no classical path that describes
the tunneling process. Let us write the tunneling amplitude as
i
T = exp
(S[x(t)] + EtT ) .
(5.4.1)
~
This expression follows from Feynman’s path integral in the same way as Bohr-Sommerfeld
quantization. In that case the action was to be evaluated for a classical path xc (t) with
period T . Here a tunneling path x(t) occurs instead, and the period T is replaced by
the tunneling time tT . Note again that in this calculation T represents the tunneling
amplitude and not a time period. How is the tunneling path determined? Let us just
consider energy conservation, and then let mathematics be our guide
M
E=
2
dx(t)
dt
2
+ V (x(t)).
(5.4.2)
This classical equation has no physically meaningful solution because now E < V (x(t)),
and hence formally the kinetic energy would be negative. Mathematically we can make
sense of
M dx(t) 2
= E − V (x(t)) < 0,
(5.4.3)
2
dt
if we replace real time t by imaginary “time” it. Then the above equation takes the form
M
−
2
dx(t)
d(it)
2
= E − V (x(t)) < 0,
(5.4.4)
which is perfectly consistent. The effect of using imaginary time is the same as changing
E − V (x(t)), which is negative, into V (x(t)) − E. Hence, the tunneling path in imaginary
time would be the classical path in an inverted potential. Next we write
# Z
Z tT " tT
M dx(t) 2
S[x(t)] + EtT =
dt
− V (x(t)) + E =
dt 2[E − V (x(t)]. (5.4.5)
2
dt
0
0
5.5. REFLECTION AT A POTENTIAL STEP
67
Here we have just used the energy conservation equation from above. Next we turn the
integral over t into an integral over x between the classical turning points x(0) and x(tT )
at which V (x) = E. Then
Z
x(tT )
1
2[E − V (x(t)].
dx(t)/dt
(5.4.6)
r
2
2
[E − V (x(t))] = i
[V (x(t)) − E],
M
M
(5.4.7)
dx
S[x(t)] + EtT =
x(0)
Using
dx(t)
=
dt
r
we thus obtain
!
Z
p
i
1 x(tT )
dx 2M [V (x) − E] .
T = exp
(S[x(t)] + EtT ) = exp −
~
~ x(0)
(5.4.8)
For the square well potential from before the classical turning points are x(0) = 0 and
x(tT ) = a. Hence, the tunneling rate then takes the form
Z
p
2a p
2 x(tT )
2
dx 2M [V (x) − E]) = exp −
2M [V0 − E] ,
(5.4.9)
|T | = exp(−
~ x(0)
~
in complete agreement with our previous approximate result eq.(5.3.13).
5.5
Reflection at a Potential Step
We have seen that quantum mechanics differs from classical mechanics because a quantum
particle can tunnel through a classically forbidden region. A similar non-classical behavior
occurs when a particle is reflected from a potential step. Let us consider a potential which
is zero for x ≤ 0, and constant and repulsive for x ≥ 0, i.e. V (x) = V0 > 0. Let us
consider energies E > V0 that are sufficient to step over the barrier classically. Then
classically a particle that is coming from −∞ would just slow down at x = 0 and continue
to move to +∞ with a reduced kinetic energy. Classically there would be no reflection
in this situation. Still, quantum mechanically there is a probability for the particle to
be reflected despite the fact that it has enough energy to continue to +∞. We make the
following ansatz for the wave function at x ≤ 0
Ψ(x) = exp(ikx) + R exp(−ikx),
(5.5.1)
Ψ(x) = T exp(iqx).
(5.5.2)
while for x ≥ 0 we write
We have not included a wave exp(−iqx) at x ≥ 0 because we are describing an experiment
in which the particle is coming in only from the left. The energy is given by
E=
~2 k 2
~2 q 2
=
+ V0 .
2M
2M
(5.5.3)
68
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
Next we impose continuity conditions for both the wave function and its first derivative
1 + R = T, ik(1 − R) = iqT ⇒ 1 − R =
q
T.
k
(5.5.4)
Solving these equations yields
T =
2k
k−q
, R=
.
k+q
k+q
(5.5.5)
The reflection rate |R|2 vanishes only when k = q. This happens either if the potential
vanishes (V0 = 0) or if the energy goes to infinity. Otherwise, there is a non-zero probability
for the particle to be reflected back to −∞.
Let us look at the above problem from the point of view of probability conservation.
For time-independent 1-dimensional problems the continuity equation reduces to
∂x j(x) = 0 ⇒ j(x) = j,
(5.5.6)
i.e. the current
~
[Ψ(x)∗ ∂x Ψ(x) − Ψ(x)∂x Ψ(x)∗ ] ,
(5.5.7)
2iM
should then be a constant j (independent of x). Let us calculate the current for the
problem from before. First, we consider x ≤ 0 and we obtain
j(x) =
j(x) =
−
=
~k
{[exp(−ikx) + R∗ exp(ikx)][exp(ikx) − R exp(−ikx)]
2M
[exp(ikx) + R exp(−ikx)][− exp(−ikx) + R∗ exp(ikx)]}
~k
(1 − |R|2 ).
M
(5.5.8)
For x ≥ 0, on the other hand, we find
j(x) =
~q 2
~q ∗
[T exp(−ikx)T exp(ikx) − T exp(ikx)T ∗ exp(−ikx)] =
|T | .
2M
M
(5.5.9)
Indeed, both currents are independent of x. In addition, they are equal (as they should)
because
"
#
k−q 2
4k 2 q
2
k(1 − |R| ) = k 1 −
=
= q|T |2 .
(5.5.10)
k+q
(k + q)2
5.6
Quantum Height Anxiety Paradox
We have seen that quantum mechanically particles behave differently than in classical
physics. For example, a classical billiard ball will never tunnel through a region of potential
energy higher than its total energy, and it will never get reflected from a region of potential
energy lower than its total energy. Let us go back to the problem with the potential step
from before. However, now we want the particle to step down in energy, in other words,
5.6. QUANTUM HEIGHT ANXIETY PARADOX
69
now V0 < 0. Certainly, a classical billiard ball incident from −∞ would enter the region
of decreased potential energy, thereby increasing its kinetic energy, and would then move
to +∞. In particular, it would not be reflected back to −∞. Let us see what happens
quantum mechanically. In the previous calculation with the potential step we have never
used the fact that V0 > 0. Therefore our result is equally valid for V0 < 0. Hence, again
T =
2k
k−q
, R=
.
k+q
k+q
(5.6.1)
Let us take the limit of a very deep potential step (V0 → −∞). Then q → ∞ and thus
T → 0, R → −1,
(5.6.2)
such that the transmission rate is zero and the reflection rate is |R|2 = 1. This is the
exact opposite of what happens classically. The quantum particle stops at the top of the
potential step and turns back, instead of entering the region of very low potential energy,
as a classical particle would certainly do. One may want to call this the “quantum height
anxiety paradox” because the quantum particle steps back into the region of high potential
energy, while a classical billiard ball would happily jump down the cliff. It is obvious that
something is wrong in this discussion. Quantum mechanics may give different results than
classical physics, but the results should not differ as much as in this paradox. In particular,
in the classical limit ~ → 0 quantum mechanics should give the same result as classical
mechanics. This is certainly not the case in this paradox. In fact, our result for R does
not even depend on ~. Hence, taking the classical limit does not change the answer: the
particle would still turn around instead of going forward.
The above paradox gets resolved when we consider a potential that interpolates more
smoothly between the regions of large and small potential energies. As we will see, the
abrupt step function change in the potential is responsible for the paradox, and we should
hence be careful in general when we use square well potentials. Still, they are the simplest
potentials for which the Schrödinger equation can be solved in closed form, and they
therefore are quite useful for learning quantum mechanics. To understand why the step
function is special, let us smear out the step over some length scale a by writing the
potential as
V (x) = V0 f (x/a),
(5.6.3)
where f is a smooth function that interpolates between f (−∞) = 0 and f (+∞) = −1,
which in the limit a → 0 approaches the step function. The Schrödinger equation with
the smooth potential takes the form
−
~2 d2 Ψ(x)
+ V0 f (x/a)Ψ(x) = EΨ(x).
2M dx2
Dividing the equation by V0 and introducing the dimensionless variable
p
y = 2M V0 /~2 x,
one obtains
−
p
d2 Ψ(y)
E
+
f
(y/
2M a2 V0 /~2 )Ψ(y) =
Ψ(y).
2
dy
V0
(5.6.4)
(5.6.5)
(5.6.6)
70
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
This rescaled form of the Schrödinger equation shows that the reflection rate |R|2 only
depends on E/V0 and on the combination 2M a2 V0 /~2 . In particular, now the limit V0 →
∞ is equivalent to the classical limit ~ → 0 as well as to the limit a → ∞ of a very
wide potential step. This is exactly what one would expect. The paradox arises only
in the case of a true step function which has a = 0. Only then ~ disappears from the
rescaled equations, and we do not get a meaningful result in the classical limit. In fact,
an arbitrarily small amount of smearing of the potential step is sufficient to eliminate the
paradox.
5.7
Free Particle on a Circle
Let us apply the time-independent Schrödinger equation to a simple problem — a particle
of mass M moving on a circle of radius R. The position of the particle can then be
characterized by an angle ϕ and we can identify x = Rϕ. The Hamilton operator hence
takes the form
~2 2
~2
~2 2
2
H=−
∂x = −
∂
=
−
∂ .
(5.7.1)
ϕ
2M
2M R2
2I ϕ
Here we have introduced I = M R2 — the moment of inertia of the quantum mechanical
rotor represented by the particle. We can now write the time-independent Schrödinger
equation as
~2
(5.7.2)
− ∂ϕ2 Ψ(ϕ) = EΨ(ϕ).
2I
Since the particle is moving on a circle, the angles ϕ = 0 and ϕ = 2π correspond to the
same point. Hence, we must impose periodic boundary conditions for the wave function,
i.e.
Ψ(ϕ + 2π) = Ψ(ϕ).
(5.7.3)
Now we make the ansatz
Ψ(ϕ) = A exp(imϕ).
(5.7.4)
Periodicity of the wave function implies that m ∈ Z. Inserting the ansatz in the timeindependent Schrödinger equation yields
E=
~2 2
m ,
2I
(5.7.5)
i.e. only a discrete set of energies is allowed for stationary states. Also it is interesting
that the two states with quantum numbers m and −m have the same energy — they
are degenerate with each other. What is the probability density to find the particle at a
particular angle ϕ when it is in the stationary state with quantum number m? We simply
square the wave function and find
|Ψ(ϕ)|2 = |A exp(imϕ)|2 = |A|2 .
(5.7.6)
5.8. FREE PARTICLE ON A DISCRETIZED CIRCLE
71
This is independent of the angle ϕ and hence independent of the position on the circle.
This result is plausible because no point on the circle is singled out, the particle appears
everywhere with the same probability. Properly normalizing the wave function, we obtain
Z 2π
1
dϕ |Ψ(ϕ)|2 = 2π|A|2 = 1 ⇒ A = √ .
(5.7.7)
2π
0
The quantum number m is related to the angular momentum of the particle. The
operator for the linear momentum is px = −i~∂x . Using x = Rϕ we identify
pϕ = −
i~
∂ϕ .
R
(5.7.8)
For circular motion the angular momentum is given by L = Rpϕ . Hence, the quantum
mechanical angular momentum operator is identified as
L = −i~∂ϕ .
(5.7.9)
When we operate with L on the wave function with quantum number m we find
LΨ(ϕ) = −i~∂ϕ A exp(imϕ) = m~Ψ(ϕ).
(5.7.10)
We see that Ψ(ϕ) is an eigenvector of the angular momentum operator with eigenvalue
m~, i.e. in quantum mechanics also angular momentum is quantized.
5.8
Free Particle on a Discretized Circle
In order to understand better why the time-independent Schrödinger equation corresponds
to an eigenvalue problem, let us now discuss the dynamics of a free particle on a discretized
circle. We allow the particle to sit only at the discrete angles
ϕn = nδϕ =
2πn
2π
, δϕ =
,
N
N
(5.8.1)
where N is some positive integer. In the limit N → ∞ we recover the case of the continuous
circle that we discussed before. With only discrete points available, the particle is now
hopping from one point to a neighboring point. The wave function then also has values
only at the discrete angles ϕn , i.e. Ψ(ϕ) → Ψ(ϕn ). The Hamilton operator that contains a
second derivative with respect to the angle ϕ cannot be applied to such a discretized wave
function. Consequently, the second derivative — and hence the Hamiltonian — must be
discretized as well. A discretized second derivative takes the form
∆ϕ Ψ(ϕn ) =
Ψ(ϕn+1 ) − 2Ψ(ϕn ) + Ψ(ϕn−1 )
.
δϕ2
(5.8.2)
Thus we can write the discretized time-independent Schrödinger equation as
−
~2 Ψ(ϕn+1 ) − 2Ψ(ϕn ) + Ψ(ϕn−1 )
~2
∆ϕ Ψ(ϕn ) = −
= EΨ(ϕn ).
2I
2I
δϕ2
(5.8.3)
72
CHAPTER 5. SQUARE-WELL POTENTIALS AND TUNNELING EFFECT
This equation can be expressed in matrix form as


2 −1 0 ... −1
Ψ(ϕ1 )
 −1 2 −1 ... 0   Ψ(ϕ2 )


 0 −1 2 ... 0   Ψ(ϕ3 )

~2 
 .

.
.
. 
.


2
2Iδϕ 


.
.
. 
.
 .
 .
.
.
. 
.
−1 0
0 ... 2
Ψ(ϕN )










=E








Ψ(ϕ1 )
Ψ(ϕ2 )
Ψ(ϕ3 )
.
.
.
Ψ(ϕN )





.




(5.8.4)
The entries −1 in the upper right and lower left corners of the matrix are due to periodic
boundary conditions on the circle. We see that the Hamilton operator is represented by
an N × N matrix, while the wave function Ψ(ϕ) has turned into an N -component vector.
In the continuum limit N → ∞ the dimension of the corresponding vector space goes to
infinity and the vector space turns into a so-called Hilbert space.
The above matrix equation only has trivial solutions Ψ(ϕn ) = 0 for generic values of
E. This can be understood as follows. Let us write
HΨ = EΨ,
(5.8.5)
as










H11 − E
H12
H13
H21
H22 − E
H23
H31
H32
H33 − E
.
.
.
.
.
.
.
.
.
HN 1
HN 2
HN 3
...
...
...
H1N
H2N
H3N
.
.
.
... HN N − E










Ψ(ϕ1 )
Ψ(ϕ2 )
Ψ(ϕ3 )
.
.
.
Ψ(ϕN )





 = 0.




(5.8.6)
This equation implies that the columns of the matrix (H − E 1) are linearly dependent.
This is possible only if the determinant of the matrix is zero, i.e.
det(H − E 1) =
N
X
hn E n = 0.
(5.8.7)
n=0
Here the coefficients hn depend on the matrix elements of H and define the characteristic
polynomial of H. For generic values of E the characteristic polynomial — and hence
the determinant of (H − E 1) — does not vanish and hence the equation HΨ = EΨ
does not have a non-trivial solution. Only for special discrete values of E the above
determinant vanishes and non-trivial solutions exist. These eigenvalues are the zeros of
the characteristic polynomial of H. The corresponding non-trivial solutions of the above
matrix problem are the eigenvectors.
Chapter 6
Spin, Precession, and the
Stern-Gerlach Experiment
Until now we have discussed how quantum physics manifests itself for point particles that
can be described classically by their position and momentum. In this chapter, we will
discuss quantum systems that have no classical counterpart, because they involve a fundamental discrete degree of freedom, known as spin, which is of intrinsically quantum
and thus of non-classical nature. Spin is an intrinsic quantum mechanical angular momentum of elementary particles, which gives rise to a magnetic moment that can interact
with an external magnetic field. This results in spin precession and manifests itself in the
Stern-Gerlach experiment.
6.1
Quantum Spin
Elementary particles fall into two categories: bosons and fermions. As we already learned
in chapter 2, bosons and fermions obey different forms of quantum statistics: an arbitrary
number of bosons but at most a single fermion can occupy the same quantum state. Besides
in their statistics, bosons and fermions also differ in their spin, which is a discrete quantum
degree of freedom, that plays the role of an intrinsic angular momentum. While bosons
have integer spin 0, 1, 2, . . . (in units of ~), fermions have half-integer spin 12 , 32 , 52 , . . . .
In particular, all known truly elementary fermions including electrons, neutrinos, and
quarks have spin 12 . Even composite objects like protons and neutrons, which consist
of three quarks, have spin 12 , while some extremely short-lived elementary particles —
known as ∆-isobars — which also consist of three quarks, have spin 23 . All currently
known truly fundamental bosons, namely photons, W- and Z-bosons, and gluons have
spin 1. The Higgs particle, which plays a central role in the standard model of particle
physics, and has recently been detected at the LHC at CERN, has spin 0, but may not be
truly fundamental. Finally, gravitons, the hypothetical quanta of the gravitational field,
should have spin 2, but interact so weakly that there is little hope to detect them in the
73
74CHAPTER 6. SPIN, PRECESSION, AND THE STERN-GERLACH EXPERIMENT
foreseeable future. In this section, we limit ourselves to the simplest non-trivial spin
which is relevant for elementary fermions.
1
2
As we have learned already, we should not think of elementary particles as tiny billiard
balls. This implies that one should not think of elementary particles with spin as tiny
spinning tops. Such classical pictures do not help in the quantum world and often are
misleading. A correct way of thinking about spin is to view it as a discrete quantum
degree of freedom, which has no classical analog. While at the classical level an electron
is described just by its position and momentum, at the quantum level a new degree of
freedom — the electron’s spin — reveals itself. An electron can exist in two states: spin
up (↑) and spin down (↓). However, as a quantum object, an electron can also exist in a
state that is a superposition of spin up and spin down.
In this chapter, we concentrate entirely on the discrete spin degree of freedom of
an elementary particle, and we decouple the classical degrees of freedom, such as the
continuous particle coordinate ~x. Discussing spin and position independently is completely
justified in non-relativistic quantum mechanics, because the interactions between orbital
motion and spin — the so-called spin-orbit couplings — are of relativistic nature. Indeed
the nature of the spin as an intrinsic angular momentum of elementary particles manifests
itself only through relativistic effects. For this reason, one sometimes reads that the spin
itself is of relativistic origin. This is somewhat misleading because, first and foremost,
spin is a most important quantum degree of freedom that has drastic implications even
in non-relativistic physics. However, it is true that relativity theory sheds an entirely
new light on spin, by revealing its nature as an intrinsic angular momentum. We will
address these issues when we discuss the Pauli equation (which is an approximation to
the fully relativistic Dirac equation). In the context of relativistic quantum field theories,
which goes far beyond the scope of this text, one can derive the spin-statistics theorem
which shows that particles with Bose statistics necessarily have integer spin, while Fermi
statistics is associated with half-integer spin. Here we treat spin from a non-relativistic
point of view. Then it just describes a new quantum degree of freedom, not obviously
related to angular momentum.
It is best to speak about spin in the appropriate language, which clearly is mathematics.
Hence, let us now define mathematically what spin is. Until now, ignoring spin, we
have described the quantum state of an electron by a wave function Ψ(~x), which is a
probability amplitude depending on the continuous classical position degree of freedom ~x
of the electron. Since electrons are fundamental fermions, they have an additional discrete
quantum degree of freedom — the spin s ∈ {↑, ↓}. Hence, the quantum state of an electron
is actually described by a 2-component wave function known as a Pauli spinor
Ψ(~x) =
Ψ↑ (~x)
Ψ↓ (~x)
.
(6.1.1)
The component Ψs (~x) is the probability amplitude for finding the electron at the position
~x with the spin s. Correspondingly, the correct normalization condition for a Pauli spinor
6.1. QUANTUM SPIN
75
is given by
Z
3
Z
†
d x Ψ(~x) Ψ(~x) =
d3 x |Ψ↑ (~x)|2 + |Ψ↓ (~x)|2 = 1.
(6.1.2)
Here we have introduced the complex conjugated and transposed spinor
Ψ(~x)† = (Ψ↑ (~x)∗ , Ψ↓ (~x)∗ ) .
(6.1.3)
For example, the probability to find the electron anywhere in space with spin up is given
by
Z
P↑ = d3 x|Ψ↑ (~x)|2 ,
(6.1.4)
while the probability to find it in a finite region Ω with spin down is
Z
P↓ (Ω) =
d3 x|Ψ↓ (~x)|2 .
(6.1.5)
Ω
On the one hand, spin is an additional label s =↑, ↓ on the wave function of an electron.
On the other hand, there is also a spin operator
~ = (Sx , Sy , Sz ) = ~ ~σ = ~ (σx , σy , σz ),
S
2
2
(6.1.6)
representing a vector of physical observables, whose components are the 2×2 Pauli matrices
1 0
0 −i
0 1
.
(6.1.7)
, σz =
, σy =
σx =
0 −1
i 0
1 0
The Pauli matrices are invariant under transposition combined with complex conjugation.
The spin operator acts on a Pauli spinor by matrix multiplication. The expectation value
of the spin operator in the state Ψ(~x) is given by
Z
~
~ x).
hSi = d3 x Ψ(~x)† SΨ(~
(6.1.8)
For example, the third component of the spin operator acts on a Pauli spinor as
~ 1 0
~
Ψ↑ (~x)
Ψ↑ (~x)
Sz Ψ(~x) =
=
,
Ψ↓ (~x)
2 0 −1
2 −Ψ↓ (~x)
such that
~
hSz i =
2
Z
d3 x |Ψ↑ (~x)|2 − |Ψ↓ (~x)|2 .
(6.1.9)
(6.1.10)
Eigenstates of the operator Sz must obey Sz χ(~x) = sz χ(~x), where sz is the corresponding
eigenvalue. This implies
~
χ↑ (~x)
χ↑ (~x)
Sz χ(~x) =
= sz
.
(6.1.11)
χ↓ (~x)
2 −χ↓ (~x)
76CHAPTER 6. SPIN, PRECESSION, AND THE STERN-GERLACH EXPERIMENT
The two possible eigenvalues are sz = ±~/2 and the corresponding eigenstates are
χ↑ (~x)
1
0
0
χ+ (~x) =
= χ↑ (~x)
, χ− (~x) =
= χ↓ (~x)
. (6.1.12)
0
0
χ↓ (~x)
1
Irrespective of the form of the spatial wave functions χ↑ (~x) and χ↓ (~x), the Pauli spinors
χ+ (~x) and χ− (~x) are always eigenstates of Sz . The eigenstates of Sz are not simultaneously
eigenstates of Sx or Sy , in particular,
~ 0 1
~ 0
1
1
=
Sx
=
,
0
0
2 1 0
2 1
~ 0 1
~ 1
0
0
=
Sx
=
,
1
1
2 1 0
2 0
~ 0 −i
~ 0
1
1
Sy
=
=
,
0
i 0
0
i
2
2
~ 0 −i
~ −i
0
0
Sy
=
=
.
(6.1.13)
1
i 0
1
0
2
2
This is due to the fact that the different components of the spin operator do not commute
with each other. We define the so-called commutator of two quantum mechanical operators
A and B as
[A, B] = AB − BA.
(6.1.14)
By straightforward matrix multiplication one then obtains
~2
0
0 −i
0 −i
0 1
−
[Sx , Sy ] =
1
i 0
i 0
1 0
4
2
~
1 0
= i
= i~Sz ,
0 −1
2
~2
1 0
1 0
0 −i
[Sy , Sz ] =
−
0 −1
0 −1
i 0
4
2
~
0 1
= i
= i~Sx ,
1 0
2
~2
1 0
0 1
0 1
1
[Sz , Sx ] =
−
0 −1
1 0
1 0
0
4
2
~
0 −i
= i
= i~Sy .
i 0
2
1
0
0 −i
i 0
0
−1
(6.1.15)
Using the totally anti-symmetric Levi-Civita tensor, these relations can be summarized as
[Si , Sj ] = i~ijk Sk .
(6.1.16)
As we will see later, these are the same commutation relations as the ones of the quantum
mechanical orbital angular momentum operator. This is consistent with the fact that spin
will turn out to be an intrinsic (non-orbital) angular momentum of elementary particles.
This aspect of spin is, however, only revealed through relativistic effects.
6.2. NEUTRON SPIN PRECESSION IN A MAGNETIC FIELD
6.2
77
Neutron Spin Precession in a Magnetic Field
In the same way as the momentum of a particle can be affected by external forces, the spin
of a particle can be affected by an external magnetic field. Of course, via the Lorentz force,
magnetic fields also affect the momentum of a charged particle. To simplify the situation,
here we limit ourselves to neutral particles with spin. The neutron, which together with
protons forms atomic nuclei, indeed is a neutral particle with spin 12 . The Hamilton
~
operator that describes a non-relativistic neutron in a constant external magnetic field B
is given by
p~ 2
~ · S.
~
H=
− µB
(6.2.1)
2M
~ is the neutron’s spin vector. The
Here µ is the magnetic moment of the neutron and S
magnetic moment of the neutron is given by
µ = gn
e~
,
2M
(6.2.2)
where M is the neutron mass, e is the elementary unit of electric charge, and gn = −1.913
is the so-called anomalous g-factor of the neutron. For truly elementary neutral particles,
the relativistic Dirac theory predicts g = 0, while for charged particles it predicts g =
2. The electron — which is a truly elementary fermion as far as we know today —
indeed has ge = 2.002319304362. The tiny deviation from 2 arises mostly from Quantum
Electrodynamics (QED), but also from other parts of the standard model of particle
physics. The anomalous magnetic moment of the electron is the most accurately measured
quantity in all of physics, with 13 digits accuracy. On the other hand, it is also very
accurately described by the theory. The muon, an about 200 times heavier cousin of the
electron, is another elementary fermion which has gµ = 2.002331842. Again, this number is
theoretically very well understood. By working out the theory to even higher accuracy, one
hopes to eventually find some hints to potential new physics beyond the standard model.
In contrast to the electron and the muon, the proton has gp = 2.793. The deviations of
gp from 2 and of gn from 0 indicate that proton and neutron are not truly elementary
objects. Indeed, these particles are known to consist of quarks and gluons, which in turn
are truly elementary as far as we can tell today.
Let us return to the neutron in a constant magnetic field, which we assume to point
~ = (0, 0, B), such that
in the z-direction, i.e. B
H = T + HS = −
~2
∆ − µBSz .
2M
(6.2.3)
The Hamilton operator of the neutron in a magnetic field is a sum of the kinetic energy
T and the magnetic spin energy HS . In particular, due to our non-relativistic treatment,
there is no coupling of the spin to the orbital motion. The wave function of an energy
eigenstate then factorizes into an orbital and a spin contribution
i
i
1
0
χ+ (~x) = exp
p~ · ~x
, χ− (~x) = exp
p~ · ~x
,
(6.2.4)
0
1
~
~
78CHAPTER 6. SPIN, PRECESSION, AND THE STERN-GERLACH EXPERIMENT
with the corresponding energies
E± =
p~ 2
µB~
∓
.
2M
2
(6.2.5)
Let us now ignore the kinetic energy and concentrate entirely on the spin Hamiltonian
~ ·S
~ = −µBSz .
HS = −µB
(6.2.6)
Next, we solve the time-dependent Schrödinger equation
i~∂t Ψ(t) = HS Ψ(t),
(6.2.7)
for a spin that is initially described by the quantum state
Ψ(0) =
Ψ↑ (0)
Ψ↓ (0)
.
(6.2.8)
The solution of the time-dependent Schrödinger equation is then given by
Ψ(t) =
Ψ↑ (0) exp
Ψ↓ (0) exp
− ~i E+ t
− ~i E− t

Ψ↑ (0) exp i µB
t
2 .
=
Ψ↓ (0) exp −i µB
2 t

(6.2.9)
We can now determine the expectation value of the spin operator as a function of time
~
hSi(t)
~
~
S
= Ψ(t)† Ψ(t)
~


µB
t
Ψ
(0)
exp
i
↑
µB
µB
~σ 
2 
=
Ψ↑ (0)∗ exp −i
t , Ψ↓ (0)∗ exp i
t
2
2
2
Ψ↓ (0) exp −i µB
2 t
= (Ψ↑ (0)∗ Ψ↓ (0) exp(−iωt) + Ψ↓ (0)∗ Ψ↑ (0) exp(iωt),
|Ψ↑ (0)|2 − |Ψ↓ (0)|2
)
2
|Ψ↑ (0)|2 − |Ψ↓ (0)|2
).
= (<[Ψ↑ (0)∗ Ψ↓ (0) exp(−iωt)], =[Ψ↑ (0)∗ Ψ↓ (0) exp(−iωt)],
2
(6.2.10)
−iΨ↑ (0)∗ Ψ↓ (0) exp(−iωt) + iΨ↓ (0)∗ Ψ↑ (0) exp(iωt),
The spin expectation value thus rotates around the magnetic field direction, i.e. the spin
precesses in the magnetic field, with the Larmor frequency
ω = µB = gn
eB
.
2M
(6.2.11)
Larmor precession is important, for example, in nuclear magnetic resonance (NMR) applications.
6.3. THE STERN-GERLACH EXPERIMENT
6.3
79
The Stern-Gerlach Experiment
In 1922 Otto Stern and Walther Gerlach performed an experiment that reveals the spin
degree of freedom. They sent a beam of silver atoms through an inhomogeneous magnetic
field and observed a splitting of the beam into two components corresponding to spin up
and spin down. Silver atoms contain many electrons. Most of them combine their orbital
angular momenta and spins to a total angular momentum zero, but the spin of one electron
remains uncanceled and produces the effect. It was not immediately clear in 1922 that
Stern and Gerlach had observed an effect of the electron’s spin. In 1924 Wolfgang Pauli
introduced a new discrete quantum degree of freedom in order to explain atomic spectra,
which was identified as the electron’s spin by George Uhlenbeck and Samuel Goudsmit in
1925.
Due to the inhomogeneity of the magnetic field, it is not so easy to compute the
results of the Stern-Gerlach experiment. To illustrate the main effect, we will thus make
an oversimplified model. Actually, we will replace space by just two discrete points x1 and
x2 with two different magnitudes of the magnetic field B1 and B2 in order to mimic the
inhomogeneity. Let us assume that a spin 12 particle can hop between the two points with
a hopping amplitude t, without changing its spin. The wave function and the Hamilton
operator then take the form




0
t
0
− µB21 ~
Ψ1↑


µB1 ~
 Ψ1↓ 
0
0
t 
2
, H = 
Ψ=
(6.3.1)
.

µB
~
 Ψ2↑ 

t
0
− 22
0 
µB2 ~
Ψ2↓
0
t
0
2
The problem decouples into two separate problems, one for spin up and one for spin down
(Sz = ± ~2 )
−µB1 Sz
t
Ψ1Sz
, HSz =
.
(6.3.2)
ΨSz =
t
−µB2 Sz
Ψ2Sz
The time-independent Schrödinger equation then yields
HSz ΨSz = ESz ΨSz ⇒ ESz ,± = −µ
B1 + B2
1p 2
Sz ±
µ (B1 − B2 )2 Sz2 + 4t2 .
2
2
(6.3.3)
In the ground state with energy ESz ,− the probabilities to find the particle in position x1
or x2 have the ratio
2
P2Sz
1
B1 − B2
1p 2
2
2
2
= 2 µ
Sz ±
µ (B1 − B2 ) Sz + 4t
.
(6.3.4)
P1Sz
t
2
2
For a homogeneous magnetic field (B1 = B2 ) one obtains P1Sz = P2Sz , such that spin
up and spin down are not spatially separated. For an inhomogeneous field, on the other
hand, since P2Sz /P1Sz depends on Sz , we see that up and down spin particles appear at x1
and x2 with different probabilities. This is exactly what is observed in the Stern-Gerlach
experiment.
80CHAPTER 6. SPIN, PRECESSION, AND THE STERN-GERLACH EXPERIMENT
Chapter 7
The Formal Structure of Quantum
Mechanics
In this chapter we investigate the mathematical structure of quantum mechanics. In particular, we will understand that wave functions can be viewed as abstract vectors in an
infinite-dimensional vector space — the so-called Hilbert space. Physical observables are
represented by Hermitean operators acting in the Hilbert space. The normalized eigenvectors of such an operator form a complete set of orthogonal basis states in the Hilbert space.
Quantum mechanical measurements of a physical observable return one of the eigenvalues
of the corresponding Hermitean operator as a result of the measurement. Immediately after the measurement, the physical system will find itself in the state corresponding to the
eigenvector associated with the measured eigenvalue. Two different physical observables
are simultaneously measurable with absolute precision only if the corresponding operators
commute with each other.
7.1
Wave Functions as Vectors in a Hilbert Space
As we have seen in the example of a free particle on a discretized circle with N points, the
quantum mechanical wave function resembles an N -component vector, and the Hamilton
operator takes the form of an N ×N matrix. In the continuum limit N → ∞ the dimension
of the corresponding vector space diverges, and we end up in a so-called Hilbert space.
Let us consider the space of all normalizable wave functions Ψ(~x). We can consider them
as vectors in the Hilbert space, i.e. we can add them up and multiply them with complex
factors such that, for example, λΨ(~x) + µΦ(~x) is also an element of the Hilbert space. The
scalar product of ordinary 3-component vectors ha|bi = ~a · ~b = a1 b1 + a2 b2 + a3 b3 has an
analog in the Hilbert space. The scalar product of two wave functions is defined as
Z
hΦ|Ψi =
d3 x Φ(~x)∗ Ψ(~x).
81
(7.1.1)
82
CHAPTER 7. THE FORMAL STRUCTURE OF QUANTUM MECHANICS
For ordinary 3-component vectors the scalar product induces the norm ||a||2 = ha|ai =
~a · ~a = a21 + a22 + a23 while for wave functions
Z
Z
||Ψ||2 = hΨ|Ψi = d3 x Ψ(~x)∗ Ψ(~x) = d3 x |Ψ(~x)|2 .
(7.1.2)
The physical interpretation of the wave function as a probability amplitude implied the
normalization condition ||Ψ||2 = 1, i.e. physical wave functions are unit-vectors in the
Hilbert space. Two ordinary vectors are orthogonal when ha|bi = ~a · ~b = 0. Similarly, two
wave functions are orthogonal when
Z
hΦ|Ψi = d3 x Φ(~x)∗ Ψ(~x) = 0.
(7.1.3)
7.2
Observables as Hermitean Operators
In quantum mechanics physical observable quantities (so-called observables) are represented by operators. For example, we have seen that the momentum is represented by the
derivative operator (times −i~), while the energy is represented by the Hamilton operator.
Also the angular momentum and other physical observables are represented by operators.
As we have seen, we can think of these operators as matrices with infinitely many rows
and columns acting on the infinite dimensional vectors in the Hilbert space. A general
operator A that acts on a wave function Ψ(~x) generates a new vector AΨ(~x) in the Hilbert
space. We can, for example, evaluate the scalar product of this vector with any other wave
function Φ(~x) and obtain
Z
hΦ|AΨi = d3 x Φ(~x)∗ AΨ(~x).
(7.2.1)
For example, the expectation value of an operator A is given by
Z
hAi = hΨ|AΨi = d3 x Ψ(~x)∗ AΨ(~x).
(7.2.2)
We also have seen that eigenvalue problems play a central role in quantum mechanics.
For example, the time-independent Schrödinger equation is the eigenvalue problem of the
Hamilton operator. As we will soon learn, the eigenvalues of an operator are the only
possible results of measurements of the corresponding physical observable. Since physical
observables are real numbers, they are represented by operators with real eigenvalues.
These operators are called Hermitean. To formally define Hermiticity we first need to
define the Hermitean conjugate (or adjoint) A† of a general (not necessarily Hermitean)
operator A. The operator A† is defined such that
hA† Φ|Ψi = hΦ|AΨi,
for all wave functions Φ(~x) and Ψ(~x). This means
Z
Z
3
†
∗
d x (A Φ(~x)) Ψ(~x) = d3 x Φ(~x)∗ AΨ(~x).
(7.2.3)
(7.2.4)
7.3. EIGENVALUES AND EIGENVECTORS
83
An operator is called Hermitean if
A† = A,
which means
Z
d3 x (AΦ(~x))∗ Ψ(~x) =
Z
(7.2.5)
d3 x Φ(~x)∗ AΨ(~x),
(7.2.6)
for all Φ(~x) and Ψ(~x). In fact, physical observables are represented by operators that are
not only Hermitean but even self-adjoint. The subtle differences between Hermiticity and
self-adjointness are rarely emphasized in the textbook literature. We will return to this
issue later.
For the momentum operator we have
Z
Z
3
∗
~
~ x)∗ Ψ(~x)
d x (−i~∇Φ(~x)) Ψ(~x) =
d3 x i~∇Φ(~
Z
~
=
d3 x Φ(~x)∗ (−i~∇)Ψ(~
x),
(7.2.7)
which shows that it is indeed Hermitean. Here we have used partial integration. There are
no boundary terms because a normalizable wave function necessarily vanishes at infinity.
It is obvious that a Hermitean operator always has real expectation values because
Z
†
hAi = hΨ|AΨi = hA Ψ|Ψi = hAΨ|Ψi = d3 x (AΨ(~x))∗ Ψ(~x) = hAi∗ .
(7.2.8)
7.3
Eigenvalues and Eigenvectors
Let us derive some facts about the eigenvalues and eigenvectors of Hermitean operators.
We consider the eigenvalue problem of a Hermitean operator A
Aχn (~x) = an χn (~x).
(7.3.1)
Here the an are the eigenvalues and the χn (~x) are the corresponding eigenvectors (sometimes also called eigenfunctions). For example, A could be the Hamilton operator. Then
the an would be the allowed energy values and χn (~x) would be the corresponding timeindependent part of the wave function. Let us consider the scalar product
Z
Z
hχm |Aχn i = d3 x χm (~x)∗ Aχn (~x) = an d3 x χm (~x)∗ χn (~x) = an hχm |χn i.
(7.3.2)
On the other hand, because A is Hermitean, we also have
Z
hχm |Aχn i = hAχm |χn i = d3 x (Aχm (~x))∗ χn (~x) = a∗m hχm |χn i.
(7.3.3)
Let us first consider the case m = n. Then comparing the above equations implies an = a∗n ,
i.e. the eigenvalues of a Hermitean operator are always real. Now consider the case m 6= n
and use a∗m = am . Then
(an − am )hχm |χn i = 0.
(7.3.4)
84
CHAPTER 7. THE FORMAL STRUCTURE OF QUANTUM MECHANICS
Assuming that an 6= am (non-degenerate eigenvalues) this implies
hχm |χn i = 0,
(7.3.5)
i.e. the eigenvectors corresponding to two non-degenerate eigenvalues are orthogonal. For
degenerate eigenvalues (i.e. for an = am ) the corresponding eigenvectors are not always
automatically orthogonal, but one can always find a set of orthogonal eigenvectors. Normalizing the eigenvectors one hence obtains the orthonormality relation
hχm |χn i = δmn .
7.4
(7.3.6)
Completeness
The eigenvectors of a Hermitean operator form an orthogonal basis of the Hilbert space.
This basis is complete, i.e. every wave function in the Hilbert space can be written as a
linear combination of eigenvectors
X
Ψ(~x) =
cn χn (~x).
(7.4.1)
n
The expansion coefficients are given by
cn = hχn |Ψi.
(7.4.2)
An equivalent way to express the completeness of the eigenvectors is
X
χn (~x)∗ χn (~x0 ) = δ(~x − ~x0 ),
(7.4.3)
n
where
δ(~x − ~x0 ) = δ(x1 − x01 )δ(x2 − x02 )δ(x3 − x03 )
(7.4.4)
is the 3-dimensional Dirac δ-function. Indeed one then finds
X
X
XZ
d3 x0 χn (~x0 )∗ Ψ(~x0 )χn (~x)
cn χn (~x) =
hχn |Ψiχn (~x) =
n
Zn
=
n
d3 x0 Ψ(~x0 )δ(~x − ~x0 ) = Ψ(~x).
(7.4.5)
The normalization condition for the wave function implies
!∗
Z
Z
X
X
d3 x |Ψ(~x)|2 =
d3 x
cm χm (~x)
cn χn (~x)
m
Z
=
d3 x
X
c∗m χm (~x)∗
m
=
X
m,n
n
c∗m cn hχm |χn i =
X
cn χn (~x)
n
X
m,n
c∗m cn δmn =
X
n
|cn |2 = 1.
(7.4.6)
7.5. IDEAL MEASUREMENTS
85
Let us now consider the expectation value of a Hermitean operator A in the state described
by the above wave function Ψ(~x)
!∗
Z
Z
X
X
3
∗
3
cn χn (~x)
hAi =
d x Ψ(~x) AΨ(~x) = d x
cm χm (~x) A
n
m
Z
=
d3 x
X
c∗m χm (~x)∗
X
m
=
X
c∗m cn an hχm |χn i =
m,n
Z
cn Aχn (~x) =
d3 x
X
c∗m χm (~x)∗
m
n
X
c∗m cn an δmn =
X
|cn |2 an .
X
cn an χn (~x)
n
(7.4.7)
n
m,n
The expectation value of A gets contributions from each eigenvalue with the weight |cn |2
which determines the contribution of the eigenvector χn (~x) to the wave function Ψ(~x).
7.5
Ideal Measurements
An important part of the formal structure of quantum mechanics concerns the interpretations of the various objects in the theory, as well as their relation with real physical
phenomena. Quantum mechanics makes predictions about the wave function, which contains information about probabilities. Such a prediction can be tested only by performing
many measurements of identically prepared systems, and averaging over the results. The
result of a specific single measurement is not predicted by the theory, only the probability
to find a certain result is. Still, of course every single measurement gives a definite answer. Although the outcome of a measurement is uncertain until it is actually performed,
after the measurement we know exactly what has happened. In other words, by doing the
measurement we generally increase our knowledge about at least certain aspects of the
quantum system. This has drastic consequences for the state of the system. In classical
physics the influence of a measurement device on an observed phenomenon is assumed to
be arbitrarily small, at least in principle. In quantum physics, on the other hand, this is
not possible. A measurement always influences the state of the physical system after the
measurement. In a real measurement the interaction of the quantum system and the measuring device may be very complicated. Therefore we want to discuss ideal measurements,
whose outcome is directly related to the objects in the theory that we have discussed
before. Let us consider the following relations from before. First, we have an expansion
of the physical state
X
Ψ(~x) =
cn χn (~x),
(7.5.1)
n
in terms of the eigenvectors χn (~x) of an operator A describing an observable that we want
to measure. The expansion coefficients are given by
cn = hχn |Ψi,
(7.5.2)
X
(7.5.3)
and they are normalized by
n
|cn |2 = 1.
86
CHAPTER 7. THE FORMAL STRUCTURE OF QUANTUM MECHANICS
The expectation value of our observable — i.e. the average over the results of a large
number of measurements — is given by
X
|cn |2 an ,
(7.5.4)
hAi =
n
where the an are the eigenvalues of the operator A. These equations suggest the following
interpretation. The possible outcomes of a measurement of the observable A are just its
eigenvalues an . The probability to find a particular an is given by |cn |2 , which determines
the contribution of the corresponding eigenvector χn (~x) to the wave function Ψ(~x) that
describes the state of the physical system. Note that the interpretation of the |cn |2 as
probabilities for the results of measurements is consistent because they are correctly normalized and indeed summing over all n with the appropriate probability gives the correct
expectation value hAi.
It is important that the theory can be tested only by performing many measurements
of identically prepared systems. This implies that before each measurement we have to
restart the system from the same initial conditions. We cannot simply perform a second
measurement of the same system that we have just measured before, because the first
measurement will have influenced the state of the system. In general the state of the
system after the first measurement is not described by Ψ(~x) any longer. Then what is the
state after the measurement? Let us assume that the actual measurement of the observable
A gave the result an (one of the eigenvalues of A). Then after the measurement we know
exactly that the system is in a state in which the observable A has the value an . What was
a possibility before the measurement, has been turned into reality by the measurement
itself. We may say that our decision to perform the measurement has forced the quantum
system to “make up its mind” and decide which an it wants to pick. The system “chooses”
a value an from the allowed set of eigenvalues with the appropriate probability |cn |2 . Once
the value an has been measured the new state of the system is χn (~x). In this state the
only possible result of a measurement of A is indeed an . Would we decide to measure
A again immediately after the first measurement, we would get the same result an with
probability 1. If we wait a while until we perform the second measurement, the system
evolves from the new initial state χn (~x) following the Schrödinger equation. In particular,
it will no longer be in the state Ψ(~x). The possible results of a second measurement hence
depend on the outcome of the first measurement.
The above concept of an ideal measurement has led to many discussions, sometimes
of philosophical nature. To define an ideal measurement we have made a clear cut between the quantum system we are interested in and the measuring device. For example,
the decision to perform the measurement has nothing to do with the quantum system
itself, and is not influenced by it in any way. The Schrödinger equation that governs the
time evolution of the quantum system does not know when we will decide to do the next
measurement. It only describes the system between measurements. When a measurement
is performed the ordinary time evolution is interrupted and the wave function Ψ(~x) is
replaced by the eigenvector χn (~x) corresponding to the eigenvalue an that was obtained
in the actual measurement. In a real experiment it may not always be clear where one
should make the cut between the quantum system and the observer of the phenomenon.
7.6. SIMULTANEOUS MEASURABILITY AND COMMUTATORS
87
In particular, should we not think of ourselves as part of a big quantum system — known
as the Universe? Then aren’t our “decisions” to perform a particular measurement just
consequences of solving the Schrödinger equation for the wave function of the Universe?
If so, how should we interpret that wave function? After all, in the past our Universe has
behaved in a specific way. Who has decided to make the “measurements” that forced the
Universe to “make up its mind” how to behave? These questions show that the above
interpretation of the formal structure of quantum mechanics does not allow us to think of
quantum mechanics as a theory of everything including the observer. Fortunately, when
we observe concrete phenomena at microscopic scales it is not very important where we
draw the line between the quantum system and the observer or the measuring device. Only
when the microscopic physics has macroscopic consequences like in the thought experiments involving Schrödiger’s cat, which may be in a quantum mechanical superposition of
dead and alive, we are dragged into the above philosophical discussions. These discussions
are certainly interesting and often very disturbing, and they may even mean that quantum
mechanics is not the whole story when the physics of everything (the whole Universe) is
concerned. In any case, if we think that quantum mechanics is truly fundamental we must
address another big question:
What could possibly be the meaning of the wave function of the Universe?
From a practical point of view, however, the above assumptions about ideal measurements are not unreasonable, and the above interpretation of the formalism of quantum
mechanics has led to a correct description of all experiments performed at the microscopic
level.
7.6
Simultaneous Measurability and Commutators
As we have learned earlier, in quantum mechanics position and momentum of a particle
cannot be measured simultaneously with arbitrary precision. This follows already from
Heisenberg’s uncertainty relation. Now that we know more about measurements we can
ask which observables can be measured simultaneously with arbitrary precision. According
to our discussion of ideal measurements this would require that the operators describing
the two observables have the same set of eigenvectors. Then after a measurement of one
observable that reduces the wave function to a definite eigenstate, a measurement of the
other observable will also give a unique answer (not just a set of possible answers with
various probabilities). The quantity that decides about simultaneous measurability of two
observables is the so-called commutator of the corresponding Hermitean operators A and
B
[A, B] = AB − BA.
(7.6.1)
The two observables are simultaneously measurable with arbitrary precision if
[A, B] = 0,
(7.6.2)
88
CHAPTER 7. THE FORMAL STRUCTURE OF QUANTUM MECHANICS
i.e. if the order in which the operators are applied does not matter. Let us prove that this
is indeed equivalent to A and B having the same set of eigenvectors. We assume that the
eigenvectors are the same and write the two eigenvalue problems as
Aχn (~x) = an χn (~x), Bχn (~x) = bn χn (~x).
(7.6.3)
Then indeed
[A, B]χn (~x) = (AB − BA)χn (~x) = Abn χn (~x) − Ban χn (~x)
= (an bn − bn an )χn (~x) = 0.
(7.6.4)
Now let us proceed in the other direction by assuming [A, B] = 0. Then we want to show
that A and B indeed have the same set of eigenvectors. First, we write very generally
Aχn (~x) = an χn (~x), Bφn (~x) = bn φn (~x).
(7.6.5)
BAφn (~x) = ABφn (~x) = bn Aφn (~x),
(7.6.6)
Then
which implies that Aφn (~x) is an eigenvector of B with eigenvalue bn . Assuming nondegenerate eigenvalues this implies
Aφn (~x) = λφm (~x),
(7.6.7)
i.e. up to a factor λ the vector Aφn (~x) is one of the φm (~x). Then λ is an eigenvalue of A
— say an — and hence indeed
φm (~x) = χn (~x).
(7.6.8)
Let us evaluate the commutator for position and momentum operators
[xi , pj ]Ψ(~x) = −i~(xi ∂j − ∂j xi )Ψ(~x) = i~δij Ψ(~x),
(7.6.9)
[xi , pj ] = i~δij .
(7.6.10)
such that
This is consistent with Heisenberg’s uncertainty relation. The commutator does not vanish,
and hence position and momentum of a particle (or more precisely their components in the
same direction) cannot be measured simultaneously with arbitrary precision. Can energy
and momentum always be measured simultaneously? To answer that question we evaluate
[~
p, H]Ψ(~x) = [~
p,
p~ 2
~ V (~x)]Ψ(~x) = −i~(∇V
~ (~x))Ψ(~x),
+ V (~x)]Ψ(~x) = −i~[∇,
2M
(7.6.11)
and we obtain
~ (~x) = i~F~ (~x).
[~
p, H] = −i~∇V
(7.6.12)
Here F~ (~x) is the force acting on the particle. Only if the force vanishes — i.e. only for
a free particle — energy and momentum can be simultaneously measured with arbitrary
precision.
7.7. COMMUTATION RELATIONS OF COORDINATES, MOMENTA, AND ANGULAR MOMENTA89
7.7
Commutation Relations of Coordinates, Momenta, and
Angular Momenta
Now that we understand the importance of commutation relations, let us work them out
for a number of important observables: the position ~x, the momentum p~, and the angular
~ = ~x × p~ of a single particle. First of all, we note that the components of the
momentum L
position operator commute with each other, i.e.
[xi , xj ] = 0,
(7.7.1)
because they just act by multiplying a coordinate space wave function. Similarly, the
components of the momentum operator also commute with each other, i.e.
[pi , pj ] = −~2 [∂i , ∂j ] = 0,
(7.7.2)
because differentiation does not depend on the order either. However, as we saw already,
the components of position and momentum do not necessarily commute
[xi , pj ] = [xi , −i~∂j ] = i~∂j xi = i~δij .
(7.7.3)
Next, let us consider the commutation relations of components of the angular momentum vector
[Lx , Ly ] = [ypz − zpy , zpx − xpz ] = y[pz , x]px + x[z, pz ]py
= i~(xpy − ypx ) = i~Lz .
(7.7.4)
Using cyclic permutations one also obtains [Ly , Lz ] = i~Lx and [Lz , Lx ] = i~Ly . Adding
the trivial relation [Li , Li ] = 0, this can be summarized as
[Li , Lj ] = i~ijk Lk .
(7.7.5)
Here we sum over the repeated index k and we have used the totally antisymmetric LeviCivita tensor ijk , which vanishes except for
123 = 231 = 312 = −132 = −213 = −321 = 1.
(7.7.6)
Next we consider
~ 2 ] = [Lx , L2x + L2y + L2z ]
[Lx , L
= Ly [Lx , Ly ] + [Lx , Ly ]Ly + Lz [Lx , Lz ] + [Lx , Lz ]Lz
= i~(Ly Lz + Lz Ly − Lz Ly − Ly Lz ) = 0.
Similarly, for general Li we obtain
~ 2 ] = 0.
[Li , L
(7.7.7)
Interestingly, although the components of the angular momentum vector do not commute
~ 2.
with each other, they do commute with L
90
CHAPTER 7. THE FORMAL STRUCTURE OF QUANTUM MECHANICS
Let us also commute the components of the angular momentum with those of the
position vector
[Lx , x] = [ypz − zpy , x] = 0,
[Lx , y] = [ypz − zpy , y] = i~z.
(7.7.8)
Applying cyclic permutations, these relations can be summarized as
[Li , xj ] = i~ijk xk .
(7.7.9)
Furthermore, one can show that
[Li , ~x 2 ] = 0,
~ 2 ] = 0.
[xi , L
(7.7.10)
Finally, let us also commute the components of momenta and angular momenta
[Lx , px ] = [ypz − zpy , px ] = 0,
[Lx , py ] = [ypz − zpy , py ] = i~pz .
(7.7.11)
These relations as well as their partners under cyclic permutations can be summarized as
[Li , pj ] = i~ijk pk .
(7.7.12)
It is straightforward to convince oneself that
[Li , p~ 2 ] = 0.
(7.7.13)
~ result again in linear
Remarkably, all commutators of the components of ~x, p~, and L
~
~ form
combinations of ~x, p~, and L. Mathematically speaking, this means that ~x, p~, and L
a closed algebra whose “multiplication” rule is the commutator. Such algebras were first
discussed by the mathematician Marius Sophus Lie (1842-1899). The angular momentum
operators form a closed algebra by themselves which is characterized by the commutation
relations [Li , Lj ] = i~ijk Lk . This algebra is known as the Lie algebra SU (2). It characterizes the rotation properties of a single particle moving in 3-dimensional space. The
~ form what one might call the Galilean
nine components of the three vectors ~x, p~, and L
algebra. It is characterized by the commutation relations listed above, and it describes
the rotation and translation properties of a non-relativistic particle. Symmetry algebras
play a central role in modern physics. In particular, the Lie algebras SU (2) and SU (3)
govern the fundamental weak and strong nuclear forces.
7.8
Time Evolution
As we have discussed before, a measurement has an effect on the time evolution of a
quantum system. In fact, immediately after the measurement it is in the eigenstate of the
observable that corresponds to the eigenvalue that was actually measured. Between measurements the time evolution of a quantum system is determined by the time-dependent
Schrödinger equation
i~∂t Ψ(~x, t) = HΨ(~x, t).
(7.8.1)
7.8. TIME EVOLUTION
91
We will now see that it is indeed sufficient to solve the time-independent Schrödinger
equation
Hχn (~x) = En χn (~x).
(7.8.2)
The information about the time-dependence can be extracted from there. In the above
equation the χn (~x) are eigenvectors of the Hamilton operator, i.e. wave functions of stationary states, and the En are the corresponding energy eigenvalues. Since H is Hermitean,
its eigenvectors form a complete basis of orthonormal vectors in the Hilbert space. Consequently, we can write the time-dependent wave function
X
Ψ(~x, t) =
cn (t)χn (~x),
(7.8.3)
n
as a linear combination of eigenvectors. Now, however, the coefficients cn (t) will be timedependent. To figure out the time-dependence we plug the above expression into the
time-dependent Schrödinger equation and obtain
X
X
X
i~
∂t cn (t)χn (~x) =
cn (t)Hχn (~x) =
cn (t)En χn (~x).
(7.8.4)
n
n
n
Since the eigenvectors are orthogonal to one another, we can identify their coefficients in
a linear combination and conclude that
i~∂t cn (t) = cn (t)En .
(7.8.5)
This implies
i
cn (t) = cn (0) exp − En t ,
~
and hence
Ψ(~x, t) =
X
n
i
cn (0) exp − En t χn (~x).
~
Still, we need to determine cn (0). At t = 0 the above equation reads
X
Ψ(~x, 0) =
cn (0)χn (~x),
(7.8.6)
(7.8.7)
(7.8.8)
n
which implies
cn (0) = hχn |Ψ(0)i.
(7.8.9)
In fact, solving the time-dependent Schrödinger equation requires to specify an initial
condition for the wave function. We have to pick Ψ(~x, 0) and then the Schrödinger equation
determines Ψ(~x, t) for any later time t. All we have to do is to decompose the initial wave
function into a linear combination of eigenvectors of the Hamilton operator, and then
X
i
Ψ(~x, t) =
hχn |Ψ(0)i exp − En t χn (~x).
(7.8.10)
~
n
92
CHAPTER 7. THE FORMAL STRUCTURE OF QUANTUM MECHANICS
Chapter 8
Contact Interactions in One
Dimension
In this chapter we consider simple toy models for atoms, molecules, and electrons in a
crystal. For simplicity we treat these systems in one instead of three dimensions. In
addition, we replace the long-ranged electrostatic Coulomb potential by an ultra-shortranged contact interaction described by a δ-function potential. We also discuss the physical
consequences of the parity (i.e. reflection) and translation symmetries of these models.
8.1
Parity
Symmetries are always very important in a physical problem. In fact, using a symmetry
will often simplify the solution considerably. When we are dealing with potentials that
are reflection symmetric, i.e. when
V (−x) = V (x),
(8.1.1)
the corresponding symmetry is called parity. The parity operator P acts on a wave function
as
P Ψ(x) = Ψ(−x).
(8.1.2)
For reflection symmetric potentials the parity operator commutes with the Hamilton operator because
~2 d2 Ψ(x)
~2 d2
[P, H]Ψ(x) = P −
+ V (x)Ψ(x) − −
+ V (x) Ψ(−x)
2M dx2
2M dx2
~2 d2 Ψ(−x)
+ V (−x)Ψ(−x)
= −
2M dx2
~2 d2 Ψ(−x)
+
− V (x)Ψ(−x) = 0.
(8.1.3)
2M dx2
93
94
CHAPTER 8. CONTACT INTERACTIONS IN ONE DIMENSION
Since [P, H] = 0 both P and H have the same eigenvectors. Let us consider the eigenvalue
problem for the parity operator
P Ψ(x) = λΨ(x).
(8.1.4)
Acting with the parity operator twice implies
P 2 Ψ(x) = λ2 Ψ(x).
(8.1.5)
P 2 Ψ(x) = P Ψ(−x) = Ψ(x),
(8.1.6)
P Ψ(x) = ±Ψ(x).
(8.1.7)
On the other hand
such that λ = ±1, and thus
The eigenvectors with eigenvalue 1 are even functions
Ψ(−x) = Ψ(x),
(8.1.8)
while the eigenvectors with eigenvalue −1 are odd
Ψ(−x) = −Ψ(x).
(8.1.9)
Since the Hamilton operator has the same eigenvectors as P , we can assume that the wave
functions of stationary states are either even or odd for reflection symmetric potentials.
Making an appropriate ansatz for the wave function will simplify the solution of the
Schrödinger equation. Of course, even without noticing the symmetry one will get the
same answer, but in a more complicated way. It is important to note that symmetry
of the potential (V (−x) = V (x)) does not imply symmetry of the wave function in the
sense of Ψ(−x) = Ψ(x). In fact, half of the solutions have Ψ(−x) = −Ψ(x). This is
understandable because only |Ψ(x)|2 has the physical meaning of probability density, and
in both cases |Ψ(−x)|2 = |Ψ(x)|2 .
8.2
A Simple Toy Model for “Atoms” and “Molecules”
While real physics takes place in three dimensions, one dimensional models often shed some
light on the physical mechanisms responsible for a certain physical phenomenon. Here we
discuss a very simple 1-d model for atoms and molecules, that certainly oversimplifies the
real 3-d physics, but still captures some essential properties of chemical binding. The
model uses so-called δ-function potentials, which are easy to handle analytically. The
δ-function potential models a positively charged ion that attracts a negatively charged
electron, thus forming an electrically neutral atom. Of course, the interaction between
real ions and electrons is via the long-ranged Coulomb potential, not via an ultra shortranged δ-function potential. In this respect the model is certainly unrealistic. Still, as we
will see, it has some truth in it.
Let us first write down the potential as
V (x) = −V0 aδ(x).
(8.2.1)
8.2. A SIMPLE TOY MODEL FOR “ATOMS” AND “MOLECULES”
95
Here V0 > 0 such that the potential is attractive. The parameter a has the dimension
of a length to compensate for the dimension of the δ-function. As a consequence, V0 has
the dimension of an energy. The fact that the δ-function has the dimension of an inverse
length already follows from its most important property
Z ∞
dx f (x)δ(x) = f (0),
(8.2.2)
−∞
which holds for any smooth function f (x). Let us now write the Schrödinger equation
with the above δ-function potential
−
~2 d2 Ψ(x)
− V0 aδ(x)Ψ(x) = EΨ(x).
2M dx2
Integrating this equation over the interval [−, ] and using
Z dx f (x)δ(x) = f (0),
(8.2.3)
(8.2.4)
−
yields
~2
−
2M
dΨ() dΨ(−)
−
= V0 aΨ(0).
dx
dx
(8.2.5)
Here we have also taken the limit → 0. The above equation implies a discontinuity in the
first derivative of the wave function, that is proportional to the strength of the potential
and to the value of the wave function at the location of the δ-function.
How can we solve the above Schrödinger equation? Because we are interested in
binding of model electrons to model ions we restrict ourselves to bound state solutions
with E < 0. For x 6= 0 the δ-function vanishes, and thus we are then dealing with a free
particle Schrödinger equation. Its solutions with E < 0 are exponentially rising or falling.
The rising solution is unphysical because it cannot be properly normalized and hence
Ψ(x) = A exp(−κ|x|),
(8.2.6)
with
~2 κ2
.
(8.2.7)
2M
We have used parity symmetry to motivate the above ansatz. Then by construction the
wave function is automatically continuous at x = 0. However, we still must impose the
appropriate discontinuity of the derivative of the wave function. We find
~2
V0 aM
~2 dΨ() dΨ(−)
−
−
=
2Aκ = V0 aΨ(0) = V0 aA ⇒ κ =
,
(8.2.8)
2M
dx
dx
2M
~2
E=−
and hence
V02 a2 M
.
(8.2.9)
2~2
In our model this is the binding energy of an atom consisting of electron and ion (represented by the δ-function potential). There are also solutions with positive energy, but we
are not interested in them at the moment.
E=−
96
CHAPTER 8. CONTACT INTERACTIONS IN ONE DIMENSION
Instead, we now want to make a model for a pair of ions that share a single electron.
It is important to note that we do not consider two electrons yet. Up to now we have
always dealt with the Schrödinger equation for a single particle, and we will continue to
do so until we will talk about the hydrogen atom. We will see that the model electron
that is shared by two model ions induces an attractive interaction (chemical binding) of
two atoms. Hence, we are now considering a pair of δ-functions at a distance R such that
R
R
V (x) = −V0 a δ(x − ) + δ(x + ) .
(8.2.10)
2
2
Again, parity symmetry allows us to consider even and odd wave functions separately.
Since we are interested in E < 0, and especially in the ground state, we first consider even
wave functions. We know that away from the points x = ±R/2 we are dealing with the
free particle Schrödinger equation. For |x| > R/2 normalizability again singles out the
exponentially decaying solution, i.e. then
Ψ(x) = A exp(−κR |x|).
(8.2.11)
In this case κR will depend on R. In particular, it will in general be different from the κ
of a single δ-function potential. Of course, then also the energy
E(R) = −
~2 κ2R
,
2M
(8.2.12)
will be a function of the separation R. For |x| < R/2 normalizability does not impose
any restrictions, i.e. both exponentially rising and falling solutions are allowed. The even
combination of the two is a hyperbolic cosine, such that for |x| < R/2
Ψ(x) = B cosh(κR x).
(8.2.13)
Note that κR has the same value as for |x| > R/2 because it is related to the energy just
as before. To find the physical solution we first require continuity of the wave function at
x = R/2. Parity symmetry then guarantees that the wave function is continuous also at
x = −R/2. We demand
κR R
κR R
A exp(−
) = B cosh(
).
(8.2.14)
2
2
Also we must impose the appropriate discontinuity of the derivative of the wave function
~2 dΨ(R/2 + ) dΨ(R/2 − )
−
−
2M
dx
dx
2
~
κR R
κR R
=
BκR sinh
+ AκR exp −
2M
2
2
κR R
= V0 aΨ(R/2) = V0 aA exp −
,
(8.2.15)
2
and hence
B sinh
κR R
2
2V0 aM
κR R
=
− 1 A exp −
.
~2 κR
2
(8.2.16)
8.2. A SIMPLE TOY MODEL FOR “ATOMS” AND “MOLECULES”
Combining the two conditions implies
2V0 aM
κR R
= 2
− 1.
tanh
2
~ κR
97
(8.2.17)
Using a graphical method it is easy to see that this equation always has a solution. Furthermore, using tanh x ≤ 1 for x > 0 one obtains
2V0 aM
V0 aM
− 1 ≤ 1 ⇒ κR ≥
= κ,
2
~ κR
~2
(8.2.18)
i.e. the value κR for two δ-functions at a distance R is always bigger than the value κ for
a single δ-function. Hence
E(R) ≤ E,
(8.2.19)
i.e. the energy of an electron in the field of two ions a distance R apart, is smaller than
the binding energy of a single atom. In other words, it is energetically favorable for the
two ions to share the electron. This is indeed the mechanism responsible for chemical
binding. Of course, real molecules are more complicated than the one in our model, for
example, because the direct Coulomb interaction between the two ions leads to a repulsive
contribution as well. Although we have ignored this effect in our model, it still captures
an essential quantum aspect of chemistry. Sharing an electron, in other words delocalizing
it and thus spreading its wave function over a larger region in space, allows a molecule to
have less energy than two separated atoms.
We have not yet discussed the parity odd solutions in the double δ-function potential.
The appropriate ansatz is
Ψ(x) = ±A exp(−κR x),
(8.2.20)
for |x| ≥ R/2 and
Ψ(x) = B sinh(κR x),
(8.2.21)
for |x| ≤ R/2. The matching conditions then take the form
κR R
A exp −
2
= B sinh
κR R
2
,
(8.2.22)
and
B cosh
κR R
2
2V0 aM
κR R
=
− 1 A exp −
,
~2 κR
2
such that now
tanh
κR R
2
=
−1
2V0 aM
−1
.
~2 κR
(8.2.23)
(8.2.24)
Again, one can use a graphical method to solve this equation. A solution exists only if
R
~2
~2
≥
⇒ V0 ≥
.
2
2V0 aM
aM R
(8.2.25)
98
CHAPTER 8. CONTACT INTERACTIONS IN ONE DIMENSION
This condition is satisfied if the potential is strong enough, or if the two δ-functions are
separated by a large distance. For too weak potentials no bound state exists besides the
ground state. Again, we can use tanh x ≤ 1, such that now
−1
2V0 aM
V0 aM
−1
≤ 1 ⇒ κR ≤
= κ,
(8.2.26)
2
~ κR
~2
i.e. in this energetically excited state the energy of the molecule is larger than that of two
separated atoms. This is understandable because an odd wave function goes through zero
between the ions, and consequently the electron is then more localized than in the case of
two separated atoms. In chemistry this would correspond to an anti-binding orbital.
8.3
Shift Symmetry and Periodic Potentials
We have seen that parity considerations simplify calculations involving potentials with a
reflection symmetry. In crystals a periodic structure of ions generates a periodic potential
for the electrons
V (x + a) = V (x),
(8.3.1)
which is symmetric under translations by multiples of the lattice spacing a. In the next
section we will study a simple model for electrons in a crystal, and the shift symmetry will
simplify the solution of that model. The analog of the parity operator P in a translation
invariant problem is the shift operator
i
T = exp
pa ,
(8.3.2)
~
where p is the momentum operator. The shift operator acts on a wave function as
T Ψ(x) = Ψ(x + a).
(8.3.3)
Let us consider the commutator of the shift operator with the Hamilton operator
~2 d2 Ψ(x)
~2 d2
[T, H]Ψ(x) = T −
+ V (x)Ψ(x) − −
+ V (x) Ψ(x + a)
2M dx2
2M dx2
~2 d2 Ψ(x + a)
+ V (x + a)Ψ(x + a)
= −
2M
dx2
~2 d2 Ψ(x + a)
+
− V (x)Ψ(x + a) = 0.
(8.3.4)
2M
dx2
Since [T, H] = 0 both T and H have the same eigenvectors. The eigenvalue problem for
the shift operator takes the form
T Ψ(x) = λΨ(x).
The Hermitean conjugate of the shift operator is
i
†
T = exp − pa ,
~
(8.3.5)
(8.3.6)
8.4. A SIMPLE TOY MODEL FOR AN ELECTRON IN A CRYSTAL
99
which describes a shift backwards, i.e. T † Ψ(x) = Ψ(x − a). The eigenvalue problem of T †
takes the form
T † Ψ(x) = λ∗ Ψ(x),
(8.3.7)
with the same λ and Ψ(x) as in the eigenvalue problem of T . Acting with T † T implies
T † T Ψ(x) = T † λΨ(x) = λλ∗ Ψ(x) = |λ|2 Ψ(x).
(8.3.8)
T † T Ψ(x) = T † Ψ(x + a) = Ψ(x),
(8.3.9)
On the other hand
such that |λ|2 = 1 and hence
λ = exp(iqa),
(8.3.10)
is a phase factor. Here the phase was arbitrarily written as qa. One obtains
T Ψ(x) = Ψ(x + a) = exp(iqa)Ψ(x),
(8.3.11)
i.e. the eigenvectors of T are periodic up to a phase factor. It is important to note that
Ψ(x) itself is not periodic. Only the probability density is periodic, i.e.
|Ψ(x + a)|2 = |Ψ(x)|2 .
8.4
(8.3.12)
A Simple Toy Model for an Electron in a Crystal
We have seen that a δ-function potential provides a simple model for an ion in an atom
or molecule. The same model can also describe the ions in a crystal with lattice spacing
a, which generate a periodic potential
X
V (x) = −V0 a
δ(x − na),
(8.4.1)
n∈Z
for an electron. Let us look for positive energy solutions of the Schrödinger equation with
this potential. We make an ansatz for the wave function in each interval [na, (n + 1)a]
separately. For x ∈ [na, (n + 1)a] we write
Ψ(x) = An exp(ik(x − na)) + Bn exp(−ik(x − na)).
(8.4.2)
The energy is then given by
~2 k 2
.
2M
By definition, for x ∈ [(n + 1)a, (n + 2)a] the wave function is given by
E=
(8.4.3)
Ψ(x) = An+1 exp(ik(x − (n + 1)a)) + Bn+1 exp(−ik(x − (n + 1)a)).
(8.4.4)
Therefore for x ∈ [na, (n + 1)a] we have
Ψ(x + a) = An+1 exp(ik(x − na)) + Bn+1 exp(−ik(x − na)).
(8.4.5)
100
CHAPTER 8. CONTACT INTERACTIONS IN ONE DIMENSION
Since we are dealing with a periodic potential, the Hamilton operator commutes with the
shift operator. Hence, the eigenvectors of H can simultaneously be chosen as eigenvectors
of T . As such they obey
Ψ(x + a) = exp(iqa)Ψ(x).
(8.4.6)
Comparing this with eq.(8.4.2) and eq.(8.4.5) yields
An+1 = exp(iqa)An , Bn+1 = exp(iqa)Bn .
(8.4.7)
Next we impose continuity and discontinuity conditions for the wave function and its first
derivative. The continuity condition at x = (n + 1)a takes the form
An+1 + Bn+1 = An exp(ika) + Bn exp(−ika) ⇒
An [exp(iqa) − exp(ika)] + Bn [exp(iqa) − exp(−ika)] = 0.
(8.4.8)
The discontinuity equation reads
~2
[ikAn+1 − ikBn+1 − ikAn exp(ika) + ikBn exp(−ika)]
2M
= V0 a[An exp(ika) + Bn exp(−ika)] ⇒
2M V0 a
An exp(iqa) − exp(ika) + 2
exp(ika)
~ ik
2M V0 a
+Bn − exp(iqa) + exp(−ika) + 2
exp(−ika) = 0.
~ ik
−
The two conditions can be summarized in matrix form
iqa
iqa
eika
e − e−ika
An
iqa e ika −2M
= 0.
V0 a
V0 a
Bn
e +e
−1
−eiqa + e−ika 2M
+1
~2 ik
~2 ik
(8.4.9)
(8.4.10)
We can read this as an eigenvalue problem with eigenvalue zero. A non-zero solution for
An and Bn hence only exists if the above matrix has a zero eigenvalue. This is the case
only if the determinant of the matrix vanishes, i.e. if
2M V0 a
[exp(iqa) − exp(ika)] − exp(iqa) + exp(−ika)
+1
~2 ik
2M V0 a
− [exp(iqa) − exp(−ika)] exp(iqa) + exp(ika)
−1
~2 ik
M V0 a
(8.4.11)
= 4 exp(iqa) − cos(qa) + cos(ka) − 2 sin(ka) = 0,
~ k
and thus
M V0 a
sin(ka).
(8.4.12)
~2 k
A graphical method reveals that there are bands of allowed energies separated by forbidden
energy regions. This follows when we plot the function cos(ka) − (M V0 a/~2 k) sin(ka)
noticing that −1 ≤ cos(qa) ≤ 1. For example, when ka is slightly less than π we have
cos(qa) = cos(ka) −
8.4. A SIMPLE TOY MODEL FOR AN ELECTRON IN A CRYSTAL
101
cos(ka) ≈ −1 and sin(ka) > 0 such that cos(ka) − (M V0 a/~2 k) sin(ka) < −1. In that case
no solution to the above equation exists, and the corresponding energy is forbidden.
This simple model sheds some light on what happens in real 3-d crystals. In fact, also
there finite bands of allowed energy states exist. In a conductor the energy levels of an
allowed band are not all occupied by electrons. Therefore an externally applied electric
field can raise the energy of the electrons by accelerating them within the crystal, thus
inducing an electric current. In an insulator, on the other hand, an allowed energy band is
completely filled. In that case the external field cannot increase the energy of the electrons
unless it provides enough energy to lift them to the next allowed energy band. Hence a
small external field does not induce an electric current. A large field, however, may cause
the insulator to break down, and become a conductor.
102
CHAPTER 8. CONTACT INTERACTIONS IN ONE DIMENSION
Chapter 9
The Harmonic Oscillator
9.1
Solution of the Schrödinger Equation
The harmonic oscillator plays an important role in many areas of physics. It describes small
oscillations of a physical system around an equilibrium configuration. For example, the
ions in a crystal oscillate around their equilibrium positions, and thus for small oscillations
represent a coupled system of harmonic oscillators. In quantum mechanics the energy of
such oscillations is quantized, and the corresponding quanta are called phonons. Phonons
play an important role, for example, in superconductors, because they provide a mechanism that binds electrons into Cooper pairs, which can undergo so-called Bose-Einstein
condensation and therefore carry electric current without resistance. Also photons — the
quantized oscillations of the electromagnetic field — are directly related to harmonic oscillators. In fact, one can view the electromagnetic field as a system of coupled oscillators,
one for each space point. Here we consider the quantum mechanics of a single harmonic
oscillator first in one dimension. A classical realization of this situation is, for example,
a particle attached to a spring with spring constant k. The potential energy of such an
oscillator is
1
1
V (x) = kx2 = M ω 2 x2 .
(9.1.1)
2
2
Here ω is the angular frequency of the oscillator. Solving the Schrödinger equation
−
~2 d2 Ψ(x) 1
+ M ω 2 x2 Ψ(x) = EΨ(x),
2M dx2
2
(9.1.2)
for the harmonic oscillator is more complicated than for the potentials that we have
previously discussed because now we are no longer just patching together solutions of
the free particle Schrödinger equation in various regions. Let us divide the equation by
~ω and use the rescaled dimensionless position variable
r
Mω
.
(9.1.3)
y = αx, α =
~
103
104
CHAPTER 9. THE HARMONIC OSCILLATOR
Then the Schrödinger equation takes the form
−
1 d2 Ψ(y) 1 2
+ y Ψ(y) = Ψ(y),
2 dy 2
2
(9.1.4)
where = E/~ω is a rescaled dimensionless energy variable. Let us make the following
ansatz for the wave function
1 2
Ψ(y) = ϕ(y) exp − y .
(9.1.5)
2
Inserting this in the Schrödinger equation yields the following equation for ϕ(y)
dϕ(y)
1 d2 ϕ(y)
1
+y
ϕ(y).
−
= −
2 dy 2
dy
2
(9.1.6)
Since this problem has a parity symmetry, we look separately for even and odd solutions.
The above equation has an even solution ϕ(y) = 1 with = 1/2. The corresponding wave
function
1 2 2
Ψ0 (x) = A exp − α x ,
(9.1.7)
2
with energy E = ~ω/2 represents the ground state of the harmonic oscillator. It should be
noted that, unlike a classical oscillator, the quantum oscillator cannot have zero energy.
In fact, this would imply that the particle is at rest at x = 0, which is in contradiction
with the Heisenberg uncertainty principle. Next we look for the first excited state, which
must be odd, by putting ϕ(y) = y and we realize that this is a solution for = 3/2. This
solution corresponds to the wave function
1 2 2
Ψ1 (x) = Bαx exp − α x ,
(9.1.8)
2
with energy E = 3~ω/2. The second excited state is again even, and follows from the
ansatz ϕ(y) = y 2 + c. Inserting this in the above equation one obtains = 5/2 and
c = −1/2, such that
1 2 2
2 2
Ψ2 (x) = C(2α x − 1) exp − α x ,
(9.1.9)
2
with energy E = 5~ω/2. One can construct all higher excited states Ψn (x) by writing
ϕ(y) as a polynomial of degree n, and inserting it in the equation for ϕ(y). The resulting
polynomials are the so-called Hermite polynomials, and the resulting energy is
1
En = ~ω n +
.
(9.1.10)
2
9.2
Operator Formalism
An alternative and more elegant method to solve the harmonic oscillator problem uses an
operator formalism that we will now discuss. The Hamilton operator for the harmonic
9.2. OPERATOR FORMALISM
oscillator takes the form
105
H
1 d2
1
= − 2 2 + α 2 x2 .
~ω
2α dx
2
(9.2.1)
Let us introduce the operators
1 d
1
1 d
1
†
a= √
αx +
, a =√
αx −
,
α dx
α dx
2
2
(9.2.2)
which are Hermitean conjugates of one another. Acting with them on a wave function one
obtains
1 d
1 dΨ(x)
1
†
αx −
αxΨ(x) +
a aΨ(x) =
2
α dx
α dx
2
1 d Ψ(x) 1 2 2
1
= − 2
+ α x Ψ(x) − Ψ(x)
2
2α
dx
2
2
1
H
−
Ψ(x),
(9.2.3)
=
~ω 2
such that
H
1
= a† a + .
(9.2.4)
~ω
2
We can solve the eigenvalue problem of H very elegantly, just by working out various
commutation relations. First of all
[a, a† ]Ψ(x) = (aa† − a† a)Ψ(x)
1
1 d
=
αx +
αxΨ(x) −
2
α dx
1
1 d
αx −
αxΨ(x) +
−
2
α dx
1 dΨ(x)
α dx
1 dΨ(x)
= Ψ(x),
α dx
(9.2.5)
such that
[a, a† ] = 1.
(9.2.6)
Also we need the commutation relations with the Hamilton operator
H
1
, a] = [a† a + , a] = [a† , a]a = −a,
~ω
2
H
1
[ , a† ] = [a† a + , a† ] = a† [a, a† ] = a† .
~ω
2
[
(9.2.7)
We will now prove that a† and a act as raising and lowering operators between the various
eigenstates of the harmonic oscillator. First, we act with a† on an eigenstate Ψn (x) and
obtain a new state
Φ(x) = a† Ψn (x),
(9.2.8)
which, as we will now show, is also an eigenstate of H. For this purpose we evaluate
H
H †
H †
† H
Φ(x) =
a Ψn (x) = [ , a ] + a
Ψn (x)
~ω
~ω
~ω
~ω
= (εn + 1)a† Ψn (x) = (εn + 1)Φ(x).
(9.2.9)
106
CHAPTER 9. THE HARMONIC OSCILLATOR
Thus, Φ(x) is an eigenstate of H with eigenvalue ~ω(εn + 1). This means it must be
proportional to Ψn+1 (x), i.e.
Φ(x) = λΨn+1 (x).
(9.2.10)
To determine the proportionality constant λ we compute the normalization of Φ(x) as
hΦ|Φi = ha† Ψn |a† Ψn i = haa† Ψn |Ψn i = h(a† a + 1)Ψn |Ψn i = n + 1.
(9.2.11)
On the other hand
such that λ =
√
hΦ|Φi = |λ|2 hΨn+1 |Ψn+1 i = |λ|2 ,
(9.2.12)
n + 1 and thus
a† Ψn (x) =
√
n + 1Ψn+1 (x).
(9.2.13)
Iterating this equation one can construct the n-th excited state by acting with the raising
operator a† on the ground state n times, i.e.
1
Ψn (x) = √ (a† )n Ψ0 (x).
n!
(9.2.14)
Next we act with the lowering operator a on an eigenstate Ψn (x) and obtain
Φ(x) = aΨn (x),
which again is an eigenstate of H because now
H
H
H
H
Φ(x) =
aΨn (x) = [ , a] + a
Ψn (x)
~ω
~ω
~ω
~ω
= (εn − 1)aΨn (x) = (εn − 1)Φ(x).
(9.2.15)
(9.2.16)
Thus, Φ(x) is an eigenstate of H with eigenvalue ~ω(εn − 1), and hence it is proportional
to Ψn−1 (x), i.e. now
Φ(x) = λΨn−1 (x).
(9.2.17)
The constant λ again follows from
hΦ|Φi = haΨn |aΨn i = ha† aΨn |Ψn i = n,
(9.2.18)
hΦ|Φi = |λ|2 hΨn−1 |Ψn−1 i = |λ|2 ,
(9.2.19)
and
such that now λ =
√
n and
aΨn (x) =
√
nΨn−1 (x).
(9.2.20)
Indeed a acts as a lowering operator. When a acts on the ground state the above equation
implies
aΨ0 (x) = 0,
(9.2.21)
which is consistent because there is no eigenstate below the ground state. The operator
a† a is called the number operator because
√
a† aΨn (x) = a† nΨn−1 (x) = nΨn (x),
(9.2.22)
i.e. it measures the number of energy quanta in the state Ψn (x).
9.3. COHERENT STATES AND THE CLASSICAL LIMIT
9.3
107
Coherent States and the Classical Limit
Classically a harmonic oscillator changes its position and momentum periodically such
that
dx(t)
x(t) = A cos(ωt), p(t) = M
= −M Aω sin(ωt).
(9.3.1)
dt
The eigenstates of the quantum harmonic oscillator, on the other hand, have expectation
values of x and p, which are zero independent of time. Hence, the question arises how the
classical oscillations can emerge from the quantum theory in the limit ~ → 0. Obviously,
the states that resemble classical oscillatory behavior cannot be the energy eigenstates.
Instead, we will discuss so-called coherent states, which are linear combinations of energy
eigenstates, and which indeed have the expected behavior in the classical limit. The
coherent states are eigenstates of the lowering operator a, i.e.
aΦλ (x) = λΦλ (x),
(9.3.2)
For the eigenvalue λ we have the corresponding eigenstate Φλ (x). Since a is not a Hermitean operator, the eigenvalue λ will in general be complex. Let us now construct the
coherent state as a linear combination of energy eigenstates
Φλ (x) =
X
cn (λ)Ψn (x).
(9.3.3)
n
The eigenvalue equation takes the form
aΦλ (x) =
X
cn (λ)aΨn (x) =
X
n
X
√
cn (λ) nΨn−1 (x) = λ
cn (λ)Ψn (x).
n
(9.3.4)
n
Hence, we can read off a recursion relation
λ
cn (λ) = √ cn−1 (λ).
n
(9.3.5)
Iterating the recursion relation we find
λn
cn (λ) = √ c0 (λ).
n!
(9.3.6)
We still need to normalize the state Φλ (x). The normalization condition takes the form
hΦλ |Φλ i =
X
n
such that
|cn (λ)|2 =
X |λ|2n
n
n!
|c0 (λ)|2 = exp(|λ|2 )|c0 (λ)|2 = 1,
λn
1 2
√
cn (λ) =
exp − |λ| .
2
n!
(9.3.7)
(9.3.8)
108
CHAPTER 9. THE HARMONIC OSCILLATOR
By construction it is clear that the coherent state is not an energy eigenstate. Still we can
compute the expectation value of the energy
X
1
1 X
†
n+
|cn (λ)|2
cn (λ)Ψn i = ~ω
hΦλ |HΦλ i = hΦλ |~ω a a +
2 n
2
n
2n
X
1 |λ|
1
n+
= ~ω
exp(−|λ|2 ) = ~ω |λ|2 +
.
(9.3.9)
2
n!
2
n
We see explicitly that we are not in an energy eigenstate when we calculate the variance
of the energy
2
hH i =
=
=
=
=
1 2
1
†
2 2
†
†
†
hΦλ |~ ω a a +
Φλ i = ~ ω hΦλ | a aa a + a a +
Φλ i
2
4
1
2 2
†
†
†
†
~ ω hΦλ | a ([a, a ] + a a)a + a a +
Φλ i
4
1
Φλ i
~2 ω 2 hΦλ | (a† )2 a2 + 2a† a +
4
1
2 2
2
2
~ ω ha Φλ |a Φλ i + 2haΦλ |aΦλ i +
4
1
~2 ω 2 |λ|4 + 2|λ|2 +
,
(9.3.10)
4
2 2
and thus
∆H =
p
hH 2 i − hHi2 = ~ω|λ|.
(9.3.11)
Only for λ = 0 there is no fluctuation in the energy, and then indeed the coherent state
reduces to the oscillator ground state, which of course has a sharp energy. Let us now
9.3. COHERENT STATES AND THE CLASSICAL LIMIT
109
consider the following position and momentum expectation values
1
1
hxi = hΦλ | √ [a + a† ]Φλ i = √ [hΦλ |aΦλ i + haΦλ |Φλ i]
2α
2α
1
= √ [λ + λ∗ ],
2α
α~
α~
hpi = hΦλ | √ [a − a† ]Φλ i = √ [hΦλ |aΦλ i − haΦλ |Φλ i]
2i
2i
α~
= √ [λ − λ∗ ],
2i
1
hx2 i = hΦλ | 2 [a2 + aa† + a† a + (a† )2 ]Φλ i
2α
1
=
hΦλ |[a2 + 2a† a + 1 + (a† )2 ]Φλ i
2α2
1
=
[hΦλ |a2 Φλ i + 2haΦλ |aΦλ i + hΦλ |Φλ i + ha2 Φλ |Φλ i]
2α2
1
=
[λ2 + 2|λ|2 + 1 + λ∗2 ],
2α2
α2 ~2 2
[a − aa† − a† a + (a† )2 ]Φλ i
hp2 i = hΦλ | −
2
α2 ~2
= −
hΦλ |[a2 − 2a† a − 1 + (a† )2 ]Φλ i
2
α2 ~2
= −
[hΦλ |a2 Φλ i − 2haΦλ |aΦλ i − hΦλ |Φλ i + ha2 Φλ |Φλ i]
2
α2 ~2 2
= −
[λ − 2|λ|2 − 1 + λ∗2 ].
2
Thus, we obtain
(∆x)2 = hx2 i − hxi2 =
1
α2 ~2
2
2
2
,
(∆p)
=
hp
i
−
hpi
=
,
2α2
2
(9.3.12)
such that
∆x∆p =
~
,
2
(9.3.13)
i.e. the coherent state saturates Heisenberg’s inequality. This is a first indication that this
state is as classical as possible, it has a minimal uncertainty product. From the proof
of the Heisenberg uncertainty relation of chapter 3 it follows that the inequality can be
saturated for a state Ψ(x) only if
d
1
hxi
i
Ψ(x) +
xΨ(x) −
+ hpi Ψ(x) = 0.
(9.3.14)
dx
2(∆x)2
2(∆x)2 ~
Inserting the above results for the coherent state this equation takes the form
√
d
Φλ (x) + α2 xΦλ (x) − 2αλΦλ (x) = 0.
dx
(9.3.15)
110
CHAPTER 9. THE HARMONIC OSCILLATOR
Using
1
a= √
2
1 d
αx +
,
α dx
(9.3.16)
we indeed identify this as the eigenvalue equation for a.
As we learned earlier, a Gaussian wave packet saturates the Heisenberg uncertainty
relation. As we now know, the same is true for an oscillator coherent state, and as we will
see next, the two are actually the same thing. In fact, let us consider a shifted Gaussian
wave packet that also has an average momentum hpi
1 2
i
2
Φ(x) = A exp − α (x − hxi) exp
hpix .
(9.3.17)
2
~
Such a wave packet spreads during its time evolution if we are dealing with free particles.
Here, however, we have a harmonic oscillator potential. As we will see later, the wave
packet then does not spread. Instead it oscillates back and forth without changing its
shape. Let us first convince ourselves that the above wave packet is indeed equal to one
of our coherent states. For that purpose we act on it with the lowering operator
1
1 d
aΦ(x) = √
Φ(x)
αx +
α dx
2
1
i
= √
αx − α(x − hxi) +
hpi Φ(x)
α~
2
1
i
= √
αhxi +
hpi Φ(x) = λΦ(x).
(9.3.18)
α~
2
We see that the Gaussian wave packet Φ(x) is indeed an eigenstate of the lowering operator,
and hence equal to the coherent state with eigenvalue
1
i
λ= √
αhxi +
hpi .
(9.3.19)
α~
2
In order to understand that the Gaussian wave packet equivalent to the coherent
state does not spread, we now investigate its time evolution. We know that each energy
eigenstate Ψn (x) simply picks up the phase exp(−iEn t/~) during its time evolution. Hence
we obtain
X
i
Φλ (x, t) =
cn (λ)Ψn (x) exp − En t
~
n
X λn
1 2
1
√ exp − |λ| Ψn (x) exp −iω n +
=
t
2
2
n!
n
ωt X [λ exp(−iωt)]n
1 2
√
= exp −i
exp − |λ| Ψn (x)
2
2
n!
n
ωt
= exp −i
Φλ(t) (x).
(9.3.20)
2
9.4. THE HARMONIC OSCILLATOR IN TWO DIMENSIONS
111
We see that up to an irrelevant global phase exp(−i ω2 t) the time evolution only manifests
itself in a time-dependent eigenvalue
λ(t) = λ exp(−iωt).
(9.3.21)
In particular, a coherent state remains coherent, i.e. the corresponding Gaussian wave
packet does not change its shape (in coordinate space). However, it oscillates back and
worth exactly like a classical harmonic oscillator. This become clear when we remind ourselves that the real part of λ is proportional to hxi, while the imaginary part is proportional
to hpi. Since λ(t) has a periodic behavior described by exp(−iωt), the same is true for its
real and imaginary parts. Consequently, both average position and average momentum of
the Gaussian wave packet change periodically, but its shape remains unchanged. This is
as close as one can get to a quasi classical behavior of the quantum oscillator.
9.4
The Harmonic Oscillator in Two Dimensions
Let us now turn to the harmonic oscillator in more than one dimension. Here we consider
two dimensions, but it will become clear that we could go to three (and even higher)
dimensions in exactly the same way. In two dimensions the harmonic oscillator potential
takes the form
1
V (~r) = V (x, y) = M ω 2 (x2 + y 2 ).
(9.4.1)
2
We see that the potential separates into an x-dependent and a y-dependent piece. This is
a special property of the harmonic oscillator, which will allow us to reduce the harmonic
oscillator problem in higher dimensions to the one dimensional problem. Alternatively we
can write the potential in polar coordinates
1
V (~r) = V (r, ϕ) = M ω 2 r2 ,
2
(9.4.2)
x = r cos ϕ, y = r sin ϕ.
(9.4.3)
where
It is obvious that the problem is rotation invariant because the potential is ϕ-independent.
This symmetry leads to the conservation of angular momentum, both classically and at
the quantum level. In quantum mechanics symmetries often lead to degeneracies in the
spectrum. In fact, as we will see, the harmonic oscillator spectrum is highly degenerate in
more than one dimension. We can understand part of the degeneracy based on rotational
invariance. However, there is more degeneracy in the harmonic oscillator spectrum than
one would expect based on rotation invariance. In fact, there are other less obvious
symmetries in the problem. These symmetries are sometimes called accidental because it
was for some time not well understood where the symmetry came from. Now we know
that the symmetry reflects a special property of the harmonic oscillator force law. At the
classical level, the symmetry implies that all orbits in a harmonic oscillator potential are
closed curves in space. This is a very special property of the harmonic oscillator. In fact,
a generic rotation invariant potential V (r) will not have closed classical orbits. Another
112
CHAPTER 9. THE HARMONIC OSCILLATOR
important problem with closed classical orbits is given by the Coulomb potential. In that
case all bound classical orbits are closed, and indeed we will see later that the discrete
spectrum of the hydrogen atom again has more degeneracy than one would expect based
on rotation symmetry alone.
Let us first look at the classical solutions of the two dimensional harmonic oscillator.
The most general solution of Newton’s equation is
x(t) = A cos(ωt), y(t) = B cos(ωt − ϕ0 ),
(9.4.4)
which parameterizes an ellipse. After one period T = 2π/ω both x(t) and y(t) return to
their original positions and the classical orbit closes. In fact, the particle returns to its
original position with its original momentum because
px (t) = −M Aω sin(ωt), py (t) = −M Bω sin(ωt − ϕ0 ),
(9.4.5)
are also periodic functions. Consequently, the whole motion repeats itself after one period.
Let us now consider the angular momentum
Lz = xpy − ypx = M ABω[sin(ωt) cos(ωt − ϕ0 ) − cos(ωt) sin(ωt − ϕ0 )]
= M ABω sin ϕ0 ,
(9.4.6)
which is time-independent and thus conserved. We will see later that the conservation of
angular momentum also manifests itself in quantum mechanics.
We are now ready to solve the Schrödinger equation of the two dimensional harmonic
oscillator. It takes the form
−
~2
∆Ψ(~r) + V (~r)Ψ(~r) = EΨ(~r),
2M
(9.4.7)
or equivalently in Cartesian coordinates
−
1
~2 2
∂x Ψ(x, y) + ∂y2 Ψ(x, y) + M ω 2 (x2 + y 2 )Ψ(x, y) = EΨ(x, y).
2M
2
(9.4.8)
Since the potential separates into an x-dependent and a y-dependent piece it is natural to
try a separation of variables also for the wave function. This motivates the ansatz
Ψ(x, y) = φ(x)χ(y).
Inserting this in the Schrödinger equation and dividing by χ(y) implies
~2
φ(x) 2
1
2
∂x φ(x) +
∂y χ(y) + M ω 2 (x2 + y 2 )φ(x) = Eφ(x),
−
2M
χ(y)
2
(9.4.9)
(9.4.10)
or
−
~2 2
1
~2 φ(x) 2
1
∂x φ(x) + M ω 2 x2 φ(x) − Eφ(x) =
∂ χ(y) − M ω 2 y 2 φ(x).
2M
2
2M χ(y) y
2
(9.4.11)
9.4. THE HARMONIC OSCILLATOR IN TWO DIMENSIONS
113
The left hand side of this equation is y-independent, while the right-hand side seems to
be y-dependent. However, if the two sides are equal, also the right hand side must be
y-independent. Calling the function on the left-hand side f (x) we can write
~2 φ(x) 2
1
∂y χ(y) − M ω 2 y 2 φ(x) = −f (x),
2M χ(y)
2
(9.4.12)
~2 2
1
f (x)
∂ χ(y) + M ω 2 y 2 χ(y) =
χ(y),
2M y
2
φ(x)
(9.4.13)
such that
−
Now the left hand side is x-independent, and therefore the same must be true for the right
hand side. This implies that
f (x)
= Ey ,
(9.4.14)
φ(x)
is some constant. Then the above equation is just the Schrödinger equation of a one
dimensional harmonic oscillator with energy Ey
−
~2 2
1
∂ χ(y) + M ω 2 y 2 χ(y) = Ey χ(y).
2M y
2
(9.4.15)
Inserting this back into the original Schrödinger equation we also obtain
−
1
~2 2
∂x φ(x) + M ω 2 x2 φ(x) = Ex φ(x),
2M
2
(9.4.16)
with
E = Ex + Ey .
(9.4.17)
This shows that the two dimensional (and in fact any higher dimensional) harmonic oscillator problem is equivalent to a set of one dimensional oscillator problems. This allows
us to write down the solutions of the two dimensional oscillator problem as
Ψ(x, y) = Ψnx (x)Ψny (y),
(9.4.18)
where Ψnx (x) and Ψny (y) are eigenfunctions of one dimensional harmonic oscillators. The
corresponding energy eigenvalue is
En = Ex + Ey = ~ω(nx + ny + 1) = ~ω(n + 1).
(9.4.19)
The n-th excited state is (n+1)-fold degenerate because there are n+1 ways of writing the
integer n as a sum of non-negative integers nx and ny . This high degeneracy is only partly
due to the rotational symmetry of the problem. The rest of the degeneracy is related to
the symmetry that ensures that all classical orbits of the oscillator are closed curves.
114
CHAPTER 9. THE HARMONIC OSCILLATOR
Chapter 10
The Hydrogen Atom
10.1
Separation of the Center of Mass Motion
The hydrogen atom is the simplest atom consisting of a single electron of charge −e and
a proton of charge +e that represents the atomic nucleus. The two particles are bound
together via the attractive Coulomb interaction. In the following we want to discuss
a slight generalization of the hydrogen problem. So-called hydrogen-like atoms consist
again of a single electron, but may have a more complicated atomic nucleus consisting
of Z protons (and hence with charge +Ze) and A − Z neutrons. A neutron has almost
the same mass as a proton, and thus the mass of the atomic nucleus is then given by
MA ≈ AMp , where Mp is the mass of the proton. As far as atomic physics is concerned we
can consider the nucleus as a single particle of mass MA and charge +Ze. Hence hydrogenlike atoms can be viewed as two-particle systems consisting of an electron and an atomic
nucleus. So far we have only considered single particles moving under the influence of
an external potential. Now we are dealing with two particles influenced by their mutual
Coulomb attraction. We do not consider any external forces in this case, which implies
that the system is translation invariant. This invariance leads to the conservation of the
total momentum. We can then go to the center of mass frame, and reduce the two-particle
problem formally to the problem of a single particle moving in an external potential.
The two-particle Schrödinger equation for hydrogen-like atoms takes the form
~2
~2
−
∆e −
∆A + V (~re − ~rA ) Ψ(~re , ~rA ) = EΨ(~re , ~rA ),
(10.1.1)
2Me
2MA
where Me is the electron mass, MA is the mass of the nucleus, and V (~r) is the Coulomb
potential depending of the separation ~r = ~re − ~rA of the two particles. The Laplace
operators ∆e and ∆A represent second derivatives with respect to the position of the
electron and the nucleus respectively. Although we now have a two-particle Schrödinger
equation we still have a single wave function Ψ(~re , ~rA ), which, however, now depends
on the positions of both particles. Ψ(~re , ~rA ) specifies the probability amplitude to find
115
116
CHAPTER 10. THE HYDROGEN ATOM
the electron at position ~re and simultaneously the nucleus at position ~rA . The proper
normalization of the wave function therefore is
Z
d3 re d3 rA |Ψ(~re , ~rA )|2 = 1.
(10.1.2)
Next we introduce the center of mass coordinate
~ = Me~re + MA~rA .
R
Me + M A
(10.1.3)
The kinetic energy of the two particles can then be expressed as
−
~2
~2
~2
~2
∆e −
∆A = −
∆R −
∆r ,
2Me
2MA
2M
2µ
(10.1.4)
where M = Me + MA is the total mass, and
µ=
Me MA
Me + M A
(10.1.5)
is the so-called reduced mass. Since the atomic nucleus is several thousand times heavier
than an electron, the reduced mass is very close to the electron mass. The total Hamilton
operator now takes the form
H=−
~2
~2
∆R −
∆r + V (~r).
2M
2µ
(10.1.6)
The first term represents the kinetic energy of the center of mass motion and depends
~ The second and third term represent kinetic and potential energy of the
only on R.
relative motion and depend only on ~r. This separation of the Hamilton operator suggests
a separation ansatz for the wave function
~ = ψ(~r)Φ(R).
~
Ψ(~r, R)
(10.1.7)
Inserting this ansatz in the two-particle Schrödinger equation implies
~2
~ = ER Φ(R),
~
∆R Φ(R)
(10.1.8)
2M
for the center of mass motion. This is the equation for a free particle of mass M . Its
solutions are plane waves
i
~ · P~ ,
~ = exp
R
(10.1.9)
Φ(R)
~
−
characterized by the total momentum P~ . From now on we will go to the center of mass
frame in which P~ = 0. Then the energy of the center of mass motion ER = P 2 /2M also
vanishes. The remaining equation for the relative motion takes the form
−
~2
∆r ψ(~r) + V (~r)ψ(~r) = Eψ(~r).
2µ
(10.1.10)
Formally, this is the Schrödinger equation for a single particle of the reduced mass µ
interacting with a potential V (~r). By separating the center of mass motion we have
reduced the original two-particle problem to a single-particle problem.
10.2. ANGULAR MOMENTUM
10.2
117
Angular Momentum
Up to now we have not used the fact that the Coulomb potential only depends on the
distance between the electron and the atomic nucleus. The corresponding rotational invariance implies that the angular momentum is a conserved quantity. This will allow us to
also separate the angular variables, and to reduce the problem further to a 1-dimensional
radial Schrödinger equation. In spherical coordinates r, θ and ϕ the Laplace operator
takes the form
2
1
1
∆r = ∂r2 + ∂r + 2
(10.2.1)
∂θ (sin θ∂θ ) + 2 2 ∂ϕ2 .
r
r sin θ
r sin θ
~ = ~r × p~ and the relative
Using the definitions of the angular momentum operator L
~
momentum operator p~ = ~∇r /i one can show that
~2
2
L = −~
1
1
2
∂ ,
∂θ (sin θ∂θ ) +
sin θ
sin2 θ ϕ
(10.2.2)
which implies
~2
2
L
∆r = ∂r2 + ∂r − 2 2 .
r
~ r
(10.2.3)
Therefore the Hamilton operator for the relative motion takes the form
H=−
~2
2µ
2
∂r2 + ∂r + Vef f (r),
r
(10.2.4)
where
Vef f (r) = V (r) +
~2
L
2µr2
(10.2.5)
is an effective potential consisting of the original Coulomb potential V (r) and a so-called
centrifugal barrier, which corresponds to a repulsive potential for states with non-zero
angular momentum. Since the effective potential is rotation invariant (independent of
the angles θ and ϕ) the above Hamilton operator commutes with the angular momentum
~ = 0. As we have seen earlier, the various components of the angular
operator, i.e. [H, L]
momentum vector, however, do not commute with each other. Still, they all commute
~ 2 ] = 0. We can now select Lz , L
~ 2 and
with the angular momentum squared, i.e. [Li , L
H as a set of mutually commuting operators. This means that we can diagonalize these
three operators simultaneously. In particular, we can chose the eigenfunctions of H to be
~ 2 . The eigenfunctions of Lz = −i~∂ϕ are simply
simultaneously eigenfunctions of Lz and L
exp(imϕ) with m ∈ Z as we have already seen for a particle on a circle. The eigenvalue of
this state is the z-component of angular momentum m~. The simultaneous eigenfunctions
~ 2 are known as spherical harmonics Ylm (θ, ϕ). One can show that
of Lz and L
~ 2 Ylm (θ, ϕ) = ~2 l(l + 1)Ylm (θ, ϕ), Lz Ylm (θ, ϕ) = ~mYlm (θ, ϕ).
L
(10.2.6)
118
CHAPTER 10. THE HYDROGEN ATOM
Here l ∈ Z and the possible values for m are given by m ∈ {−l, −l + 1, ..., l}. The simplest
spherical harmonics are given by
1
Y00 (θ, ϕ) = √ ,
4π
r
3
Y10 (θ, ϕ) =
cos θ,
4π
r
3
Y1±1 (θ, ϕ) = ∓
sin θ exp(±iϕ).
8π
It is beyond the scope of this course to discuss angular momentum in more detail. For
the moment we should accept the above result without a detailed proof. The proof is not
difficult, just time consuming. The states with angular momentum l = 0 are sometimes
also called s-states, states with l = 1 are called p-states, and states with l = 2 are so-called
d-states. The p-states are 3-fold, and the d-states are 5-fold degenerate because there are
2l + 1 possible m-values for a given angular momentum l. Since the Hamilton operator
commutes with the angular momentum vector, we can write its eigenfunctions as
ψ(~r) = R(r)Ylm (θ, ϕ).
Inserting this in the Schrödinger equation for the relative motion implies
2
2
~2 l(l + 1)
~
2
∂r + ∂r + V (r) +
R(r) = ER(r).
−
2µ
r
2µr2
(10.2.7)
(10.2.8)
This is a 1-dimensional Schrödinger equation for the radial motion. Note, however, that
the angular momentum l enters the equation via the centrifugal barrier term. Note also
that the z-component m of the angular momentum does not appear in the equation. This
again shows that all states with angular momentum l are (2l + 1)-fold degenerate because
there are that many spherical harmonics Ylm (θ, ϕ) all with the same eigenvalue l(l + 1).
10.3
Solution of the Radial Equation
Having reduced the problem to a 1-dimensional radial Schrödinger equation, we are now
ready to solve that equation. Up to this point we have only used the fact that the potential
V (r) is rotation invariant. From now on we will use the explicit Coulomb form
V (r) = −
Ze2
.
r
First we introduce the dimensionless variable
r
8µ|E|
ρ=
r.
~2
Then the radial equation takes the form
2
l(l + 1) λ 1
2
∂ρ + ∂ρ −
+ −
R(ρ) = 0,
ρ
ρ2
ρ 4
(10.3.1)
(10.3.2)
(10.3.3)
10.3. SOLUTION OF THE RADIAL EQUATION
where
Ze2
λ=
~
r
µ
.
2|E|
First we consider the large distance limit ρ → ∞. Then
1
2
R(ρ) = 0,
∂ρ −
4
119
(10.3.4)
(10.3.5)
such that R(ρ) ∼ exp(−ρ/2). This motivates the following ansatz
R(ρ) = G(ρ) exp(−ρ/2).
Inserting this ansatz in the above equation yields
2
l(l + 1) λ − 1
2
∂ρ +
− 1 ∂ρ −
+
G(ρ) = 0.
ρ
ρ2
ρ
At very small ρ we thus have
∂ρ2
l(l + 1)
2
G(ρ) = 0,
+ ∂ρ −
ρ
ρ2
(10.3.6)
(10.3.7)
(10.3.8)
which is solved by G(ρ) ∼ ρl . Therefore we also explicitly separate off this short distance
behavior and write
G(ρ) = H(ρ)ρl .
(10.3.9)
Inserting this in the above equation for G(ρ) we obtain
2l + 2
λ−1−l
2
∂ρ +
− 1 ∂ρ +
H(ρ) = 0.
ρ
ρ
(10.3.10)
We see that a constant H(ρ) = a0 solves the equation provided that λ = l + 1. The
corresponding wave function has no radial node, and thus corresponds to the state of
lowest energy for a given value of l. However, we can also make the ansatz H(ρ) = a0 +a1 ρ.
Then
2l + 2
λ−1−l
− 1 a1 +
(a0 + a1 ρ) = 0,
(10.3.11)
ρ
ρ
such that λ = l + 2 and (2l + 2)a1 + a0 = 0. This solution has one zero of the radial wave
function, and thus corresponds to the first excited state for a given value of l. Similarly,
one can find solutions with nr zeros of the radial wave function. In that case
λ = nr + l + 1 = n ∈ Z.
(10.3.12)
This implies for the quantized energy values
E=−
Z 2 e4 µ
.
2~2 n2
(10.3.13)
Based on rotation invariance one would expect that a state with angular momentum l is
(2l + 1)-fold degenerate. However, we can realize a given n with various nr and hence with
120
CHAPTER 10. THE HYDROGEN ATOM
various l. For example, for the first excited states with n = 2 we can have nr = 0, l = 1 or
nr = 1, l = 0. This means that an s-state and a p-state have the same energy (something
one would not expect based on rotation invariance alone). The s-state is 1-fold degenerate,
while the p-state is 3-fold degenerate, and hence the first excited state of hydrogen-like
atoms is in fact 4-fold degenerate. Generally, the state with quantum number n is n2 fold degenerate. This high degeneracy, which one would not expect based on rotation
invariance alone, is due to the fact that all bound classical orbits in the Coulomb potential
are closed curves (ellipses).
10.4
Relativistic Corrections
At this point we have reached our goal of understanding quantitatively the simplest atom
— hydrogen. We have gained some insight in the quantum world, which is indeed consistent with experiments. Still, the non-relativistic 1/r potential problem does not exactly
represent the real hydrogen atom. There are small relativistic corrections, which we want
to mention at the end of the course. The Schrödinger equation provides a non-relativistic
description, while we know that Nature is relativistic. The correct relativistic equation
for the electron is the so-called Dirac equation, which follows from the Hamilton operator
H=α
~ · p~c + βMe c2 + V (r).
(10.4.1)
Here
α
~=
0 ~σ
~σ 0
, β=
I 0
0 −I
,
are 4 × 4 matrices with
1 0
0 −i
0 1
1 0
,
, σz =
, σy =
, σx =
I=
0 −1
i 0
1 0
0 1
(10.4.2)
(10.4.3)
as 2 × 2 submatrices. The Dirac Hamilton operator acts on a 4-component wave function


Ψ1 (~r)
 Ψ2 (~r) 

Ψ(~r) = 
 Ψ3 (~r)  ,
Ψ4 (~r)
(10.4.4)
a so-called Dirac spinor. The upper components Ψ1 (~r) and Ψ2 (~r) describe electrons, and
the lower components Ψ3 (~r) and Ψ4 (~r) describe positrons — the anti-particles of the
electron. The fact that we need a 2-component wave function to describe the electron is
related to a property called spin, which represents the intrinsic angular momentum of the
electron, which is independent of its orbital motion. In fact, the above Pauli matrices ~σ
are related to the spin
~ = ~ ~σ .
S
(10.4.5)
2
10.4. RELATIVISTIC CORRECTIONS
121
~ obey angular momentum
Indeed, it is straightforward to show that the components of S
commutation relations. For example
~2
0 1
0 −i
0 −i
0 1
[Sx , Sy ] =
(
−
)
1 0
i 0
i 0
1 0
4
~2
2i 0
= i~Sz .
(10.4.6)
=
0 −2i
4
Also
2
~2 = ~ (
S
4
0 1
1 0
2
+
0 −i
i 0
2
+
1 0
0 −1
2
)=
3~2
I.
4
(10.4.7)
Writing this as s(s + 1)~2 we identify s = 1/2, i.e. the Pauli matrices describe an angular
momentum 1/2 with z-components Sz = m~ and m = ±1/2. This additional intrinsic
quantum number of the electron leads to a doubling of the whole hydrogen spectrum. In
fact, each state that we have discussed previously can now occur with spin up (m = 1/2)
and with spin down (m = −1/2). This effect survives even in the extreme non-relativistic
limit. Hence the degeneracy of hydrogen states with quantum number n is indeed 2n2 .
Spin is not only an extra label on the electron. It also influences the dynamics via
relativistic effects. The most important of these is due to the so-called spin-orbit coupling.
When one reduces the Dirac equation to the upper electron components one finds an extra
contribution to the potential which has the form
Vso (r) =
Ze2 ~ ~
S · L,
2Me2 c2 r3
(10.4.8)
involving the scalar product of spin and orbital angular momentum. The s-states are
unaffected by this term because they have l = 0, but p-states are energetically shifted due
to the spin-orbit coupling. With the above term in the Hamilton operator, the orbital
angular momentum no longer commutes with the Hamiltonian. Still, the total angular
momentum
~ +L
~
J~ = S
(10.4.9)
does commute with H reflecting the fact that the problem is still rotation invariant. The
total angular momentum J~ also obeys angular momentum commutation relations, and
hence the eigenvalues of J~2 are given by ~2 j(j + 1). For a given value l of the orbital
angular momentum one can construct states with j = l ± 1/2. We can now write
such that
~ 2 + 2S
~ ·L
~ +L
~ 2,
J~2 = S
(10.4.10)
~2
3
~
~
S·L=
j(j + 1) − l(l + 1) −
.
2
4
(10.4.11)
~ ·L
~ = ~2 l/2 and is shifted to higher energies,
Consequently, the state with j = l + 1/2 has S
2
~
~
while the state with j = l−1/2 has S · L = −~ (l+1)/2 such that its energy is lowered. The
actual spin-orbit energy shifts in the hydrogen atom are very small, but experimentally
clearly verified.
122
CHAPTER 10. THE HYDROGEN ATOM
One can solve the Dirac equation for hydrogen-like atoms just like we have solved the
Schrödinger equation. The resulting energy spectrum is given by
v
u
α2
E = Me c2 u
(10.4.12)
i2 .
t1 + h
p
n + (j + 1/2)2 − α2
Here α = e2 /~c is the fine-structure constant and again n ∈ Z. Even in this spectrum
there is more degeneracy than one would expect based on rotation invariance alone. Still,
due to spin-orbit couplings the degeneracy is now greatly reduced compared to the nonrelativistic calculation.
In the real hydrogen atom there are even more subtle effects that the above Dirac
Hamilton operator does not contain. For example, the atomic nucleus also has a spin,
which interacts with the spin of the electron. This gives rise to extremely small (but observable) energy shifts known as the hyperfine structure of the hydrogen spectrum. Finally,
a full quantum treatment of the hydrogen atom must also quantize the electromagnetic
field that mediates the Coulomb interaction between electron and proton. Then we are
entering the subject of Quantum Electrodynamics (QED) which is far beyond the scope
of this course. Still, it is worth mentioning that a tiny energy shift — the so-called Lamb
shift — is an observable consequence of QED in the hydrogen spectrum. Also this effect
finally lifts all the degeneracies that we cannot understand based on rotation invariance
alone. In the real hydrogen atom with all its tiny relativistic corrections the classical orbit
of an electron would finally no longer be closed.
Chapter 11
Einstein-Podolsky-Rosen Paradox,
Bell’s Inequality, and
Schrödinger’s Cat
In this chapter we investigate the foundations of quantum mechanics. In particular, we
discuss the Einstein-Podolsky-Rosen (EPR) paradox, we ask whether some hidden classical
variables might underlie quantum physics, and we talk about Schrödinger’s famous cat.
11.1
The Einstein-Podolsky-Rosen Paradox
By now we have a solid knowledge of quantum mechanics. We have understood the formal
structure of the theory, the statistical interpretation of the wave function, and its relation
to actual measurements. In particular, we have learned how to do quantum mechanics
by solving the Schrödinger equation for several systems of physical interest. Hopefully,
we have gained some confidence in our ability to use quantum mechanics for describing
concrete physical phenomena, and perhaps we have also gained some intuition for what
quantum mechanics does. Although we have learned how quantum mechanics works, have
we understood what it really means? As Feynman once said: “I think I can safely say
that nobody understands quantum mechanics.” So, we should not feel too bad if we don’t
either. In order to understand better what it is that is so hard to understand, we will now
discuss the Einstein-Podolsky-Rosen paradox.
Einstein was convinced that quantum mechanics cannot be the whole story. In particular, the probabilistic nature of quantum reality was unacceptable for him. Of course,
he had to acknowledge that quantum mechanics correctly describes Nature at microscopic
scales, but he believed that it must be incomplete. Instead of assuming that there is nothing more to know about a quantum system than its wave function, he believed that there
are elements of reality that quantum mechanics simply does not “know” about. Following
123
124CHAPTER 11. EPR PARADOX, BELL’S INEQUALITY, AND SCHRÖDINGER’S CAT
Einstein, people have attempted to describe such elements of reality by so-called hidden
variables. These attempts ended when Bell showed that local theories of hidden variables
are inconsistent with quantum mechanics.
In order to argue that quantum mechanics is incomplete, in 1935 Einstein, Podolsky,
and Rosen discussed the following “paradox”. Consider the decay of a spinless particle,
e.g. a neutral pion at rest, into two particles with spin 1/2, e.g. an electron and a positron.
After the decay, the electron is moving in one direction, and, as a consequence of momentum conservation, the positron is moving in the opposite direction. Angular momentum
conservation implies that the spin of the electron plus the spin of the positron add up to
the spin zero of the neutral pion. Hence, either the electron has spin up and the positron
has spin down, or the electron has spin down and the positron has spin up. The spin part
of the wave function takes the form
1
|Ψi = √ (| ↑↓i − | ↓↑i).
2
(11.1.1)
~=S
~e + S
~p ,
S
(11.1.2)
The total spin operator
~e = ~ ~σe of the electron and the spin S
~p = ~ ~σp of the positron. We
is the sum of the spin S
2
2
have
σe3 | ↑↓i = | ↑↓i, σp3 | ↑↓i = −| ↑↓i, σe3 | ↓↑i = −| ↓↑i, σp3 | ↓↑i = | ↓↑i,
(11.1.3)
and hence
1
σ3 |Ψi = (σe3 + σp3 ) √ (| ↑↓i − | ↓↑i) = 0,
2
i.e. the total spin indeed vanishes.
(11.1.4)
Let us imagine that the electron travels a large distance until it reaches a detector that
measures its spin. Simultaneously, the positron travels a large distance in the opposite
direction until it reaches another detector that measures its spin. If the electron spin is
measured to be up, the positron spin must be measured to be down, and vice versa. Experiments of this kind have actually been done, and indeed the measured spins always point in
opposite directions. What did Einstein find paradoxical about this situation? According
to the standard interpretation of quantum mechanics, the outcome of the measurement
of the electron spin is uncertain until it is actually being performed. In particular, both
spin up and spin down are equally probable. Only the measurement itself forces the electron “to make up its mind” and choose one of the two results from the given statistical
distribution. Once the electron spin has been measured, the result of the measurement of
the positron spin is completely determined. However, the positron is by now so far away
that no light signal can communicate the result of the measurement of the electron spin
before the measurement of the positron spin is actually being performed. Still, the measured positron spin is certainly opposite to the one of the electron. Einstein believed that
immediately after the pion decay, the electron and positron are equipped with a hidden
variable that determines whether they have spin up or down. Hence, before the electron
spin is actually measured, the result of the measurement is already pre-determined by the
11.2. THE QUANTUM MECHANICS OF SPIN CORRELATIONS
125
hidden variable. In particular, in order to ensure that the positron has opposite spin, no
action needs to travel faster than light. Since the hidden variable is not part of quantum
mechanics, Einstein, Podolsky, and Rosen argued that quantum mechanics is incomplete.
11.2
The Quantum Mechanics of Spin Correlations
In 1964 Bell showed that local hidden variable theories are inconsistent with quantum
mechanics. Since the predictions of quantum mechanics are indeed in agreement with
experiment, this means that Nature is fundamentally non-local. No wonder that Einstein
found that unacceptable. Still, there seems to be no way around this. In order to appreciate what Bell argued, we first need to understand what quantum mechanics has to
say about spin correlations. Bell suggested to consider measurements of the electron and
positron spins along different quantization axes defined by the unit-vectors ~a and ~b. He
then considered the expectation value of the product of the two measured spin values
P (~a, ~b) = h(~σe · ~a)(~σp · ~b)i.
(11.2.1)
Let us evaluate this expectation value for the state |Ψi from above. First, we consider
hσe3 σp3 i = −1.
(11.2.2)
hσe1 σp1 i = hσe2 σp2 i = −1.
(11.2.3)
Similarly, one obtains
Next, we consider
1
hσe1 σp3 i = h (σe+ + σe− )σp3 i = 0.
2
These results can be summarized as
(11.2.4)
hσei σpj i = −δij ,
(11.2.5)
P (~a, ~b) = −~a · ~b.
(11.2.6)
which implies
In particular, if the two quantization axes are parallel (~a = ~b), the product of the two
measured spins is always −1. If the axes are anti-parallel (~a = −~b), on the other hand,
the product is 1.
11.3
A Simple Hidden Variable Model
We will now consider a simple concrete example of a hidden variable theory. As we
will see, this hidden variable model is inconsistent with the above results of quantum
mechanics, which in turn are consistent with experiment. Although the hidden variable
model is hence ruled out by experiment, it may illustrate the general ideas behind hidden
126CHAPTER 11. EPR PARADOX, BELL’S INEQUALITY, AND SCHRÖDINGER’S CAT
variable theories. We consider a particular hidden variable model that is classical but
still probabilistic. It assumes that the quantum spins of the electron and the positron are
determined by corresponding classical spin vectors ±~λ of unit length, which are a concrete
realization of a hidden variable. We assume that in the moment of the decay the electron
is endowed with a classical hidden spin variable ~λ which points in a random direction,
while the positron is endowed with the classical spin −~λ. In this model, the measurement
of the spin of the electron along the quantization axis ~a gives
se (~a, ~λ) = sign(~a · ~λ) = ±1,
(11.3.1)
while the measurement of the spin of the positron along the quantization axis ~b gives
sp (~b, ~λ) = −sign(~b · ~λ) = ±1.
(11.3.2)
In other words, the quantum spin is +1 (in units of ~2 ) if the classical spin has an angle
less than π/2 with the quantization axis, and −1 otherwise. It is typical of a hidden
variable theory that the measurement of the quantum spin ±1 does not reveal all details
of the hidden variable — in this case the direction of the classical spin ~λ. In this sense,
if the hidden variable theory were correct, there would be certain elements of reality that
quantum mechanics does not “know” about. Let us first consider two special cases: ~a and
~b parallel or anti-parallel. For ~a = ~b, we have
se (~a, ~λ) = −sp (~a, ~λ) ⇒ P (~a, ~a) = −1,
(11.3.3)
se (~a, ~λ) = sp (−~a, ~λ) ⇒ P (~a, −~a) = 1.
(11.3.4)
and for ~a = −~b
At least these special cases are consistent with quantum mechanics and thus with experiment. It is easy to show that, for general ~a and ~b, our hidden variable model predicts
2
P (~a, ~b) = acos(~a · ~b) − 1.
π
(11.3.5)
Obviously, this contradicts quantum mechanics and is thus inconsistent with experiment.
As we will see in the next section, there is a general proof (using Bell’s inequality) which
shows that, not only this particular hidden variable model, but any local theory of hidden
variables necessarily disagrees with quantum mechanics and thus with experiment.
11.4
Bell’s Inequality
Let us assume that there are some hidden variables λ that pre-determine the spins of the
electron and the positron in the moment they are created in the decay of the neutral pion.
The result of the spin measurement of the electron, se (~a, λ) = ±1 (in units of ~2 ), depends
only on the quantization axis ~a and on the value of the hidden variables λ. Similarly, the
measured spin of the positron, sp (~a, λ) = ±1, depends on the quantization axis ~b and λ.
11.4. BELL’S INEQUALITY
127
If the quantization axes are chosen to be parallel (~a = ~b), it is an experimental fact that
the results of the two measurements are perfectly anti-correlated, i.e.
se (~a, λ) = −sp (~a, λ).
(11.4.1)
We want to assume that the hidden variables are distributed according to some arbitrary
probability distribution ρ(λ) ≥ 0 which is normalized as
Z
dλ ρ(λ) = 1,
(11.4.2)
so that
P (~a, ~b) =
Z
dλ ρ(λ)se (~a, λ)sp (~b, λ) = −
Z
dλ ρ(λ)se (~a, λ)se (~b, λ).
(11.4.3)
Next, we consider
Z
h
i
dλ ρ(λ) se (~a, λ)se (~b, λ) − se (~a, λ)se (~c, λ)
Z
h
i
=
dλ ρ(λ) 1 − se (~b, λ)se (~c, λ) se (~a, λ)se (~b, λ).
P (~a, ~c) − P (~a, ~b) =
(11.4.4)
In the last step, we have used se (~b, λ)2 = 1. Since
we have
se (~a, λ)se (~b, λ) = ±1,
(11.4.5)
h
i
ρ(λ) 1 − se (~b, λ)se (~c, λ) ≥ 0,
(11.4.6)
and hence
Z
h
i
dλ ρ(λ) 1 − se (~b, λ)se (~c, λ) |se (~a, λ)se (~b, λ)|
Z
h
i
=
dλ ρ(λ) 1 − se (~b, λ)se (~c, λ) = 1 + P (~b, ~c).
(11.4.7)
|P (~a, ~c) − P (~a, ~b)| ≤
This is Bells’ inequality.
It is remarkable that this result — derived for a general theory of local hidden variables
— is actually inconsistent with the predictions of quantum mechanics. This is easy to see.
Let us assume that the three quantization axes ~a, ~b, and ~c are in the xy-plane with a 90
degrees angle between ~a and ~b, and with 45 degrees angles between ~a and ~c and between
~b and ~c. Then
1
P (~a, ~b) = 0, P (~a, ~c) = P (~b, ~c) = − √ ,
(11.4.8)
2
so that
1
1
|P (~a, ~c) − P (~a, ~b)| = √ > 1 − √ = 1 + P (~b, ~c).
2
2
(11.4.9)
128CHAPTER 11. EPR PARADOX, BELL’S INEQUALITY, AND SCHRÖDINGER’S CAT
Obviously, this violates Bell’s inequality. Consequently, the results of quantum mechanics
(which are in excellent agreement with experiment) are inconsistent with any underlying
theory of local hidden variables. Most physicists concluded from this results that hidden
variables are simply not the correct idea. People who fell in love with hidden variables
had to admit that they cannot be local. Thus, in any case, Bell’s inequality tells us that
Nature is fundamentally non-local at the quantum level.
Einstein was not amused by the “spooky action at a distance” that happens when the
wave function collapses in the process of a measurement. Although no light signal could
possibly communicate the result of the measurement of the electron spin, the measurement
of the positron spin is always perfectly anti-correlated. Still, this apparent violation of
causality is not as catastrophic as it may seem. In particular, the observer of the electron
spin cannot use the “spooky” correlation to transmit information to the observer of the
positron with velocities faster than light. This is because she has no way to influence the
result of her spin measurement. After a long sequence of measurements, both observers
simply get a long list of results ±~/2 which are perfectly random. Only when they compare
their lists at the end of the day, they realize that both lists are identical, up to an overall
minus sign.
11.5
Schrödinger’s Cat
Let us also discuss the (in)famous experiment with Schrödinger’s cat. The cute animal
is put in a box together with a container of cyanide, a hammer, a Geiger counter, and
a single unstable atomic nucleus. The life-time of the nucleus is such that during one
hour it decays with a probability of 50 percent. When this happens, the Geiger counter
registers the event, activates the hammer, which hits the container, releases the cyanide,
and kills the cat. On the other hand, with the other 50 percent probability, nothing
happens. Before we open the box, we do not know whether the cat is dead or alive. In
fact, quantum mechanically, the cat is in a quantum state
1
|Ψi = √ (|deadi + |alivei).
2
(11.5.1)
Here we have again used Dirac’s bracket notation which will be explained in the next
section. Of course, Dirac can also not help to make sense of |Ψi. A macroscopic object
like a cat can hardly be in a linear superposition of dead and alive.
In any case, Schrödinger then puts us in the following situation. When we open the
box and look at the cat, we make the measurement that forces the cat to “make up its
mind” whether it wants to be dead or alive. In some sense, we kill the cat (with 50 percent
probability) by opening the box. The delicacy of quantum measurements becomes obvious
when situations are driven to such extremes. Is it really us who decides the fate of the
cat? Probably not. We can blame everything on the Geiger counter. Once it registers the
radioactive decay, a classical event has happened and the fate of the cat is determined.
When we open the box after one hour, the thing has long happened. However, this thought
11.5. SCHRÖDINGER’S CAT
129
experiment should still bother us. There may be something very deep about the quantum mechanical measurement process that we presently do not comprehend. This cannot
stop us from doing quantum mechanics by using what we have learned to describe the
physics at microscopic scales. However, I invite you to remain curious and open about
new ideas, even if they may sound crazy at first. In this spirit, let us allow the big question:
Are there hidden degrees of freedom underlying quantum physics?
130CHAPTER 11. EPR PARADOX, BELL’S INEQUALITY, AND SCHRÖDINGER’S CAT
Chapter 12
Abstract Formulation of Quantum
Mechanics
12.1
Dirac’s Bracket Notation
In chapter 7 we have investigated the formal structure of quantum mechanics. In particular, we found that wave functions can be viewed as unit-vectors in an abstract infinitedimensional Hilbert space, and that physical observables are described by Hermitean operators acting on those vectors. The value of a wave function Ψ(x) at the point x can be
viewed as a component of a vector with infinitely many components — one for each space
point. From ordinary vector algebra, we know that it is useful to describe vectors in a
basis-free notation ~a instead of using the Cartesian coordinate representation (a1 , a2 , a3 ).
For wave functions in the Hilbert space we have thus far used the position space representation Ψ(x) or the momentum space representation Ψ(k), and we will now proceed to
the basis-free abstract notation |Ψi introduced by Dirac. In chapter 7 we have already
written the scalar product of two wave functions Ψ(x) and Φ(x) as hΦ|Ψi. Dirac dissects
this bracket expression into a “bra”-vector hΦ| and a “ket”-vector |Ψi. As in ordinary
vector algebra, Dirac’s basis-free notation is quite useful in many calculations in quantum
mechanics. It should be pointed out that we are not really learning anything new in this
section. We are just reformulating what we already know in a more compact notation.
Let us investigate the analogy with ordinary vector algebra in some detail. The scalar
product of two ordinary vectors ~a = (a1 , a2 , a3 ) and ~b = (b1 , b2 , b3 ) takes the form
ha|bi = ~a · ~b = a1 b1 + a2 b2 + a3 b3 ,
(12.1.1)
and the norm of a vector can be written as
||a||2 = ha|ai = ~a · ~a = a21 + a22 + a23 .
131
(12.1.2)
132
CHAPTER 12. ABSTRACT FORMULATION OF QUANTUM MECHANICS
Similarly, the scalar product of two wave functions takes the form
Z
hΦ|Ψi = dx Φ(x)∗ Ψ(x),
(12.1.3)
and the norm of a wave function is given by
Z
Z
||Ψ||2 = hΨ|Ψi = dx Ψ(x)∗ Ψ(x) = dx |Ψ(x)|2 .
(12.1.4)
Introducing the ordinary unit-vectors ~i in the Cartesian i-direction one can write
ai = hi|ai = ~i · ~a,
(12.1.5)
i.e. the component ai of the basis-free abstract vector ~a in the Cartesian basis of unitvector ~i is simply the projection onto those basis vectors. Similarly, one can introduce
abstract vectors |xi as analogs of ~i in the quantum mechanical Hilbert space. The ketvectors |xi are so-called position eigenstates. They describe the wave function of a particle
completely localized at the point x. The usual wave function Ψ(x) is the analog of the
Cartesian component ai , and the abstract ket-vector |Ψi is the analog of ~a. In analogy to
eq.(12.1.5) one can now write
Ψ(x) = hx|Ψi,
(12.1.6)
i.e. the ordinary wave function Ψ(x) is a component of the abstract state vector |Ψi in the
basis of position eigenstates |xi and is again given by the projection hx|Ψi.
Obviously, the j-component of the Cartesian unit vectors ~i (in Cartesian coordinates)
is given by
ij = ~j · ~i = hj|ii = δij .
(12.1.7)
Similarly, the position eigenstates obey
hx0 |xi = δ(x − x0 ).
(12.1.8)
Since the position x is a continuous variable, the Kronecker δ for the discrete indices i
and j is replaced by a Dirac δ-function. Since the unit-vectors ~i form a complete basis of
orthogonal vectors, one can write the 3 × 3 unit-matrix as a sum of tensor products
X
X
ij ik =
δij δik = δjk = 1jk .
(12.1.9)
i
i
In a basis-free notation one can write this equation as
X
(~i ⊗ ~i) = 1.
(12.1.10)
i
Here the direct tensor product ⊗ turns two 3-component vectors into a 3 × 3 matrix. In
basis-free notation the corresponding equation for position eigenstates in the Hilbert space
takes the form
Z
dx |xihx| = 1,
(12.1.11)
12.1. DIRAC’S BRACKET NOTATION
133
where the sum over the discrete index i is replaced by an integral over the continuous
coordinate x. In the basis of position eigenstates the same equation reads
Z
Z
0
00
dx hx |xihx|x i = dx δ(x − x0 )δ(x − x00 ) = δ(x0 − x00 ) = hx0 |1|x00 i.
(12.1.12)
Using the completeness relation eq.(12.1.11) we can write
Z
Z
|Ψi = 1|Ψi = dx |xihx|Ψi = dx Ψ(x)|xi,
(12.1.13)
i.e. the wave function Ψ(x) = hx|Ψi is the component of an abstract state vector |Ψi in
the complete orthonormal basis of position eigenstates |xi. Multiplying this equation with
hx0 | one consistently obtains
Z
Z
0
0
hx |Ψi = dx Ψ(x)hx |xi = dx Ψ(x)δ(x − x0 ) = Ψ(x0 ).
(12.1.14)
The two equations for ordinary vectors corresponding to eqs.(12.1.13,12.1.14) take the
form
X
X
X
~a = 1~a =
(~i ⊗ ~i)~a =
(~i · ~a)~i =
ai~i,
(12.1.15)
i
i
i
and
~j · ~a =
X
ai~j · ~i =
X
i
ai δij = aj .
(12.1.16)
i
Just as the Cartesian basis of unit-vectors ~i is not the only basis in real space, the
basis of position eigenstates |xi is not the only basis in Hilbert space. In particular, the
momentum eigenstates |ki (with p = ~k) also form an orthonormal complete basis. In
this case we have
Z
1
dk |kihk| = 1.
(12.1.17)
2π
Hence, we can also write
1
|Ψi = 1|Ψi =
2π
Z
dk |kihk|Ψi.
(12.1.18)
This raises the question how the object hk|Ψi defined in the basis of momentum eigenstates
|ki is related to the familiar wave function Ψ(x) = hx|Ψi in the basis of position eigenstates
|xi. Expressed in the basis of position eigenstates, the momentum eigenstates take the
form
i
hx|ki = exp(ikx) = exp
px ,
(12.1.19)
~
i.e. they are the familiar wave functions of free particles with momentum p = ~k. Hence,
one obtains consistently
Z
Z
1
1
hx|Ψi = hx|1|Ψi =
dk hx|kihk|Ψi =
dk Ψ(k) exp(ikx) = Ψ(x). (12.1.20)
2π
2π
134
CHAPTER 12. ABSTRACT FORMULATION OF QUANTUM MECHANICS
We identify the object hk|Ψi as the momentum space wave function Ψ(k) that we encountered earlier as the Fourier transform of the position space wave function Ψ(x). Similarly,
one obtains
hk|xi = hx|ki∗ = exp(−ikx).
(12.1.21)
and thus
hk|Ψi = hk|1|Ψi =
Z
Z
dx hk|xihx|Ψi =
dx Ψ(x) exp(−ikx) = Ψ(k).
(12.1.22)
We are already familiar with Hermitean operators A representing physical observables.
We are also familiar with the object
Z
hΦ|AΨi = dx Φ(x)∗ AΨ(x).
(12.1.23)
Now we consider the operator A in a basis-free notation and write
Z
Z
hΦ|A|Ψi = hΦ|1A|Ψi = dx hΦ|xihx|A|Ψi = dx Φ(x)∗ hx|A|Ψi.
(12.1.24)
Hence, we easily identify hx|A|Ψi = AΨ(x).
As we have seen earlier, the eigenfunctions χn (x) that solve the eigenvalue problem
Aχn (x) = an χn (x) form a complete orthonormal basis in Hilbert space. In basis-free
notation this is expressed as
A|χn i = an |χn i,
(12.1.25)
with
hχm |χn i = δmn ,
and
(12.1.26)
|χn ihχn | = 1.
(12.1.27)
An arbitrary state |Ψi can be decomposed as
X
|Ψi =
cn |χn i.
(12.1.28)
X
n
n
Multiplying this equation with hχm | we obtain
X
X
hχm |Ψi =
cn hχm |χn i =
cn δmn = cm .
n
(12.1.29)
n
We now consider the eigenvalue problem of the Hamilton operator
H|χn i = En |χn i,
(12.1.30)
which is nothing but the time-independent Schrödinger equation. For example, the Hamilton operator for a single particle moving in a potential V (x)
H = T + V,
(12.1.31)
12.1. DIRAC’S BRACKET NOTATION
135
separates into the kinetic energy operator T and the potential energy operator V. In a
basis of coordinate eigenstates one has
hx|T|χn i = −
~2 d2 χn (x)
,
2M dx2
(12.1.32)
as well as
hx|V|x0 i = V (x)δ(x − x0 ) ⇒ hx|V|χn i =
Z
dx hx|V|x0 ihx|χn i = V (x)χn (x),
(12.1.33)
and hence
hx|H|χn i = hx|T + V|χn i = −
~2 d2 χn (x)
+ V (x)χn (x) = En χn (x).
2M dx2
(12.1.34)
In basis-free Dirac notation the time-dependent Schrödinger equation (with a Hamiltonian
that is time-independent) takes the form
i~∂t |Ψ(t)i = H|Ψ(t)i.
(12.1.35)
X
(12.1.36)
Expanding
|Ψ(t)i =
cn (t)|χn i,
n
in the basis of stationary energy eigenstates |χn i one obtains
cn (t) = hχn |Ψ(t)i.
(12.1.37)
Thus, multiplying the time-dependent Schrödinger equation with hχn | one finds
i~∂t cn (t) = hχn |H|Ψ(t)i =
X
hχn |H|χm ihχm |Ψ(t)i
m
=
X
Em δmn hχm |Ψ(t)i = En cn (t),
(12.1.38)
m
which implies
i
cn (t) = cn (0) exp − En t ,
~
(12.1.39)
and hence
|Ψ(t)i =
X
n
i
hχn |Ψ(0)i exp − En t |χn i.
~
(12.1.40)
It should be pointed out again that all results of this section have been derived in earlier
chapters. We have just presented them in the more compact basis-free Dirac notation.
136
CHAPTER 12. ABSTRACT FORMULATION OF QUANTUM MECHANICS
12.2
Unitary Time-Evolution Operator
The time-dependent Schrödinger equation can be solved formally by writing
i
|Ψ(t)i = exp − H(t − t0 ) |Ψ(t0 )i,
~
(12.2.1)
because then indeed
i
i~∂t |Ψ(t)i = H exp − H(t − t0 ) |Ψ(t0 )i = H|Ψ(t)i.
~
(12.2.2)
The operator
i
U(t, t0 ) = exp − H(t − t0 ) ,
(12.2.3)
~
is known as the time-evolution operator because it determines the time-evolution
|Ψ(t)i = U(t, t0 )|Ψ(t0 )i,
(12.2.4)
of the physical state |Ψ(t)i starting from an initial state |Ψ(t0 )i. The time-evolution
operator is unitary, i.e.
U(t, t0 )† U(t, t0 ) = U(t, t0 )U(t, t0 )† = 1.
(12.2.5)
This property follows from the Hermiticity of the Hamiltonian because H† = H implies
i
i †
†
(12.2.6)
U(t, t0 ) = exp H (t − t0 ) = exp H(t − t0 ) = U(t, t0 )−1 .
~
~
Unitarity of the time-evolution operator guarantees conservation of probability because
hΨ(t)|Ψ(t)i = hΨ(t0 )|U(t, t0 )† U(t, t0 )|Ψ(t0 )i = hΨ(t0 )|Ψ(t0 )i = 1.
12.3
(12.2.7)
Schrödinger versus Heisenberg Picture
S far we have formulated quantum mechanics in the way it was first done by Schrödinger. In his approach the Hermitean operators that describe physical observables are
time-independent and the time-dependence of the wave function is obtained by solving
the time-dependent Schrödinger equation. At the end, the results of measurements of an
observable quantity A are given as expectation values hΨ(t)|A|Ψ(t)i which inherit their
time-dependence from the wave function while the operator A itself is time-independent.
Heisenberg had a different (but equivalent) picture of quantum mechanics. He prefered
to think of the wave function as being time-independent, i.e. |Ψ(t)i = |Ψ(0)i and attribute
the time-dependence to the operators. In particular, a time-dependent Hermitean operator A(t) in the Heisenberg picture is related to the time-independent operator A in the
Schrödinger picture by a unitary transformation with the time-evolution operator
A(t) = U(t, t0 )† AU(t, t0 ).
(12.3.1)
12.4. TIME-DEPENDENT HAMILTON OPERATORS
137
In the Heisenberg picture, the result of a measurement is given by
hΨ(t0 )|A(t)|Ψ(t0 )i = hΨ(t0 )|U(t, t0 )† AU(t, t0 )|Ψ(t0 )i = hΨ(t)|A|Ψ(t)i,
(12.3.2)
which agrees with Schrödinger’s result. Since Heisenberg worked with time-dependent operators he needed an analog of the Schrödinger equation to describe the operator dynamics.
One finds
∂t A(t) = ∂t [U(t, t0 )† AU(t, t0 )]
= ∂t U(t, t0 )† AU(t, t0 ) + U(t, t0 )† A∂t U(t, t0 )
i
i
i
=
HU(t, t0 )† AU(t, t0 ) − U(t, t0 )† A HU(t, t0 ) = [H, A(t)].
~
~
~
(12.3.3)
This is the so-called Heisenberg equation of motion. It is physically completely equivalent
to the Schrödinger equation. It should be noted that the Hamilton operator itself remains
time-independent in the Heisenberg picture, i.e.
H(t) = U(t, t0 )† HU(t, t0 ) = H.
(12.3.4)
This is a consequence of energy conservation.
It should be noted that the time-dependent operator A(t) in the Heisenberg picture is
Hermitean at all times if the time-independent operator in the Schrödinger picture A is
Hermitean, i.e.
A(t)† = [U(t, t0 )† AU(t, t0 )]† = U(t, t0 )† A† U(t, t0 ) = U(t, t0 )† AU(t, t0 ) = A(t).
(12.3.5)
Furthermore, the commutation relations of two operators A(t) and B(t) are covariant under
time-evolution, i.e.
[A(t), B(t)] = [U(t, t0 )† AU(t, t0 ), U(t, t0 )† BU(t, t0 )]
= U(t, t0 )† [A, B]U(t, t0 ) = [A, B](t).
12.4
(12.3.6)
Time-dependent Hamilton Operators
Until now, we have only considered closed systems with time-independent Hamilton operators. Interestingly, as a consequence of energy conservation, the Hamilton operator
remained time-independent even in the Heisenberg picture. For open quantum systems
that are influenced by external forces, on the other hand, time-translation invariance is
broken and, consequently, energy is in general no longer conserved. In that case one has
a time-dependent Hamilton operator already in the Schrödinger picture. The Schrödinger
equation then takes the form
i~∂t |Ψ(t)i = H(t)|Ψ(t)i.
(12.4.1)
Again, we introduce a time-evolution operator U(t, t0 ) that evolves the quantum state
|Ψ(t)i = U(t, t0 )|Ψ(t0 )i,
(12.4.2)
138
CHAPTER 12. ABSTRACT FORMULATION OF QUANTUM MECHANICS
from an initial time t0 to a later time t. Inserting eq.(12.4.2) into the Schrödinger equation
one obtains
i~∂t U(t, t0 ) = H(t)U(t, t0 ).
(12.4.3)
A formal integration of this equation yields
Z
i t
U(t, t0 ) = 1 −
dt1 H(t1 )U(t1 , t0 ).
~ t0
Iterating this equation one obtains
Z tn−1
Z t1
Z
∞
X
i n t
dtn H(t1 )H(t2 )...H(tn ).
dt2 ...
dt1
U(t, t0 ) = 1 +
(− )
~
t0
t0
t0
(12.4.4)
(12.4.5)
n=1
In particular, for a time-independent Hamilton operator H one again obtains
Z
Z t1
Z tn−1
∞
X
i n t
U(t, t0 ) = 1 +
(− )
dt1
dt1 ...
dtn Hn
~
t0
t0
t0
n=1
Z
Z
Z t
∞
t
t
X i 1
= 1+
(− )n
dt1
dt2 ...
dtn Hn
~ n! t0
t0
t0
n=1
i
= exp − H(t − t0 ) .
~
(12.4.6)
Even for general time-dependent (Hermitean) Hamilton operators the time-evolution operator is unitary. Taking the Hermitean conjugate of eq.(12.4.3) one finds
−i~∂t U(t, t0 )† = U(t, t0 )† H(t).
(12.4.7)
Thus, we have
i~∂t [U(t, t0 )† U(t, t0 )] = −U(t, t0 )† H(t)U(t, t0 ) + U(t, t0 )† H(t)U(t, t0 ) = 0,
and, consequently,
12.5
U(t, t0 )† U(t, t0 )) = U(t0 , t0 )† U(t0 , t0 )) = 1.
(12.4.8)
(12.4.9)
Dirac’s Interaction picture
In the Schrödinger picture of quantum mechanics the time-dependence resides in the wave
function and possibly in an explicitly time-dependent Hamiltonian. In the Heisenberg
picture, on the other hand, the time-dependence resides in the operators. Dirac introduced
the so-called interaction picture which is a hybrid of the Schrödinger and Heisenberg
pictures and has time-dependence in both the wave function and the operators. The
Dirac picture is useful when a closed system with time-independent Hamilton operator H0
is influenced by time-dependent external forces described by the interaction Hamiltonian
V(t) such that the total Hamilton operator
H(t) = H0 + V(t),
(12.5.1)
12.5. DIRAC’S INTERACTION PICTURE
139
is indeed time-dependent. In this case, one may perform a unitary transformation with
the time-evolution operator
i
U0 (t, t0 ) = exp[− H0 (t − t0 )],
~
(12.5.2)
of the unperturbed closed system. This transformation turns the state |Ψ(t)i of the
Schrödinger picture into the state
|ΨD (t)i = U0 (t, t0 )† |Ψ(t)i,
(12.5.3)
in Dirac’s interaction picture. Similarly, an operator A in the Schrödinger picture turns
into the operator
AD (t) = U0 (t, t0 )† AU0 (t, t0 ),
(12.5.4)
in the Dirac picture. For example, one finds
H0D (t) = U0 (t, t0 )† H0 U0 (t, t0 ) = H0 , VD (t) = U0 (t, t0 )† V(t)U0 (t, t0 ).
(12.5.5)
In the interaction picture the Schrödinger equation takes the form
i~∂t |ΨD (t)i = i~∂t [U0 (t, t0 )† |Ψ(t)i]
= −U0 (t, t0 )† H0 |Ψ(t)i + U0 (t, t0 )† i~∂t |Ψ(t)i
= U0 (t, t0 )† (H(t) − H0 )|Ψ(t)i = VD (t)|ΨD (t)i,
(12.5.6)
i.e. the equation of motion for the wave function is driven by the interaction VD (t). Similarly, in the Dirac picture the analog of the Heisenberg equation for the time-dependence
of operators takes the form
∂t AD (t) = ∂t [U0 (t, t0 )† AU0 (t, t0 )]
i
i
= U0 (t, t0 )† H0 AU0 (t, t0 ) − U0 (t, t0 )† A H0 U0 (t, t0 )
~
~
i
i
†
=
U0 (t, t0 ) [H0 , A]U0 (t, t0 ) = [H0D , AD (t)],
~
~
(12.5.7)
i.e. the equation of motion for operators is driven by H0D . In the Dirac picture the timeevolution operator is defined by
|ΨD (t)i = UD (t, t0 )|ΨD (t0 )i.
(12.5.8)
It takes the form
UD (t, t0 ) = 1 +
∞
X
i
(− )n
~
n=1
Z
t
Z
t1
dt1
t0
Z
tn−1
dt2 ...
t0
dtn VD (t1 )VD (t2 )...VD (tn ).
(12.5.9)
t0
In the absence of time-dependent external forces, i.e. for V(t) = 0, the Dirac picture
reduces to the Heisenberg picture.
140
CHAPTER 12. ABSTRACT FORMULATION OF QUANTUM MECHANICS
Chapter 13
Quantum Mechanical
Approximation Methods
13.1
The Variational Approach
In many cases of physical interest it is impossible to find an analytic solution of the
Schrödinger equation. Numerical methods are a powerful tool to find accurate approximate
solutions. A thorough discussion of numerical methods would deserve a whole course and
will not be attempted here. Instead, we will concentrate on approximate analytic methods.
An interesting method is the variational approach. It is based on the observation that the
time-independent Schrödinger equation
Hχn (x) = En χn (x),
(13.1.1)
can be derived from a variational principle. The Schrödinger equation results when one
varies the wave function χn (x) such that
Z
hχn |H|χn i − En hχn |χn i =
dx χn (x)∗ (H − En )χn (x),
(13.1.2)
is minimized. Here the energy En appears as a Lagrange multiplier that enforces the
constraint
Z
hχn |χn i = dx χn (x)∗ χn (x) = 1.
(13.1.3)
It is not difficult to derive the Schrödinger equation from this variational principle. For
example, using
H=−
~2 d2
+ V (x),
2M dx2
141
(13.1.4)
142
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
one can write
hχn |(H − En )|χn i =
Z
~2 dχn (x)∗ dχn (x)
dx [
+ χn (x)∗ (V (x) − En )χn (x)] =
2M
dx
dx
Z
dχn (x)
dx L(χn (x),
).
dx
(13.1.5)
Here we have introduced the Lagrange density
L(χn ,
dχn
~2 dχn (x)∗ dχn (x)
)=
+ χn (x)∗ (V (x) − En )χn (x).
dx
2M
dx
dx
(13.1.6)
The Euler-Lagrange equation that results from the variational principle then takes the
form
δL
δL
~2 d2 χn (x)
d
−
=
− (V (x) − En )χn (x) = 0,
(13.1.7)
dx δ(dχ∗n /dx) δχ∗n
2M dx2
which is indeed nothing but the Schrödinger equation. Varying the Lagrange density with
respect to χn (x) yields the same equation in complex conjugated form.
The variational approximation now proceeds as follows. When one cannot find the
analytic solution χ0 (x) for a ground state wave function, one makes an ansatz for a trial
function χ̃0 (x) that depends on a number of variational parameters αi . Then one computes
hχ̃0 |H|χ̃0 i as well as hχ̃0 |χ̃0 i and one puts
d hχ̃0 |H|χ̃0 i
= 0.
dαi hχ̃0 |χ̃0 i
(13.1.8)
The resulting equations for the parameters αi determine the best approximate solution
χ̃0 (x) given the chosen ansatz. The approximate ground state energy results as
Ẽ0 =
hχ̃0 |H|χ̃0 i
.
hχ̃0 |χ̃0 i
(13.1.9)
The true ground state energy E0 is the absolute minimum of the right-hand side under
arbitrary variations of χ̃0 (x). Hence, Ẽ0 ≥ E0 , i.e. the variational result for the ground
state energy is an upper limit on the true value.
Let us illustrate the variational method with an example — a modified 1-dimensional
harmonic oscillator problem with the potential
1
V (x) = M ω 2 x2 + λx4 .
2
(13.1.10)
In the presence of the quartic term λx4 it is impossible to solve the Schrödinger equation
analytically. We now make the variational ansatz
1 2 2
χ̃0 (x) = A exp − α x ,
(13.1.11)
2
13.2. NON-DEGENERATE PERTURBATION THEORY TO LOW ORDERS
143
for the ground state wave function. This particular ansatz depends on only one variational
parameter α. In this case, one finds
√ 3λ
|A|2 π ~2 α2 M ω 2
+ 4 ,
hχ̃0 |H|χ̃0 i =
+
α
4M
4α2
4α
√
2
|A| π
hχ̃0 |χ̃0 i =
.
(13.1.12)
α
Hence, the variation with respect to α takes the form
~2 α M ω 2 3λ
d hχ̃0 |H|χ̃0 i
=
−
− 5 = 0.
dα hχ̃0 |χ̃0 i
2M
2α3
α
(13.1.13)
For λ = 0 this equation implies α2 = M ω/~ in agreement with our earlier results for
the harmonic oscillator. Using this optimized value for the variational parameter α one
obtains
~2 α2 M ω 2
~ω
hχ̃0 |H|χ̃0 i
=
+
=
,
(13.1.14)
Ẽ0 =
2
hχ̃0 |χ̃0 i
4M
4α
2
which is indeed the exact ground state energy of a harmonic oscillator. In general, one
cannot expect to obtain the exact result from a variational calculation. However, for λ = 0
our family of ansatz wave functions χ̃0 (x) indeed contains the exact wave function. Hence,
in this case the variational calculation gives the exact result.
For λ 6= 0 the equation for α can be written as
α6 −
M 2 ω 2 α2 6λM
− 2 = 0.
~2
~
(13.1.15)
This cubic equation for α2 can still be solved in closed form, but the result is not very
transparent. Hence, we specialize on small values of λ and we write
α2 =
Mω
+ cλ + O(λ2 ),
~
(13.1.16)
which implies c = 3/M ω 2 . The variational result for the ground state energy then becomes
Ẽ0 =
13.2
hχ̃0 |H|χ̃0 i
~2 α2 M ω 2
3λ
~ω
3λ~2
=
+
+
=
+
.
hχ̃0 |χ̃0 i
4M
4α2
4α4
2
4M 2 ω 2
(13.1.17)
Non-Degenerate Perturbation Theory to Low Orders
Perturbation theory is another powerful approximation method. It is applicable when
a quantum system for which an analytic solution of the Schrödinger equation exists is
influenced by a small additional perturbation. First, we consider time-independent perturbations V and we write the total Hamilton operator as
H = H0 + λV,
(13.2.1)
144
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
where λ is a small parameter that controls the strength of the perturbation. The Hamiltonian H0 is the one of the unperturbed system for which an analytic solution of the
time-independent Schrödinger equation
(0) (0)
H0 |χ(0)
n i = En |χn i,
(0)
(13.2.2)
(0)
is known. Here |χn i and En are the eigenstates and energy eigenvalues of the unperturbed system. The full time-independent Schrödinger equation (including the perturbation) takes the form
H|χn i = (H0 + λV)|χn i = En |χn i.
(13.2.3)
Now we expand the full solution as a superposition of unperturbed states
X
cmn (λ)|χ(0)
|χn i =
m i.
(13.2.4)
m
The expansion coefficients are given by
cmn (λ) = hχ(0)
m |χn i.
(13.2.5)
In perturbation theory these coefficients are expanded in a power series in the strength
parameter λ, i.e.
∞
X
k
cmn (λ) =
c(k)
(13.2.6)
mn λ .
k=0
Similarly, the energy eigenvalues of the perturbed problem are written as
En (λ) =
∞
X
En(k) λk .
(13.2.7)
k=0
First, we consider non-degenerate perturbation theory, i.e. we assume that the unperturbed
(0)
(0)
states |χn i that we consider have distinct energy eigenvalues En . In that case, the
(0)
perturbed state |χn i reduces to the unperturbed state |χn i as the perturbation is switched
off, and hence
cmn (0) = c(0)
(13.2.8)
mn = δmn .
Let us consider non-degenerate perturbation theory up to second order in λ, such that
En = En(0) + λEn(1) + λ2 En(2) + ...,
(13.2.9)
and
|χn i = |χ(0)
n i+λ
X
(0)
2
c(1)
mn |χm i + λ
m
X
(0)
c(2)
mn |χm i + ...
(13.2.10)
m
Inserting this into the Schrödinger equation one obtains
"
#
X
X
(0)
2
(0)
(H0 + λV) |χ(0)
c(1)
c(2)
n i+λ
mn |χm i + λ
mn |χm i + ... =
m
En(0)
+
λEn(1)
2
+λ
En(2)
m
+ ... ×
"
#
|χ(0)
n i+λ
X
m
(0)
2
c(1)
mn |χm i + λ
X
m
(0)
c(2)
mn |χm i + ... .
(13.2.11)
13.2. NON-DEGENERATE PERTURBATION THEORY TO LOW ORDERS
145
Let us consider this equation order by order in λ. To zeroth order we simply obtain the
unperturbed Schrödinger equation
(0) (0)
H0 |χ(0)
n i = En |χn i,
while to first order in λ one obtains
X
X
(0)
(0)
(0)
(0)
(1) (0)
H0
c(1)
c(1)
mn |χm i + V|χn i = En
mn |χm i + En |χn i,
m
(13.2.12)
(13.2.13)
m
which implies
X
(0)
(0)
(0)
(0)
(1) (0)
c(1)
mn (Em − En )|χm i + V|χn i = En |χn i.
(13.2.14)
m
(0)
Projecting this equation on the state hχl | one obtains
(1)
(0)
cln (El
(0)
(1)
− En(0) ) + hχl |V|χ(0)
n i = En δln .
(13.2.15)
Hence, putting l = n one obtains
(0)
En(1) = hχ(0)
n |V|χn i,
(13.2.16)
i.e. the leading order correction to the energy is given by the expectation value of the
(0)
perturbation V in the unperturbed state |χn i. Putting l 6= n and using the fact that the
(0)
(0)
energies are non-degenerate, i.e. El 6= En , one also finds
(1)
cln =
(0)
(0)
(0)
(0)
hχl |V|χn i
.
(13.2.17)
En − El
(1)
It should be noted that the coefficient cnn is not determined by this equation. In fact, it
is to some extent arbitrary. Through a choice of phase for the state |χn i one can always
(1)
make cnn real. Then it can be determined from the normalization condition
X
hχn |χn i =
|cmn (λ)|2 = 1 + 2λcnn + O(λ2 ) = 1 ⇒ c(1)
(13.2.18)
nn = 0.
m
As an example, we consider again an anharmonic perturbation V (x) = λx4 to a 1dimensional harmonic oscillator. The unperturbed ground state wave function is then
given by
1
(0)
hx|χ0 i = A exp − α2 x2 ,
(13.2.19)
2
√
with α2 = M ω/~ and |A|2 = α/ π. We thus obtain
Z
3λ~2
(1)
(0)
(0)
E0 = hχ0 |V|χ0 i = dx λx4 |A|2 exp(−α2 x2 ) =
.
(13.2.20)
4M 2 ω 2
This result happens to agree with the one of the previous variational calculation expanded
to first order in λ. In general, one would not necessarily expect agreement between a
146
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
variational and a perturbative calculation. The perturbative calculation is based on a
systematic power series expansion in the strength λ of the perturbation. Hence, it gives
correct results order by order. The precision of a variational calculation, on the other
hand, depends on the choice of variational ansatz for the wave function. In our example,
the variational ansatz contained the exact wave function of the unperturbed problem.
Since that wave function alone determines the first order correction to the energy, in this
particular case the variational calculation gives the correct result to first order in λ.
Next, we consider second order perturbation theory. The terms of second order in λ
in eq.(13.2.11) imply
X
X
(0)
(0)
H0
c(2)
c(1)
mn |χm i + V
mn |χm i =
m
En(0)
m
X
(0)
c(2)
mn |χm i
+ En(1)
m
X
(0)
(2) (0)
c(1)
mn |χm i + En |χn i.
(13.2.21)
m
(0)
Projecting this equation on the state hχn | one finds
X
(0)
(0)
(1) (1)
En(2) =
c(1)
mn hχn |V|χm i − En cnn
m
X
=
(0)
(0)
c(1)
mn hχn |V|χm i =
m6=n
(2)
It should be noted that E0
13.3
(0) 2
X |hχ(0)
n |V|χm i|
(0)
m6=n
(0)
.
(13.2.22)
En − Em
for the ground state is always negative.
Degenerate Perturbation Theory to First Order
Until now we have assumed that the unperturbed level is not degenerate. Now we consider
(0)
the degenerate case, i.e. we assume that there are N states |χt i (t ∈ {1, 2, ..., N }) with the
(0)
same energy En . This degeneracy may be lifted partly or completely by the perturbation
V. As in the non-degenerate case we write
X
|χn i =
cmn (λ)|χ(0)
(13.3.1)
m i.
m
(0)
Since the states |χt i are now degenerate, any linear combination of them is also an energy
eigenstate. Consequently, in contrast to the non-degenerate case, one can no longer assume
(0)
that cmn (0) = cmn = δmn . Instead, in the degenerate case one has
(0)
ctn = Utn ,
(13.3.2)
for t, n ∈ {1, 2, ..., N }. Here U is a unitary transformation that turns the arbitrarily
chosen basis of unperturbed degenerate states into the basis of (not necessarily degenerate)
perturbed states in the limit λ → 0. Hence, to first order in λ we have
En = En(0) + λEn(1) + ...,
(13.3.3)
13.4. THE HYDROGEN ATOM IN A WEAK ELECTRIC FIELD
and
|χn i =
N
X
(0)
X
|χt iUtn + λ
147
(0)
c(1)
mn |χm i + ...
(13.3.4)
m
t=1
Again, inserting into the Schrödinger equation one obtains
#
"N
X
X (0)
(1)
(0)
cmn |χm i + ... =
(H0 + λV)
|χt iUtn + λ
m
t=1
"
(En(0) + λEn(1) + ...)
N
X
#
(0)
|χt iUtn + λ
X
(0)
c(1)
mn |χm i + ... .
(13.3.5)
m
t=1
To first order in λ one now obtains
H0
X
(0)
c(1)
mn |χm i + V
m
N
X
(0)
|χ(0)
n iUtn = En
X
m
t=1
N
X
(0)
(13.3.6)
|χt iUtn .
(13.3.7)
(0)
(1)
c(1)
mn |χm i + En
|χt iUtn ,
t=1
which implies
X
(0)
(0)
(0)
c(1)
mn (Em − En )|χm i + V
m
N
X
(0)
|χt iUtn = En(1)
t=1
N
X
(0)
t=1
(0)
Projecting on hχs | with s ∈ {1, 2, ..., N } one obtains
N
X
(0)
(1)
hχ(0)
s |V|χt iUtn = En Usn .
(13.3.8)
t=1
Multiplying from the left by Uls† , summing over s, and using Uls† Usn = δln , one obtains
En(1) δln
=
N
X
Uls† hχ(0)
s |V|χt iUtn ,
(0)
(13.3.9)
s,t=1
(1)
i.e. to first order in λ, the perturbed energies En result from diagonalizing the perturbation V in the space of degenerate unperturbed states by the unitary transformation
U.
13.4
The Hydrogen Atom in a Weak Electric Field
Let us apply perturbation theory to the problem of the hydrogen atom in a weak external electric field. We consider a homogeneous electric field Ez in the z-direction. The
corresponding potential energy for an electron at the position ~r is then given by
V (~r) = eEz z.
(13.4.1)
148
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
The unperturbed Hamilton operator is the one of the hydrogen atom without the external
field, i.e.
~2
Ze2
H0 = −
−
.
(13.4.2)
2µ
|~r|
The unperturbed energy spectrum of bound states is
En(0) = −
Z 2 e4 µ
.
2~2 n2
(13.4.3)
The corresponding wave functions are
ψnlm (~r) = Rnl (r)Ylm (θ, ϕ),
(13.4.4)
where Ylm (θ, ϕ) are the spherical harmonics. In particular, the ground state is a nondegenerate s-state with a spherically symmetric wave function
1
ψ100 (~r) = R10 (r)Y00 (θ, ϕ) = √ R10 (r).
4π
To first order in the perturbation, the shift of the ground state energy,
Z
Z
eEz
eEz
(1)
3
2
En = hψ100 |V|ψ100 i =
d r R10 (r) z =
d3 r R10 (r)2 r cos θ = 0,
4π
4π
(13.4.5)
(13.4.6)
vanishes. The first excited state of the unperturbed hydrogen atom is 4-fold degenerate.
There are one s-state
1
ψ200 (~r) = R20 (r)Y00 (θ, ϕ) = √ R20 (r).
4π
(13.4.7)
and three p-states
r
3
R20 (r) cos θ,
4π
r
3
ψ21±1 (~r) = R20 (r)Y1±1 (θ, ϕ) = ∓
R20 (r) sin θ exp(±iϕ).
8π
ψ210 (~r) = R20 (r)Y10 (θ, ϕ) =
(13.4.8)
We must now evaluate the perturbation in the space of degenerate unperturbed states.
The perturbation V (~r) = eEz z is invariant against rotations around the z-axis. Consequently, the perturbed Hamiltonian still commutes with Lz (but no longer with Lx and
Ly ). Thus, the perturbation cannot mix states with different m quantum numbers. The
four degenerate states have m = ±1 for two of the p-states and m = 0 for the other p-state
and the s-state. Hence, only the m = 0 states can mix. The m = ±1 states are not even
shifted in their energies because
hψ21±1 |V|ψ21±1 i = 0.
The energy shifts of the two m = 0 states are the eigenvalues of the matrix
hψ200 |V|ψ200 i hψ200 |V|ψ210 i
V =
.
hψ210 |V|ψ200 i hψ210 |V|ψ210 i
(13.4.9)
(13.4.10)
13.5. NON-DEGENERATE PERTURBATION THEORY TO ALL ORDERS
149
Parity symmetry implies that the diagonal elements vanish
hψ200 |V|ψ200 i = hψ210 |V|ψ210 i = 0.
(13.4.11)
In addition, for the off-diagonal matrix elements one obtains
hψ200 |V|ψ210 i = hψ210 |V|ψ200 i = −3
~2 Ez
.
Zem
(13.4.12)
Hence, the energy shifts of the two m = 0 states are
E (1) = ±3
13.5
~2 Ez
.
ZeM
(13.4.13)
Non-Degenerate Perturbation Theory to All Orders
In the next step we investigate non-degenerate perturbation theory to all orders. In this
calculation we put λ = 1 and we thus write H = H0 + V. Let us consider the so-called
resolvent
1
,
(13.5.1)
R(z) =
z−H
(0)
as well as its expectation value in the unperturbed state |χn i, i.e. the function
Rn (z) = hχ(0)
n |
1
z−H
|χ(0)
n i.
(13.5.2)
One pole of Rn (z) is located at the energy eigenvalue En of the perturbed Hamiltonian
H. We write
1
z−H
=
1
z − H0
(z − H + V)
1
z−H
=
1
z − H0
+
1
V
1
z − H0 z − H
.
(13.5.3)
Iterating this equation one obtains
1
z−H
=
1
z − H0
∞ X
V
k=0
1
k
,
z − H0
(13.5.4)
which implies
Rn (z) =
1
z−
(0)
En
1
=
z−
(0)
En
1
=
z−
(0)
En
+
+
+
1
z−
(0)
En
1
z−
(0)
En
1
z−
(0)
2 hχn |V
∞
X
"
z−
(0)
hχn |A
(0) 2
En
k=0
"
V
P0
(
k=0
∞
X
k
z − H0
k=0
(0)
2 hχn |V
1
∞ X
(0)
En
P0
z−
(0)
En
|χ(0)
n i
Q0
+
)V
z − H0
#k
|χ(0)
n i
#k
A
|χ(0)
n i.
(13.5.5)
150
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
Here we have introduced the projection operators
(0)
P0 = |χ(0)
n ihχn |, Q0 = 1 − P0 ,
(13.5.6)
(0)
on the unperturbed state |χn i and on the rest of the Hilbert space of unperturbed states.
We have also defined the operator
A=V
k
∞ X
Q0
V ,
z − H0
(13.5.7)
k=0
which satisfies
A=V+V
Q0
A.
z − H0
(13.5.8)
Similarly, the operator
B=V
∞
X
"
P0
(0)
k=0
z − En
Q0
+
z − H0
! #k
V
,
(13.5.9)
obeys the equation
P0
B=V+V
(0)
z − En
Q0
+
z − H0
!
B.
(13.5.10)
The last equality in eq.(13.5.5), namely
B=A+
∞ X
k=0
k
P0
A ,
z − H0
(13.5.11)
P0
B.
z − H0
(13.5.12)
is equivalent to
B=A+A
This is indeed consistent because
P0
Q0
+
z − H0
!
B
(0)
z − En
Q0
P0
P0
= A+V
(B − A) + A
B + (V − A)
B
z − H0
z − H0
z − H0
Q0
P0
= B+V
(B − A) + (V − A)
B
z − H0
z − H0
Q0
P0
Q0
P0
. = B+V
A
B−V
A
B = B.
z − H0 z − H 0
z − H0 z − H 0
B = V+V
(13.5.13)
Next we introduce the function
(0)
An (z) = hχ(0)
n |A|χn i,
(13.5.14)
13.6. DEGENERATE PERTURBATION THEORY TO ALL ORDERS
151
and we evaluate
hχ(0)
n |A
∞
X
"
(0)
k=0
An (z)
#k
P0
∞
X
!k
An (z)
z−
k=0
A
z − En
(0)
En
#k
"
∞
(0)
(0)
X
|χ
ihχ
|
n
n
(0)
|χ(0)
A |χ(0)
n i = hχn |A
n i=
(0)
z − En
k=0
= z − En(0)
An (z)
.
(0)
(13.5.15)
z − En − An (z)
Inserting this in eq.(13.5.5) we finally obtain
Rn (z) =
1
z−
(0)
En
1
+
z−
An (z)
(0)
En
z−
(0)
En
1
=
− An (z)
z−
(0)
En
.
(13.5.16)
− An (z)
(0)
Hence, there is a pole in Rn (z) if z = En − An (z). The pole position determines the
perturbed energy En = z and hence
k
∞ X
Q0
(0)
(0)
En − En = An (En ) = hχn |V
V |χ(0)
(13.5.17)
n i.
En − H0
k=0
This is an implicit equation for the perturbed energy En valid to all orders in the perturbation V. Expanding this equation to first order we obtain
(0)
En − En(0) = En(1) = hχ(0)
n |V|χn i,
(13.5.18)
in complete agreement with our previous result. To second order we find
(0)
(0)
En − En(0) = En(1) + En(2) = hχ(0)
n |V|χn i + hχn |V
Q0
V|χ(0)
n i.
En − H0
(13.5.19)
(0)
To order V2 in the term Q0 /(En − H0 ) the energy En must be replaced by En and hence
En(2) = hχ(0)
n |V
=
Q0
(0)
En
− H0
(0)
V|χ(0)
n i = hχn |V
(0) 2
X |hχ(0)
n |V|χm i|
(0)
m6=n
(0)
(0)
X |χ(0)
m ihχm |
(0)
m6=n En
−
(0)
Em
V|χ(0)
n i
,
(13.5.20)
En − Em
again in agreement with our previous result.
13.6
Degenerate Perturbation Theory to All Orders
Let us now consider degenerate perturbation theory to all orders. Again, we assume that
(0)
(0)
there are N states |χs i (s ∈ {1, 2, ..., N }) with the same energy En . We introduce the
projection operators
N
X
(0)
P0 =
|χ(0)
(13.6.1)
s ihχs |, Q0 = 1 − P0 ,
s=1
152
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
on the space of degenerate unperturbed states and its complement in the Hilbert space.
We now consider the N × N matrix-function
k
∞ X
Q0
(0)
(0)
(0)
(0)
Ast (z) = hχs |A|χt i = hχs |V
V |χt i,
(13.6.2)
z − H0
k=0
which we diagonalize by a unitary transformation
|φ(0)
n i=
N
X
(0)
(0)
|χt iUtn , hφl | =
t=1
N
X
Uls† hχ(0)
s |,
(13.6.3)
s=1
such that
(0)
hφl |V
k
∞ N
X
X
Q0
(0)
V |φn i =
Uls† Ast (z)Utn = An (z)δln .
z − H0
(13.6.4)
s,t=1
k=0
It should be noted that the projector on the set of unitarily transformed unperturbed
(0)
states |φn i is still given by
N
X
(0)
|φ(0)
n ihφn |
=
n=1
N
X
(0)
†
|χt iUtn Uns
hχ(0)
s |
n,s,t=1
=
N
X
(0)
|χt iδts hχ(0)
s |
=
s,t=1
N
X
(0)
|χ(0)
s ihχs | = P0 .
(13.6.5)
s=1
At this stage we have reduced the degenerate case to the previously discussed nondegenerate case. In particular, the perturbed energy eigenvalues are solutions of the
implicit equation
En −
En(0)
= An (En ) =
hφ(0)
n |V
∞ X
k=0
Q0
V
En − H0
k
|φ(0)
n i.
(13.6.6)
To first order in the perturbation V we obtain
(0)
En − En(0) = En(1) = hφ(0)
n |V|φn i.
(13.6.7)
(0)
To that order the perturbed wave functions are simply the states |φn i (n ∈ {1, 2, ..., N })
in which the perturbation is diagonalized. It should be noted that the original degeneracy
is often at least partially lifted by the perturbation. In particular, if the perturbation
V has fewer symmetries than the unperturbed Hamiltonian H0 some degeneracies may
disappear in the presence of the perturbation.
Sometimes the states of an unperturbed Hamiltonian are almost — but not quite
— degenerate. In this case of quasi-degeneracy one might think that non-degenerate
perturbation theory would be appropriate. However, if the perturbation gives rise to
energy shifts that are comparable with or even larger than the splittings between the almost
13.6. DEGENERATE PERTURBATION THEORY TO ALL ORDERS
153
degenerate levels, one must use quasi-degenerate perturbation theory. Let us consider a
(0)
(0)
group of quasi-degenerate states |χs i (s ∈ {1, 2, ..., N }) with energies Es that are all
close to an energy E (0) . In that case, it is useful to consider a modified unperturbed
Hamiltonian
N X
(0)
(0)
(0)
H00 = H0 +
E (0) − Et
|χt ihχt |,
(13.6.8)
t=1
and a modified perturbation
V0 = V −
N X
(0)
E (0) − Et
(0)
(0)
|χt ihχt |,
(13.6.9)
t=1
such that the whole problem
H00 + V0 = H0 + V,
(13.6.10)
(0)
remains unchanged. By construction, the unperturbed quasi-degenerate eigenstates |χt i
of the original Hamiltonian are exactly degenerate eigenstates of modified unperturbed
Hamiltonian, i.e.
H00 |χ(0)
s i
=
H0 |χ(0)
s i
=
Es(0) |χ(0)
s i
+
N X
(0)
E (0) − Et
(0)
(0)
|χt ihχt |χ(0)
s i
t=1
(0) (0)
+ E (0) − Es(0) |χ(0)
s i = E |χs i.
(13.6.11)
At this stage one can use the formalism of degenerate perturbation theory. In particular,
we consider the function
A0st (z)
=
0
hχ(0)
s |V
k
∞ X
Q0
(0)
0
V |χt i
0
z − H0
k=0
=
hχ(0)
s |
V−
N X
E
(0)
−
Eu(0)
!
(0)
|χ(0)
u ihχu |
u=1
×
∞
X
k=0
=
"
Q0
z − H00
hχ(0)
s |
V−
V−
N X
!#k
(0)
E (0) − Ev(0) |χ(0)
v ihχv |
(0)
|χt i
v=1
N X
E
(0)
−
Eu(0)
!
(0)
|χ(0)
u ihχu |
u=1
k
∞ X
Q0
(0)
= hχ(0)
|V
V
|χt i
s
z − H00
k=0
k
∞ X
Q0
(0)
(0)
(0)
(0)
+
Es − E
hχs |
V |χt i
0
z − H0
k=0
= Ast (z) + Es(0) − E (0) δst ,
k
∞ X
Q0
(0)
V |χt i
z − H00
k=0
(13.6.12)
154
CHAPTER 13. QUANTUM MECHANICAL APPROXIMATION METHODS
which we again diagonalize by a unitary transformation such that
N
X
Uls† A0st (z)Utn = A0n (z)δln .
(13.6.13)
s,t=1
Hence, the perturbed energy eigenvalues are now solutions of the implicit equation
En − E (0) = A0n (En ).
(13.6.14)
Chapter 14
Charged Particle in an
Electromagnetic Field
14.1
The Classical Electromagnetic Field
In this chapter we will study the quantum mechanics of a charged particle (e.g. an electron)
in a general classical external electromagnetic field. In principle, the electromagnetic field
itself should also be treated quantum mechanically. This is indeed possible and naturally
leads to quantum electrodynamics (QED). QED is a relativistic quantum field theory
— a subject beyond the scope of this course. Here we will limit ourselves to classical
electrodynamics. Hence, we will only treat the charged particle moving in the external
field (but not the field itself) quantum mechanically.
In previous chapters we have investigated the quantum mechanics of a point particle
with coordinate ~x moving in an external potential V (~x). In particular, we have studied
the motion of a non-relativistic electron with negative charge −e in the Coulomb field of
an atomic nucleus of positive charge Ze. In that case the potential energy is given by
V (~x) = −eΦ(~x) = −
Ze2
,
r
(14.1.1)
where r = |~x|. The electrostatic potential
Ze
,
r
is related to the static electric field of the nucleus by
Φ(~x) =
(14.1.2)
~ x) = −∇Φ(~
~ x) = Ze ~er ,
E(~
(14.1.3)
r2
where ~er = ~x/|~x| is the unit-vector in the direction of ~x. The electric field of the atomic
nucleus (and hence Φ(~x)) can be obtained as a solution of the first Maxwell equation
~ · E(~
~ x, t) = 4πρ(~x, t),
∇
155
(14.1.4)
156
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
with the point-like static charge density of the nucleus given by
ρ(~x, t) = Zeδ(~x).
(14.1.5)
Here we want to investigate the quantum mechanical motion of a charged particle in
a general classical external electromagnetic field. For this purpose, we remind ourselves
of Maxwell’s equations
~ · E(~
~ x, t) = 4πρ(~x, t),
∇
~ × E(~
~ x, t) + 1 ∂t B(~
~ x, t) = 0,
∇
c
~ · B(~
~ x, t) = 0,
∇
~ x, t) = 4π ~j(~x, t).
~ × B(~
~ x, t) − 1 ∂t E(~
∇
c
c
(14.1.6)
Adding the time-derivative of the first and c times the divergence of the last equation one
obtains the continuity equation
~ · ~j(~x, t) = 0,
∂t ρ(~x, t) + ∇
(14.1.7)
which guarantees charge conservation.
~ x, t) and B(~
~ x, t) can be expressed in terms of scalar and
The electromagnetic fields E(~
~
vector potentials Φ(~x, t) and A(~x, t) as
~ x, t) = −∇Φ(~
~ x, t) − 1 ∂t A(~
~ x, t),
E(~
c
~ x, t) = ∇
~ × A(~
~ x, t).
B(~
(14.1.8)
Then the homogeneous Maxwell equations
~ × E(~
~ x, t) + 1 ∂t B(~
~ x, t)
∇
c
~ × ∂t A(~
~ x, t) + 1 ∂t ∇
~ × A(~
~ x, t) = 0,
~ ×∇
~ · Φ(~x, t) − 1 ∇
= −∇
c
c
~ × B(~
~ x, t) = ∇
~ ·∇
~ × A(~
~ x, t) = 0,
∇
(14.1.9)
are automatically satisfied. The inhomogeneous equations can be viewed as four equations
~ x, t).
for the four unknown functions Φ(~x, t) and A(~
All fundamental forces in Nature are described by gauge theories. This includes the
electromagnetic, weak, and strong forces and even gravity. Gauge theories have a high
degree of symmetry. In particular, their classical equations of motion (such as the Maxwell
equations in the case of electrodynamics) are invariant against local space-time dependent
gauge transformations. In electrodynamics a gauge transformation takes the form
1
Φ(~x, t)0 = Φ(~x, t) + ∂t ϕ(~x, t),
c
0
~
~
~ x, t).
A(~x, t) = A(~x, t) − ∇ϕ(~
(14.1.10)
14.2. CLASSICAL PARTICLE IN AN ELECTROMAGNETIC FIELD
157
Under this transformation the electromagnetic fields
~ x, t)0 = −∇Φ(~
~ x, t)0 − 1 ∂t A(~
~ x, t)0 = −∇Φ(~
~ x, t) − 1 ∂t A(~
~ x, t)
E(~
c
c
1 ~
1~
~ x, t),
∇∂t ϕ(~x, t) + ∂t ∇ϕ(~
x, t) = E(~
−
c
c
~ x, t)0 = ∇
~ × A(~
~ x, t)0 = ∇
~ × A(~
~ x, t) − ∇
~ × ∇ϕ(~
~ x, t) = B(~
~ x, t),
B(~
(14.1.11)
remain unchanged — they are gauge invariant. As a consequence, Maxwell’s equations
themselves are gauge invariant as well. In fact, in a gauge theory only gauge invariant
~ x, t)
quantities have a physical meaning. The scalar and vector potentials Φ(~x, t) and A(~
vary under gauge transformations and are not physically observable. Instead they are
mathematical objects with an inherent unphysical gauge ambiguity. Instead, as gauge in~ x, t) and B(~
~ x, t) are physically observable.
variant quantities, the electromagnetic fields E(~
14.2
Classical Particle in an Electromagnetic Field
The motion of a point particle is governed by Newton’s equation
M~a(t) = F~ (t).
(14.2.1)
For a particle with charge −e moving in an external electromagnetic field the force is given
by
~v (t) ~
~
~
F (t) = −e E(~x(t), t) +
× B(~x(t), t) .
(14.2.2)
c
Newton’s equation can be derived from the action
Z
Z
M
1
2
3
~ y , t) ,
S[~x(t)] = dt
~v (t) − dtd y ρ(~y , t)Φ(~y , t) − ~j(~y , t) · A(~
2
c
(14.2.3)
where
ρ(~y , t) = −eδ(~y − ~x(t)),
~j(~y , t) = −e~v (t)δ(~y − ~x(t)),
(14.2.4)
are the charge and current densities of the charged particle at position ~x(t). It is easy to
show that charge is conserved, i.e.
~ · ~j(~y , t) = 0.
∂t ρ(~y , t) + ∇
(14.2.5)
Inserting eq.(14.2.4) into eq.(14.2.3), for the action one obtains
Z
S[~x(t)] =
dt
M
~v (t) ~
2
~v (t) + eΦ(~x(t), t) − e
· A(~x(t), t) .
2
c
(14.2.6)
158
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
This action is indeed invariant under gauge transformations because
Z
~v (t) ~
0
0
dt Φ(~x(t), t) −
· A(~x(t), t) =
c
Z
1
~v (t) ~
~ x(t), t)) =
dt Φ(~x(t), t) + ∂t ϕ(~x(t), t) −
· (A(~x(t), t) − ∇ϕ(~
c
c
Z
~v (t) ~
1d
dt Φ(~x(t), t) −
· A(~x(t), t) +
ϕ(~x(t), t) ,
(14.2.7)
c
c dt
and because the total derivative
d
~ x(t), t),
ϕ(~x(t), t) = ∂t ϕ(~x(t), t) + ~v · ∇ϕ(~
dt
(14.2.8)
integrates to zero as long as ϕ(~x(t), t) vanishes in the infinite past and future. Identifying
the Lagrange function
L=
M
~v (t) ~
~v (t)2 + eΦ(~x(t), t) − e
· A(~x(t), t),
2
c
(14.2.9)
it is straightforward to derive Newton’s equation as the Euler-Lagrange equation
d δL
δL
−
= 0.
dt δvi (t) δxi
(14.2.10)
The theory can also be formulated in terms of a classical Hamilton function
H = p~(t) · ~v (t) − L,
(14.2.11)
where p~ is the momentum canonically conjugate to the coordinate ~x. One finds
e~
M~v (t) = p~(t) + A(~
x(t), t),
c
(14.2.12)
and thus one obtains
H=
i2
1 h
e~
p~(t) + A(~
x(t), t) − eΦ(~x(t), t).
2M
c
(14.2.13)
This is indeed consistent because
vi (t) =
i
dxi (t)
∂H
1 h
e
=
=
pi (t) + Ai (~x(t), t) .
dt
∂pi (t)
M
c
(14.2.14)
The other equation of motion is
i
dpi (t)
∂H
e h
e
=−
=−
pj (t) + Aj (~x(t), t) ∂i Aj (~x(t), t) + e∂i Φ(~x(t), t). (14.2.15)
dt
∂xi (t)
Mc
c
It is straightforward to show that these equations of motion are again equivalent to Newton’s equation.
14.3. GAUGE INVARIANT FORM OF THE SCHRÖDINGER EQUATION
14.3
159
Gauge Invariant Form of the Schrödinger Equation
Remarkably, the gauge invariance of electrodynamics is intimately related to the phase ambiguity of the quantum mechanical wave function. As we have seen earlier, the Schrödinger
equation
~2
i~∂t Ψ(~x, t) = −
∆Ψ(~x, t) + V (~x)Ψ(~x, t),
(14.3.1)
2M
determines the wave function only up to a global phase ambiguity
Ψ(~x, t)0 = Ψ(~x, t) exp(iφ).
(14.3.2)
Here φ is a constant, independent of space and time.
We now apply the gauge principle to the Schrödinger equation, i.e. we demand that
the physics is invariant even under local transformations
Ψ(~x, t)0 = Ψ(~x, t) exp(iφ(~x, t)),
(14.3.3)
with a space-time dependent phase φ(~x, t). Of course, if the wave function Ψ(~x, t) solves
the original Schrödinger equation (14.3.1), the wave function Ψ(~x, t)0 of eq.(14.3.3) in
general does not. This is easy to see because
∂t Ψ(~x, t)0 = [∂t Ψ(~x, t) + iΨ(~x, t)∂t φ(~x, t)] exp(iφ(~x, t)),
(14.3.4)
contains the second term on the right hand side that was not present in the original
Schrödinger equation. However, if the potential energy V (~x) is replaced by a scalar potential −eΦ(~x, t), the Schrödinger equation takes the form
i~Dt Ψ(~x, t) = −
~2
∆Ψ(~x, t),
2M
(14.3.5)
with the covariant derivative
e
Dt Ψ(~x, t) = ∂t Ψ(~x, t) − i Φ(~x, t)Ψ(~x, t).
~
(14.3.6)
Using the gauge transformation property
1
Φ(~x, t)0 = Φ(~x, t) + ∂t ϕ(~x, t),
c
(14.3.7)
of the electromagnetic scalar potential, one obtains
e
Dt Ψ(~x, t)0 = ∂t Ψ(~x, t)0 − i Φ(~x, t)0 Ψ(~x, t)0
~
= [∂t Ψ(~x, t) + iΨ(~x, t)∂t φ(~x, t)] exp(iφ(~x, t))
e
1
− i Φ(~x, t) + ∂t ϕ(~x, t) Ψ(~x, t) exp(iφ(~x, t))
~
c
= Dt Ψ(~x, t) exp(iφ(~x, t)),
(14.3.8)
160
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
provided that we identify
e
ϕ(~x, t).
~c
We also introduce a space-like covariant derivative
φ(~x, t) =
~ x, t) = ∇Ψ(~
~ x, t) + i e A(~
~ x, t)Ψ(~x, t),
DΨ(~
~c
(14.3.9)
(14.3.10)
which also transforms as
~ x, t)0 = ∇Ψ(~
~ x, t)0 + i e A(~
~ x, t)0 Ψ(~x, t)0
DΨ(~
~c
i
h
~ x, t) + iΨ(~x, t)∇φ(~
~ x, t) exp(iφ(~x, t))
= ∇Ψ(~
i
e h~
~ x, t) Ψ(~x, t) exp(iφ(~x, t))
+ i
A(~x, t) − ∇ϕ(~
~c
~ x, t) exp(iφ(~x, t)),
= DΨ(~
(14.3.11)
~ one obtains
under a gauge transformation. Using p~ = (~/i)∇
~~
e~
D = p~ + A(~
x, t),
i
c
(14.3.12)
which is the quantum version of M~v from eq.(14.2.12) that we encountered in the classical
theory.
~ · ∇Ψ(~
~ x, t) with D
~ ·
Finally, in the Schrödinger equation we replace ∆Ψ(~x, t) = ∇
~ x, t) and we obtain
DΨ(~
i~Dt Ψ(~x, t) = −
~2 ~ ~
D · DΨ(~x, t).
2M
(14.3.13)
Inserting the explicit form of the covariant derivatives, the Schrödinger equation for a
charged particle in an arbitrary external electromagnetic field takes the form
h
i
i h
i
e
~2 h ~
e ~
~ + i e A(~
~ x, t) Ψ(~x, t). (14.3.14)
i~ ∂t − i Φ(~x, t) Ψ(~x, t) = −
∇ + i A(~
x, t) · ∇
~
2M
~c
~c
This equation is invariant under gauge transformations of the form
1
Φ(~x, t)0 = Φ(~x, t) + ∂t ϕ(~x, t),
c
~ x, t)0 = A(~
~ x, t) − ∇ϕ(~
~ x, t),
A(~
e
Ψ(~x, t)0 = Ψ(~x, t) exp i ϕ(~x, t) .
~c
(14.3.15)
Under this transformation, both sides of the Schrödinger equation change by a factor
exp(i(e/~c)ϕ(~x, t)). Canceling this factor out, the equation remains invariant.
As usual, the wave function Ψ(~x, t) that solves the gauged Schrödinger equation
(14.3.14) can be interpreted as the probability amplitude for finding the particle at position
~x at time t. In particular, the probability density
ρ(~x, t) = |Ψ(~x, t)|2 ,
(14.3.16)
14.4. MAGNETIC FLUX TUBES AND THE AHARONOV-BOHM EFFECT
161
is gauge invariant and hence physically meaningful. Again, probability conservation follows
from a continuity equation
~ · ~j(~x, t) = 0.
∂t ρ(~x, t) + ∇
(14.3.17)
However, in the presence of electromagnetic fields the usual probability current must be
modified by replacing ordinary with covariant derivatives such that now
~j(~x, t) =
14.4
i
~ h
∗~
∗
~
Ψ(~x, t) DΨ(~x, t) − (DΨ(~x, t)) Ψ(~x, t) .
2M i
(14.3.18)
Magnetic Flux Tubes and the Aharonov-Bohm Effect
Let us consider a charged particle moving in the electromagnetic field of an idealized
solenoid. Such a solenoid generates magnetic flux inside it but has no outside magnetic
or electric field. Hence a classical particle moving outside the solenoid is not affected by
it at all. We will see that this is not the case quantum mechanically. We consider an
infinitely thin and infinitely long solenoid oriented along the z-axis. The corresponding
vector potential is given by
Ax (~x, t) = −
y
Φ
x
Φ
, Ay (~x, t) =
, Az (~x, t) = 0,
2
2
2
2π x + y
2π x + y 2
(14.4.1)
while the scalar potential Φ(~x, t) = 0 vanishes. The corresponding electric and magnetic
fields vanish. However, there is an infinitely thin tube of magnetic flux along the zdirection. The quantity Φ (which has nothing to do with the scalar potential Φ(~x, t)) is
the magnetic flux. This follows from
Z
Z
Z
~
~
~
~
~
~ x, t) = Φ.
df · B(~x, t) =
df · ∇ × A(~x, t) =
d~l · A(~
(14.4.2)
S
S
∂S
Here S is any 2-d surface pinched by the z-axis and its boundary ∂S is a closed curve
winding around this axis. Hence, the magnetic field takes the form
Bx (~x, t) = By (~x, t) = 0, Bz (~x, t) = Φδ(x)δ(y),
(14.4.3)
i.e. it vanishes everywhere except along the z-axis. Since both the electric and the magnetic
field vanish outside the solenoid, a classical particle moving outside the solenoid is totally
unaffected by it and travels in a straight line with constant velocity. We will see that this
is not the case quantum mechanically.
To see this, we first consider a modified double slit experiment, similar to the one
discussed in chapter 2, section 2.7. Again, we consider a planar screen oriented in the
y-z-plane with two slits located at x = z = 0, the first at y1 = −d/2 and the second at
y2 = d/2. We place a source of electrons at the point x = −l, y = z = 0 a distance
l before the screen, and we detect an interference pattern at a parallel detection screen
a distance l behind the screen with the two slits. An electron detected at position y at
time T may have passed through the slit at y1 = −d/2 or through the one at y2 = d/2,
162
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
but we don’t know through which slit it went. In the first case it has traveled a total
distance squared |~x(T )|2 = 2l2 + d2 /4 + (y + d/2)2 , while in the second case we have
|~x(T )|2 = 2l2 + d2 /4 + (y − d/2)2 . Using Feynman’s path integral method, the total
probability amplitude is a superposition of contributions from two paths
iM (2l2 + d2 /4 + (y − d/2)2 )
Ψ(y) = A exp
2~T
2
2
iM (2l + d /4 + (y + d/2)2 )
+ A exp
2~T
2
2
iM (2l + y + d2 /2)
iM yd
iM yd
= A exp
exp
+ exp −
2~T
2~T
2~T
2
2
2
M yd
iM (2l + y + d /2)
2 cos
.
(14.4.4)
= A exp
2~T
2~T
To obtain the probability density we take the absolute value squared
2
2
2 M yd
2 M yd
ρ(y) = |Ψ(y)| = 4|A| cos
= 4ρ0 cos
.
2~T
2~T
(14.4.5)
This is the typical interference pattern of a double slit experiment.
Now we modify the situation by adding the solenoid discussed before. It is oriented
along the z-axis within the screen, right between the two slits. As a result, electrons
passing through the different slits pass the solenoid on different sides. Classically, this
would make no difference because there is no electromagnetic field outside the solenoid.
Quantum mechanically, however, there is an effect, as first suggested by Aharonov and
Bohm. The set-up with the two slits and the solenoid is known as an Aharonov-Bohm
experiment and its result is the Aharonov-Bohm effect. To investigate the Aharonov-Bohm
effect, we must include the electromagnetic contribution
Z
~v (t) ~
Sem [~x(t)] =
dt eΦ(~x(t), t) − e
· A(~x(t), t)
c
Z
Z
~v (t) ~
e
~ · A(~
~ x, t)
· A(~x(t), t) =
=
dt e
dl
(14.4.6)
c
c C
to the action in the Feynman path integral. Here C is the path along which the particle
travels. The relative phase between the two paths C1 and C2 contributing in the double
slit experiment, that is generated by this term, is given by
Z
Z
1
e
~ · A(~
~ · A(~
~ x, t) −
~ x, t) =
(Sem [~x1 (t)] − Sem [~x2 (t)]) =
dl
dl
~
~c
C1
C2
Z
e
~ · A(~
~ x, t) = eΦ .
dl
(14.4.7)
~c C
~c
Here C = C1 ∪ C2 is the closed curve obtained by combining the two paths C1 and C2 . By
construction, this curve winds around the solenoid and hence the corresponding integral
14.5. FLUX QUANTIZATION FOR MONOPOLES AND SUPERCONDUCTORS 163
yields the magnetic flux ϕ. The resulting interference pattern of the modified double slit
experiment is now given by
eΦ
0
2
2 M yd
.
(14.4.8)
ρ(y) = |Ψ(y)| = 4ρ0 cos
+
2~T
2~c
In other words, the presence of the magnetic flux Φ causes an observable effect in the
interference pattern. Interestingly, when the magnetic flux is quantized in integer units n,
i.e. when
~c
Φ = 2πn ,
(14.4.9)
e
the solenoid becomes invisible to the electron and thus behaves classically.
14.5
Flux Quantization for Monopoles and Superconductors
In 1931 Dirac investigated the quantum mechanics of magnetically charged particles —
so-called magnetic monopoles. There is no experimental evidence for such particles, and
~ B(~
~ x, t) = 0 indeed says that magnetic monopoles do not exist. Still,
Maxwell’s equation ∇·
there are some theories — the so-called grand unified extensions of the standard model of
particle physics — that naturally contain magnetically charged particles. The magnetic
field of a point-like magnetic charge (a so-called Dirac monopole) is determined from
~ · B(~
~ x, t) = 4πgδ(~x),
∇
(14.5.1)
where g is the magnetic charge of the monopole. The resulting magnetic field is given by
~ x, t) = g ~er ,
B(~
r2
(14.5.2)
and the corresponding magnetic flux through any surface S surrounding the monopole is
given by
Z
~ x, t) = 4πg.
Φ=
df~ · B(~
(14.5.3)
S
~ x, t) =
The magnetic field of a monopole can not be obtained from a vector potential as B(~
~ × A(~
~ x, t). In order to still be able to work with a vector potential Dirac introduced
∇
a mathematical trick — the so-called Dirac string. A Dirac string is an infinitely thin
solenoid that emanates from the monopole and carries its magnetic flux g to infinity. Of
course, the Dirac string (being just a mathematical trick) must be physically invisible.
This is the case if the magnetic charge g and thus the magnetic flux of the solenoid obeys
the Dirac quantization condition
g=
Φ
~c
~c
=
⇒ eg = .
4π
2e
2
(14.5.4)
The Dirac quantization condition implies that the units e and g of electric and magnetic
charge are related. In particular, the existence of magnetic monopoles would immediately
164
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
explain why electric charge is quantized. This is one reason why many physicists find
grand unified theories an attractive idea for going beyond the standard model of particle
physics. However, despite numerous experimental efforts, nobody has ever observed a
single magnetic charge. Grand unified theories predict that magnetic monopoles must have
existed immediately after the big bang. Then, why don’t we see any of them today? This
question has been answered by Alan Guth from MIT. In his scenario of the inflationary
universe with an exponential expansion early on magnetic monopoles get so much diluted
that it would indeed be impossible to find any in the universe today.
In the presence of magnetic monopoles flux quantization would simply follow the Dirac
quantization condition. Interestingly, magnetic flux is also quantized inside superconductors, despite the fact that no magnetic monopoles seem to exist in the universe. Superconductivity is a very interesting effect that occurs in metals at very low temperatures. Below
a critical temperature, electric currents flow in a superconductor without any resistance.
This effect is due to a pair formation of electrons. The so-called Cooper pairs (consisting of two electrons) are bosons and can thus undergo Bose-Einstein condensation. The
electrons in the condensate of Cooper pairs are in a coherent quantum state that leads
to superconductivity. In order to form a Cooper pair, the electrons must overcome their
Coulomb repulsion. This is possible inside a crystal lattice because the quantized lattice
oscillations — the phonons — mediate an attractive force between electrons. There are
other interesting materials — the more recently discovered ceramic high-temperature superconductors — for which the mechanism of Cooper pair formation is not yet understood.
Both the ordinary low-temperature superconductors and the more exotic high-temperature
superconductors show the so-called Meissner effect: magnetic fields are expelled from the
superconductor. When a ring of non-superconducting metal is placed in an external magnetic field, the magnetic flux permeates the material. When the ring is cooled down below
the critical temperature it becomes superconducting and the magnetic flux is expelled.
Some of the flux is expelled to the outside, some to the inside of the ring. It turns out
that the magnetic flux trapped inside the ring is quantized in units of
Φ = 2πn
~c
.
2e
(14.5.5)
Interestingly, the quantization is such that the trapped flux does not give rise to an observable Aharonov-Bohm phase for a Cooper pair of charge 2e moving around the ring. The
wave function of an electron (of charge e), on the other hand, would pick up a minus sign
when it moves around the ring. The flux quantization in superconductors is a dynamical
effect. If the trapped magnetic flux would not be quantized, the Aharonov-Bohm effect for
Cooper pairs would destroy their coherence and hence the superconductivity. It is energetically favorable to maintain the superconducting state and have an invisible quantized
trapped flux. It is irrelevant that the trapped flux is visible to unpaired electrons because
those do not superconduct.
14.6. CHARGED PARTICLE IN A CONSTANT MAGNETIC FIELD
14.6
165
Charged Particle in a Constant Magnetic Field
~ = B~ez
Let us now consider the case of a constant homogeneous external magnetic field B
in the z-direction which can be obtained from the potentials
Φ(~x, t) = 0, Ax (~x, t) = 0, Ay (~x, t) = Bx, Az (~x, t) = 0.
(14.6.1)
~ x, t) is to some extent ambiguous.
It should be noted that this choice of Φ(~x, t) and A(~
Other gauge equivalent choices yield the same physical results.
First, we consider a classical particle moving in the x-y-plane perpendicular to the
magnetic field. The particle experiences the Lorentz force
~v (t)
F~ (t) = −e
× B~ez ,
c
(14.6.2)
which forces the particle on a circular orbit of some radius r. It moves along the circle
with an angular velocity ω, which implies the linear velocity v = ωr and the acceleration
a = ω 2 r. Hence, Newton’s equation takes the form
mω 2 r = e
ωr
eB
B ⇒ ω=
.
c
Mc
(14.6.3)
The so-called cyclotron frequency ω is independent of the radius r.
Next, we consider the same problem semiclassically, i.e. using Bohr-Sommerfeld quantization. For this purpose, we first compute the action of a classical periodic cyclotron
orbit ~xc (t). Using eq.(14.2.9) one obtains
Z
S[~xc (t)] =
0
T
1
eBπr2
dt L = M ω 2 r2 T +
.
2
c
(14.6.4)
Similarly, using eq.(14.2.13) one finds for the energy
1
E = H = M ω2 r2 .
2
(14.6.5)
Hence, the Bohr-Sommerfeld quantization condition takes the form
S[~xc (t)] − ET = 2π~n ⇒ r2 =
2~n
.
Mω
(14.6.6)
Consequently, in quantum mechanics the allowed radii of cyclotron orbits are now quantized. Inserting the above expression for r2 into the energy one obtains
E = ~ωn.
(14.6.7)
Interestingly, up to a constant ~ω/2 the semiclassical quantized energy values are those of
a harmonic oscillator with the cyclotron frequency ω.
166
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
Finally, we consider the problem fully quantum mechanically. The Schrödinger equation (14.3.14) then takes the form
~2
eBx
eBx
2
2
i~∂t Ψ(~x, t) = −
∂ + ∂y + i
∂y + i
+ ∂z Ψ(~x, t)
2M x
~c
~c
"
#
eBx
~2
eBx 2
2
2
2
∂ + ∂y + 2i
+ ∂z Ψ(~x, t).
= −
∂y −
2M x
~c
~c
(14.6.8)
The corresponding time-independent Schrödinger equation takes the form
"
#
2
~2
eBx
eBx
−
∂ 2 + ∂y2 + 2i
∂y −
+ ∂z2 Ψ(~x) = EΨ(~x).
2M x
~c
~c
(14.6.9)
Again, we want to consider motion in the x-y-plane. We make the factorized ansatz
Ψ(~x) = ψ(x) exp(ipy y/~) (which implies pz = 0) and we obtain
py 2
~2 2 1
2
−
∂ + Mω x +
ψ(x) = Eψ(x).
(14.6.10)
2M x 2
Mω
Indeed, this is the equation of motion of a (shifted) harmonic oscillator. Hence, the
quantum mechanical energy spectrum takes the form
1
E = ~ω n +
.
(14.6.11)
2
Interestingly, the energy of the charged particle is completely independent of the transverse
momentum py . As a result, the energy levels — which are known as Landau levels — have
an infinite degeneracy.
14.7
The Quantum Hall Effect
The quantum Hall effect is one of the most remarkable effects in condensed matter physics
in which quantum mechanics manifests itself on macroscopic scales. One distinguishes
three effects: the classical Hall effect discovered by the student Hall in 1879, the integer
quantum Hall effect discovered by von Klitzing in 1980 (Nobel prize in 1985), and the
fractional quantum Hall effect discovered by Tsui, Störmer and Gossard in 1982 and
theoretically explained by Laughlin (Nobel prize in 1998).
The basic experimental set-up is the same in the three cases. Some flat (quasi 2dimensional) rectangular sample of size Lx × Ly (oriented in the x-y-plane) of some
~ = B~ez
conducting or semi-conducting material is placed in a strong magnetic field B
perpendicular to the sample. A current with density
~j = −ne~v ,
(14.7.1)
14.7. THE QUANTUM HALL EFFECT
167
is flowing through the sample in the y-direction, i.e. jy = −nev. Here n is the number of
electrons per unit area which move with the velocity ~v = v~ey . The electrons moving in
the magnetic field experience the Lorentz force
~v
~
F~ = −e × B.
c
(14.7.2)
The resulting sideways motion of the electrons leads to an excess of negative charge on
one edge of the sample which generates an electric field transverse to the current. One
then observes a voltage drop along the transverse direction, the x-direction in this case.
More electrons move to the edge of the sample until the resulting electric field
~ = − ~v × B,
~
E
c
(14.7.3)
exactly compensates the Lorentz force. Hence, in equilibrium we have
~
~
~ = j × B.
E
nec
(14.7.4)
~ = ρ~j,
E
(14.7.5)
The resistivity tensor ρ is defined by
such that we can identify
B
ρ=
nec
0 1
−1 0
.
(14.7.6)
The conductivity tensor σ is the matrix inverse of the resistivity tensor ρ and it is defined
by
~
~j = σ E,
(14.7.7)
such that
nec
σ=
B
0 −1
1 0
.
(14.7.8)
Since both the resistivity and the conductivity tensor are purely off-diagonal the material
behaves paradoxically. On the one hand, it looks insulating because σxx = 0, i.e. there is
no current in the direction of the electric field. On the other hand, it looks like a perfect
conductor because ρxx = 0. The Hall resistivity is given by
ρxy =
B
.
nec
(14.7.9)
This is indeed the result of the classical Hall effect. The resistivity is linear in the magnetic
field B and depends on the electron density n. The classical Hall effect can thus be used
to determine the electron density by measuring the Hall resistivity as well as the applied
magnetic field.
Remarkably, at sufficiently low temperatures eq.(14.7.9) is not the correct result in real
samples. In fact, one observes deviations from the linear behavior in the form of quantum
168
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
Hall plateaux, i.e. the Hall resistivity becomes independent of the magnetic field in regions
that correspond to the quantized values
h
.
(14.7.10)
νe2
Here h is Planck’s quantum, e is the electric charge, and ν is an integer. This so-called
integer quantum Hall effect was first observed by von Klitzing and his collaborators. Later,
quantum Hall plateaux were also observed at fractional values (e.g. at ν = 1/3, 2/5, 3/7)
by Tsui, Störmer and Gossard. This is now known as the fractional quantum Hall effect.
Very surprisingly, the value of the resistivity is independent of any microscopic details of
the material, the purity of the sample, the precise value of the magnetic field, etc. In fact,
the quantum Hall effect is now used to maintain the standard of resistance. At each of the
quantum Hall plateaux the dissipative resistivity ρxx practically drops to zero (by about
13 orders of magnitude).
ρxy =
In addition to the previous discussion of Landau levels we should now add the trans~ = Ex~ex generated by the sideways moving electrons. The Schrödinger
verse electric field E
equation then takes the form
py 2
~2 2 1
2
−
∂ + Mω x +
+ eEx x ψ(x) =
2M x 2
Mω
"
#
py
~2 2 1
eEx 2 py eEx
e2 Ex2
2
−
∂ + Mω x +
+
−
−
ψ(x) = Eψ(x).
2M x 2
M ω M ω2
Mω
2M ω 2
(14.7.11)
Again, this is the equation of motion of a shifted harmonic oscillator, now centered around
py
eEx
−
.
(14.7.12)
M ω M ω2
It is straightforward to construct the current density and to verify that eq.(14.7.3) is indeed
satisfied for the quantum mechanical solution.
x0 = −
Let us count the number of states in each of the Landau levels for a Hall sample of
size Lx × Ly . For simplicity, we choose periodic boundary conditions in the y-direction,
such that
2π~
py =
ny ,
(14.7.13)
Ly
with ny ∈ Z. Since the wave function must be located inside the Hall sample one must
have x0 ∈ [0, Lx ]. The number N of ny values for which this is the case, and hence the
degeneracy of a given Landau level, is
M ωLx Ly
eBLx Ly
=
.
(14.7.14)
2π~
2π~c
When ν Landau levels are completely occupied by N ν electrons, the number density (per
unit area) of electrons is given by
N=
n=
Nν
eBν
=
.
Lx Ly
2π~c
(14.7.15)
14.8. THE “NORMAL” ZEEMAN EFFECT
169
Using eq.(14.7.9) one then finds
ρxy =
h
B
= 2,
nec
νe
(14.7.16)
i.e. the quantum Hall plateaux correspond to completely filled Landau levels. It is clear
that complete filling implies a very special situation for the electron system in the Hall
sample. In particular, in that case there are no available low-lying energy levels into which
interacting electrons could scatter, and thus dissipation is extremely suppressed. A full
understanding of the integer quantum Hall effect requires the incorporation of impurities in
the sample. In particular, it is non-trivial to understand the enormous precision with which
eq.(14.7.9) is realized despite numerous imperfections in actual materials. To understand
this goes beyond the scope of the present discussion.
In contrast to the integer quantum Hall effect, the fractional quantum Hall effect relies
heavily on electron-electron correlations. It is a complicated collective phenomenon in
which electrons break up into quasiparticles with fractional charge and with with fractional
spin — so-called anyons. Again, the explanation of the fractional quantum Hall effect is
beyond the scope of this course.
14.8
The “Normal” Zeeman Effect
Let us briefly discuss the so-called “normal” Zeeman effect, which describes the shift
of energy levels of hydrogen-like atoms in a weak homogeneous external magnetic field
~ = B~ez ignoring the effects of spin. When spin is included one speaks of the “anomalous”
B
Zeeman effect. This is clearly a case of bad nomenclature because normally spin plays an
important role and the “anomalous” Zeeman effect is the more common phenomenon. In
this case, it is natural to choose the so-called symmetric gauge
Ax (~x, t) = −
By
Bx
, Ay (~x, t) =
, Az (~x, t) = 0.
2
2
(14.8.1)
Then to leading order in the magnetic field we have
e 2 e 2
eB
px + Ax + py + Ay = p2x + p2y +
(xpy − ypx ) + O(B 2 ).
c
c
c
(14.8.2)
Hence, the “normal” Zeeman effect results from the Hamilton operator
H = H0 +
eBLz
,
2µc
(14.8.3)
where H0 is the unperturbed problem (at B = 0) and µ is the reduced mass of the
electron and the atomic nucleus. Due to the rotation invariance of the unperturbed
~ = 0. The external magnetic field explicitly breaks rotation invariance.
problem [H0 , L]
Only the rotations around the z-axis remain a symmetry of the perturbed problem and
hence [H, Lz ] = 0. Since the unperturbed states of hydrogen-like atoms can be chosen as
170
CHAPTER 14. CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD
eigenstates of Lz with eigenvalue ~m, they remain eigenstates when the magnetic field is
switched on. However, the energy eigenvalue is now shifted to
E = E0 +
eB~m
.
2µc
(14.8.4)
As a result, the (2l+1)-fold degeneracy of states with angular momentum quantum number
l is lifted.
Chapter 15
Coupling of Angular Momenta
15.1
Quantum Mechanical Angular Momentum
Angular momentum is a fundamental quantity that is conserved in any known physical
process. Angular momentum conservation is a consequence of the isotropy of space —
the laws of Nature are invariant against spatial rotations. Of course, rotation invariance
may be broken explicitly under certain conditions, for example, in the presence of external
electric or magnetic fields. In that case, the subsystem without the fields is not rotation
invariant. Still, the total system behaves in the same way when everything including the
fields is spatially rotated. Angular momentum is a vector. In quantum mechanics its
components cannot be measured simultaneously because the corresponding operators do
not commute. This has interesting consequences for the physical behavior of quantum
mechanical particles under spatial rotations.
~ = ~r × p~ familiar from classical mechanics,
Besides the orbital angular momentum L
quantum mechanical particles can carry an internal angular momentum known as spin.
While orbital angular momentum is quantized in integer units, spin may be quantized in
integer or half-integer units. Interestingly, there is an intimate connection between spin
and statistics: particles with half-integer spin obey Fermi-Dirac statistics and are thus
fermions, while particles with integer spin obey Bose-Einstein statistics and are hence
bosons. This connection between spin and statistics can be understood in the framework
of relativistic quantum field theory — but not from quantum mechanics alone.
~ and spin S,
~ in general only
When particles carry both orbital angular momentum L
~
~
~
their total angular momentum J = L+ S is conserved. In that case, one is confronted with
the problem of coupling two angular momenta together. The same problem arises when
several particles add their angular momenta together to the conserved angular momentum
of the total system. Performing the corresponding angular momentum “gymnastics” is an
important tool of the quantum mechanic. Although the subject is a bit formal, its understanding is vital in atomic, molecular and particle physics, as well as in other branches of
171
172
CHAPTER 15. COUPLING OF ANGULAR MOMENTA
our field.
Let us consider the commutation relations of an arbitrary angular momentum J~ in
quantum mechanics. Here J~ may be an orbital angular momentum, a spin, or any com~ we have already derived the
bination of these. For an orbital angular momentum L
commutation relation
[Li , Lj ] = i~εijk Lk ,
(15.1.1)
~ = ~r × p~ and from the fundamental commutation relation [xi , pj ] =
from the definition L
i~δij . Now we postulate
[Ji , Jj ] = i~εijk Jk ,
(15.1.2)
for any angular momentum in quantum mechanics. In particular, different components of
the angular momentum vector do not commute with one another. However, all components
commute with the magnitude J~2 = Jx2 +Jy2 +Jz2 , i.e. [Ji , J~2 ] = 0. As a consequence, one can
construct simultaneous eigenstates of the operator J~2 and one component Ji . Usually one
chooses Jz , i.e. the arbitrary quantization direction is then the z-direction. In general, this
choice does not imply a physical violation of rotation invariance. One could have chosen
any other quantization axis without any effect on the physics. When rotation invariance
is already broken, for example, by the direction of an external electric or magnetic field,
it is very convenient to choose the quantization axis along the same direction. As usual,
we choose the z-direction as our quantization axis and we thus construct simultaneous
eigenstates |j, mi of both J~2 and Jz ,
J~2 |j, mi = ~2 j(j + 1)|j, mi, Jz |j, mi = ~m|j, mi.
(15.1.3)
It will turn out that both j and m are either integer or half-integer.
For convenience, we introduce the angular momentum raising and lowering operators
J± = Jx ± iJy .
(15.1.4)
They obey the commutation relations
[J+ , J− ] = 2~Jz , [J~2 , J± ] = 0, [Jz , J± ] = ±~J± .
(15.1.5)
Jz J± |j, mi = (J± Jz + [Jz , J± ])|j, mi = ~(m ± 1)J± |j, mi,
(15.1.6)
We have
i.e. the state J± |j, mi is also an eigenstate of Jz and has the quantum number m ± 1.
Similarly
J~2 J± |j, mi = J± J~2 |j, mi = ~2 j(j + 1)J± |j, mi,
(15.1.7)
i.e. J± |j, mi is still also an eigenstate of J~2 with the unchanged quantum number j. Since
j determines the magnitude of the angular momentum vector, one expects that for fixed j
the m quantum number that measures the z-component of the angular momentum vector
should be bounded from above and from below. On the other hand, for any state |j, mi
with quantum number m one can construct the states
J± |j, mi = Cj,m |j, m ± 1i,
(15.1.8)
15.1. QUANTUM MECHANICAL ANGULAR MOMENTUM
173
with quantum number m ± 1. The apparent contradiction is resolved only if the constant
Cj,m vanishes for a given m = mmax or m = mmin . Let us compute Cj,m from the
normalization condition hj, m ± 1|j, m ± 1i = 1. Using the relation
†
J±
J± = J∓ J± = J~2 − Jz2 ∓ ~Jz ,
(15.1.9)
we obtain
†
|Cj,m |2 = |Cj,m |2 hj, m ± 1|j, m ± 1i = hj, m|J±
J± |j, mi
2
2
= hj, m|J~ − J ∓ ~Jz |j, mi
z
2
= ~ [j(j + 1) − m(m ± 1)].
(15.1.10)
For fixed j, the maximal value mmax of the quantum number m is determined by Cj,mmax =
0, which implies mmax = j. Similarly, the minimal value is determined by Cj,mmin = 0,
which implies mmin = −j. Hence, for fixed j there are 2j + 1 possible m values
m ∈ {mmin , mmin + 1, ..., mmax − 1, mmax } = {−j, −j + 1, ..., j − 1, j}.
(15.1.11)
Since the difference mmax − mmin = 2j is an integer, j can be an integer or a half-integer.
Both possibilities are realized in Nature.
Let us now couple two angular momenta J~1 and J~2 together to a total angular momentum
J~ = J~1 + J~2 .
(15.1.12)
The angular momenta J~1 and J~2 could, for example, be orbital angular momentum and
spin of the same particle, or angular momenta of two different particles. In any case, since
they act in different Hilbert spaces the two angular momentum operators commute with
one another, i.e.
[J1i , J2j ] = 0.
(15.1.13)
As a consequence, the total angular momentum operator J~ = J~1 + J~2 indeed obeys the
usual commutation relations
[Ji , Jj ] = [J1i , J1j ] + [J2i , J2j ] = i~εijk (J1k + J2k ) = i~εijk Jk .
(15.1.14)
Let us assume that the states of subsystem 1 have a fixed quantum number j1 and
are given by |j1 , m1 i. Similarly, the states of subsystem 2 have quantum number j2 and
are given by |j2 , m2 i. Hence, the combined system has (2j1 + 1)(2j2 + 1) product states
|j1 , m1 i|j2 , m2 i which span the Hilbert space of the total system. How does this space
decompose into sectors of definite total angular momentum? It will turn out that the
possible values for j are restricted by
j ∈ {|j1 − j2 |, |j1 − j2 | + 1, ..., j1 + j2 }.
(15.1.15)
Indeed, the total number of states then is
(2|j1 − j2 | + 1) + (2|j1 − j2 | + 3) + ... + (2(j1 + j2 ) + 1) = (2j1 + 1)(2j2 + 1). (15.1.16)
174
CHAPTER 15. COUPLING OF ANGULAR MOMENTA
Also the question arises how one can construct states
|(j1 , j2 )j, mi =
X
Cm1 ,m2 |j1 , m1 i|j2 , m2 i,
(15.1.17)
m1 ,m2
as linear combinations of the product states? This is the “gymnastics” problem of coupling together two angular momenta. The factors Cm1 ,m2 are known as Clebsch-Gordan
coefficients.
15.2
Coupling of Spins
Let us consider the spins of two spin 1/2 particles, for example, a proton and a neutron
forming the atomic nucleus of heavy hydrogen (deuterium). The corresponding bound
state of proton and neutron is known as a deuteron. What are the possible total spins of
the coupled system? In this case j1 = j2 = 1/2 and thus
j ∈ {|j1 − j2 |, |j1 − j2 | + 1, ..., j1 + j2 } = {0, 1},
(15.2.1)
i.e. the total spin j is either 0 (a singlet) or 1 (a triplet). Altogether, there are four
states. This is consistent because there are also four product states |j1 , m1 i|j2 , m2 i with
m1 = ±1/2 and m2 = ±1/2. For these four states we introduce the short-hand notation
1 1 1 1
1 1 1 1
| , i| , i = | ↑↑i, | , i| , − i = | ↑↓i,
2 2 2 2
2 2 2 2
1 1 1 1
1 1 1 1
| , − i| , i = | ↓↑i, | , − i| , − i = | ↓↓i.
2 2 2 2
2 2 2 2
(15.2.2)
These product states are eigenstates of Jz = J1z + J2z ,
1 1
Jz | ↑↑i = ~( + )| ↑↑i = ~| ↑↑i,
2 2
1 1
1 1
Jz | ↑↓i = ~( − )| ↑↓i = 0, Jz | ↓↑i = ~(− + )| ↓↑i = 0,
2 2
2 2
1 1
Jz | ↓↓i = ~(− − )| ↓↓i = −~| ↓↓i.
2 2
(15.2.3)
The first and the last of the four states must belong to j = 1 because they have m = ±1,
i.e.
1 1
1 1
|( , )1, 1i = | ↑↑i, |( , )1, −1i = | ↓↓i.
(15.2.4)
2 2
2 2
The two remaining states have m = 0. One linear combination of them is the m = 0 state
of the triplet, and the orthogonal combination is the m = 0 state with j = 0. In order to
identify the m = 0 state with j = 1 we act with the lowering operator
1 1
1
1 1
1
1
|( , )1, 0i = √ J− |( , )1, 1i = √ (J1− + J2− )| ↑↑i = √ (| ↑↓i + | ↓↑i). (15.2.5)
2 2
2 2
2~
2~
2
15.2. COUPLING OF SPINS
175
The orthogonal combination
1
1 1
(15.2.6)
|( , )0, 0i = √ (| ↑↓i − | ↓↑i),
2 2
2
should hence be the state with j = 0. This can be checked explicitly, for example, by
acting with the operator J~+ = J1+ + J2+ , i.e.
1 1
~
1
J~+ |( , )0, 0i = (J1+ + J2+ ) √ (| ↑↓i − | ↓↑i) = √ (| ↑↑i − | ↑↑i) = 0.
(15.2.7)
2 2
2
2
Hence, according to the previous discussion the spin of the deuteron could be 0 or
1. The proton and neutron that form the deuteron nucleus attract each other through
the so-called strong interaction. This interaction is spin-dependent. There is a term
proportional to −J~1 · J~2 (where J~1 and J~2 are the spin operators of proton and neutron)
in the proton-neutron potential. One can write
J~2 = (J~1 + J~2 )2 = J~12 + J~22 + 2J~1 · J~2 ,
(15.2.8)
and hence
1
−J~1 · J~2 = (J~12 + J~22 − J~2 ).
2
Since both the proton and the neutron have spin 1/2 we have
1
1
3
2
2
2
J~1 = J~2 = ~
+ 1 = ~2 .
2 2
4
In the spin singlet state we have J~2 = 0 and hence
1 1
3
1 1
−J~1 · J~2 |( , )0, 0i = ~2 |( , )0, 0i.
2 2
4
2 2
2
2
~
In the spin triplet state, on the other hand, J = 2~ such that
(15.2.9)
(15.2.10)
(15.2.11)
1
1 1
1 1
(15.2.12)
−J~1 · J~2 |( , )1, mi = − ~2 |( , )1, mi.
2 2
4
2 2
In the triplet channel the proton-neutron interaction is attractive while in the singlet
channel it is repulsive. As a consequence, the deuteron has spin 1, while in the spin 0
channel there is no bound state.
The strong interaction is mediated by gluons, just as electromagnetic interactions are
mediated by photons. The strong interaction analogs of electrons and positrons are quarks
and anti-quarks — the basic building blocks of protons and neutrons. There are two uquarks and one d-quark in each proton. A u-quark has electric charge 2/3 and a d-quark
has −1/3. Hence, the charge of a proton is indeed 2 × 2/3 − 1/3 = 1. Similarly, a neutron
contains one u-quark and two d-quarks and thus has charge 2/3 − 2 × 1/3 = 0. Like
electrons, quarks are fermions with spin 1/2. What is the possible total spin of a bound
system of three quarks? As we learned before, two spin 1/2 particles can couple to a total
spin j = 0 or j = 1. When a third spin 1/2 particle is added to the j = 0 state, the total
spin is 1/2. If it is added to the j = 1 state the total spin can be either j − 1/2 = 1/2
or j + 1/2 = 3/2. Again, the strong interactions between quarks are spin-dependent and
they favor the total spin 1/2 states corresponding to proton and neutron. The spin 3/2
state also exists but is unstable. It is known in particle physics as the ∆-isobar.
176
15.3
CHAPTER 15. COUPLING OF ANGULAR MOMENTA
Coupling of Orbital Angular Momentum and Spin
Just as there are spin-dependent strong interactions between protons and neutrons or between quarks, there are also spin-dependent electromagnetic interactions between electrons
and protons. As we already discussed in the context of the relativistic Dirac equation,
there are spin-orbit coupling terms proportional to
2
~ ·S
~ = 1 (J~2 − L
~2 − S
~ 2 ) = ~ j(j + 1) − l(l + 1) − 3 .
L
(15.3.1)
2
2
4
Let us consider the coupling of an orbital angular momentum l and a spin s = 1/2 of an
electron in more detail. In this case, there are 2(2l + 1) product states |l, ml i|s, ms i. The
possible values of the total angular momentum are
1
1
j ∈ {|l − s|, l + s} = {l − , l + }.
2
2
(15.3.2)
1
1
2 l−
+1+2 l+
+ 1 = 2(2l + 1)
2
2
(15.3.3)
Indeed, there are again
states. The direct product state
1
1
1
|(l, ) l + , l + i = |l, li| ↑i
2
2
2
(15.3.4)
has m = l + 1/2 and must thus have j = l + 1/2. We can construct the other states with
j = l + 1/2 and with lower m-values by acting with J− = L− + S− . For example,
1
1
1
|(l, ) l + , l − i =
2
2
2
=
=
1
J− |l, li| ↑i
2l + 1~
1
√
(L− + S− )|l, li| ↑i
2l + 1~
√
1
√
(|l, li| ↓i + 2l|l, l − 1i| ↑i).
2l + 1
√
(15.3.5)
The orthogonal combination
√
1
1
1
1
|(l, ) l − , l − i = √
( 2l|l, li| ↓i − |l, l − 1i| ↑i),
2
2
2
2l + 1
(15.3.6)
has j = l − 1/2 and m = l − 1/2. Again, by acting with J− one can generate all other
states with j = l − 1/2 and smaller m-values.
Chapter 16
Systems of Identical Particles
16.1
Bosons and Fermions
Identical elementary particles (like two photons or two electrons) are indistinguishable from
one another — they are like perfect twins. Ultimately, this is a consequence of quantum
field theory. Elementary particles are quantized excitations of fields that fill all of space.
Since the properties of the field are the same everywhere in space, a particle created here
has exactly the same properties as a particle created somewhere else. In classical physics
one usually thinks about a particle in a different way. Starting from some initial positions,
one can use Newton’s equation to predict the particles’ paths and one can always keep
track of where a given particle came from. Classically, one can always distinguish two
particles by referring to their initial positions. One can say: “This is the same particle
that originally came from the position ~x1 ”. It is classically distinguishable from another
particle that started from the position ~x2 . However, photons or electrons are not classical
particles. As elementary particles they must be treated quantum mechanically, and at the
quantum level the concept of a classical path breaks down. In quantum mechanics it is
impossible to keep track of a particle by constantly watching it as it travels along a path.
Instead a quantum system of N particles is described by a multi-particle wave function
Ψ(~x1 , ~x2 , ..., ~xN ), where ~x1 , ~x2 , ..., ~xN are the positions of the particles. The absolute
value |Ψ(~x1 , ~x2 , ..., ~xN )|2 determines the probability density for finding the particles at the
positions ~x1 , ~x2 , ..., ~xN . Consequently, the normalization condition takes the form
Z
d3 x1 d3 x2 ...d3 xN |Ψ(~x1 , ~x2 , ..., ~xN )|2 = 1.
(16.1.1)
Similarly, the probability density for finding one of the particles at the position ~x1 while
the other particles are in arbitrary positions is given by
Z
ρ(~x1 ) = d3 x2 ...d3 xN |Ψ(~x1 , ~x2 , ..., ~xN )|2 = 1.
(16.1.2)
As we said above, identical elementary particles are absolutely indistinguishable. Hence,
when two identical particles are interchanged, this has no physically observable effects.
177
178
CHAPTER 16. SYSTEMS OF IDENTICAL PARTICLES
Let us denote the pair permutation operator of two identical particles by Pij . It acts on
a wave function by interchanging the corresponding particle coordinates. For example,
P12 Ψ(~x1 , ~x2 , ..., ~xN ) = Ψ(~x2 , ~x1 , ..., ~xN ).
(16.1.3)
For identical particles the states Ψ(~x1 , ~x2 , ..., ~xN ) and its corresponding permutation partner Ψ(~x2 , ~x1 , ..., ~xN ) are physically equivalent. However, this does not imply that the two
wave functions are the same. In fact, they may still differ by a complex phase which drops
out from physical observables.
The Hamilton operator of a system of non-relativistic identical particles can be written
as
N
~2 X
H=−
∆i + V (~x1 , ~x2 , ..., ~xN ),
2m
(16.1.4)
i=1
Here m is the particle mass and ∆i is the Laplace operator that takes second derivatives
with respect to the coordinates of ~xi . It should be noted that the masses of all the
particles are the same — otherwise they would not be indistinguishable. The potential
energy V (~x1 , ~x2 , ..., ~xN ) depends on the coordinates of all particles. For indistinguishable
particles the potential energy must be invariant against particle permutations Pij , for
example,
P12 V (~x1 , ~x2 , ..., ~xN ) = V (~x2 , ~x1 , ..., ~xN ) = V (~x1 , ~x2 , ..., ~xN ).
(16.1.5)
One obtains
#
N
~2 X
−
∆i + V (~x2 , ~x1 , ..., ~xN ) Ψ(~x2 , ~x1 , ..., ~xN )
2M
"
P12 HΨ(~x1 , ~x2 , ..., ~xN ) =
i=1
= HP12 Ψ(~x1 , ~x2 , ..., ~xN ),
(16.1.6)
and thus P12 H = HP12 . The same is true for all permutations and hence
[H, Pij ] = 0.
(16.1.7)
This means that the eigenstates of H can be chosen such that they are simultaneously
eigenstates of Pij . For example, an eigenstate of P12 obeys
P12 Ψ(~x1 , ~x2 , ..., ~xN ) = λΨ(~x1 , ~x2 , ..., ~xN ),
(16.1.8)
where λ is the corresponding eigenvalue. When we apply the permutation operator twice
we obtain
2
P12
Ψ(~x1 , ~x2 , ..., ~xN ) = P12 Ψ(~x2 , ~x1 , ..., ~xN ) = Ψ(~x1 , ~x2 , ..., ~xN )
= λ2 Ψ(~x1 , ~x2 , ..., ~xN ) ⇒ λ = ±1.
(16.1.9)
It should be noted that, in general, different pair permutation operators do not commute.
For example, for three particles the operators P12 and P23 do not commute. The combinations P312 = P12 P23 and P231 = P23 P12 correspond to two different cyclic permutations.
16.1. BOSONS AND FERMIONS
179
As a consequence, in general one cannot find simultaneous eigenstates to all permutation
operators Pij .
The permutations Pij form a finite group — the permutation group SN . Since different permutations, in general, do not commute, the group SN is non-Abelian. In this
respect it is like the group of spatial rotations in three dimensions, SO(3). However,
unlike the permutation group SN , the rotation group SO(3) is continuous: there is an
infinite (continuous) number of rotations, but there is only a finite (discrete) number of
permutations. Mathematically speaking, the angular momentum multiplets characterized
by j form (2j + 1)-dimensional irreducible representations of the group SO(3). Similarly,
the irreducible representations of the group SN have different dimensions. There are two
different irreducible representations of the permutation group of two particles S2 : the symmetric and the anti-symmetric one, which are both 1-dimensional. In fact, all permutation
groups SN have two 1-dimensional representations: the totally symmetric and the totally
anti-symmetric one. However, for N ≥ 3 the permutation group SN also has irreducible
representations of larger dimension. For example, besides the 1-dimensional symmetric
and anti-symmetric representations the group S3 has a 2-dimensional irreducible representation of mixed symmetry.
The indistinguishability of elementary particles has profound consequences for their
wave function. For example, in a quantum state in which two particles are located at
the positions ~x1 and ~x2 , it makes no sense to ask which particle is in which position.
Hence, the distinct states |~x1 i|~x2 i and |~x2 i|~x1 i cannot be correct wave functions for such
a quantum state. Only if the two particles were distinguishable it would matter if the
particle of type 1 is at position ~x1 and the particle of type 2 is at position ~x2 (|~x1 i|~x2 i) or
vice versa (|~x2 i|~x1 i). For indistinguishable particles there is only one state describing the
two identical particles at the positions ~x1 and ~x2 . Since the Hamiltonian commutes with
the permutation operator one can choose this state as a simultaneous eigenstate of H and
P12 . Since P12 has eigenvalues ±1, one can construct a symmetric state
1
|si = √ (|~x1 i|~x2 i + |~x2 i|~x1 i) ,
2
(16.1.10)
as well as an anti-symmetric state
1
|ai = √ (|~x1 i|~x2 i − |~x2 i|~x1 i) .
2
(16.1.11)
P12 |si = |si, P12 |ai = −|ai.
(16.1.12)
By construction, we have
Four three identical particles one can again construct a totally symmetric state
|si =
+
1
√ (|~x1 i|~x2 i|~x3 i + |~x2 i|~x1 i|~x3 i + |~x2 i|~x3 i|~x1 i
6
|~x3 i|~x2 i|~x1 i + |~x3 i|~x1 i|~x2 i + |~x1 i|~x3 i|~x2 i) ,
(16.1.13)
180
CHAPTER 16. SYSTEMS OF IDENTICAL PARTICLES
as well as an anti-symmetric state
|ai =
−
1
√ (|~x1 i|~x2 i|~x3 i − |~x2 i|~x1 i|~x3 i + |~x2 i|~x3 i|~x1 i
6
|~x3 i|~x2 i|~x1 i + |~x3 i|~x1 i|~x2 i − |~x1 i|~x3 i|~x2 i) .
(16.1.14)
For any pair permutation operator Pij we then have
Pij |si = |si, Pij |ai = −|ai.
(16.1.15)
Besides the totally symmetric and anti-symmetric states, for three particles there are also
states of mixed symmetry.
16.2
The Pauli Principle
Remarkably, there is an intimate relation between the statistics of particles (i.e. their properties under permutations) and their spin. Bosons obeying Bose-Einstein statistics have
a totally symmetric wave function and have integer spins. Fermions obeying Fermi-Dirac
statistics, on the other hand, have a totally anti-symmetric wave function and have halfinteger spin. This connection between spin and statistics is known as the Pauli principle.
It cannot be derived within the theoretical framework of quantum mechanics alone. However, it can be derived from quantum field theory in the form of the so-called spin-statistics
theorem.
First, we consider bosons with spin 0 whose spin wave function is trivial. The Pauli
principle implies that their orbital wave function Ψ(~x1 , ~x2 , ..., ~xN ) must be totally symmetric, i.e. for any pair permutation
Pij Ψ(~x1 , ~x2 , ..., ~xN ) = Ψ(~x1 , ~x2 , ..., ~xN ).
(16.2.1)
For example, two (or three) spinless bosons located at the positions ~x1 , ~x2 (and ~x3 ) are
described by the states |si constructed above.
For particles with spin the situation is a bit more complicated. Let us now consider
fermions with spin 1/2, whose spin wave function is non-trivial. The Pauli principle
dictates that the total wave function (orbital and spin) is anti-symmetric under particle
exchange. This does not imply that the orbital or spin wave functions alone must be
anti-symmetric. Two spin 1/2 particles can couple their spins to a total spin j = 0 or
j = 1. The singlet state with spin 0 given by
1 1
1
|( , )0, 0i = √ (| ↑↓i − | ↓↑i),
2 2
2
(16.2.2)
is anti-symmetric under the exchange of the two spins. In that case, in order to obtain a
totally anti-symmetric wave function, the orbital wave function Ψ(~x1 , ~x2 ) must be symmetric, i.e. P12 Ψ(~x1 , ~x2 ) = Ψ(~x1 , ~x2 ). On the other hand, the triplet states with spin j = 1
16.2. THE PAULI PRINCIPLE
181
given by
1 1
|( , )1, 1i = | ↑↑i,
2 2
1
1 1
|( , )1, 0i = √ (| ↑↓i + | ↓↑i),
2 2
2
1 1
|( , )1, −1i = | ↓↓i,
(16.2.3)
2 2
are symmetric under spin exchange. Hence, in that case the orbital wave function must
be anti-symmetric, i.e. P12 Ψ(~x1 , ~x2 ) = −Ψ(~x1 , ~x2 ).
Let us now consider three identical spin 1/2 fermions, for example, three quarks inside
a proton or neutron. As we have discussed before, the spins of three spin 1/2 particles
can couple to a total spin 1/2 or 3/2. For three quarks the coupling to a total spin 1/2 is
energetically favored by the spin-dependence of the strong interactions. In that case, the
spin wave function is of mixed symmetry. Here we consider the simpler case of coupling
to total spin 3/2 which leads to the unstable ∆-isobar. For example, if we combine three
u-quarks we obtain the ∆++ particle with electric charge 3 × 2/3 = 2. In this state the
quark spins are coupled to a total spin j = 3/2. The corresponding 2j + 1 = 4 states
1 1 1 3 3
|( , , ) , i = | ↑↑↑i,
2 2 2 2 2
1
1 1 1 3 1
|( , , ) , i = √ (| ↑↑↓i + | ↑↓↑i + | ↓↑↑i),
2 2 2 2 2
3
1 1 1 3 1
1
|( , , ) , − i = √ (| ↓↓↑i + | ↓↑↓i + | ↑↓↓i),
2 2 2 2 2
3
1 1 1 3 3
|( , , ) , − i = | ↓↓↓i,
(16.2.4)
2 2 2 2 2
are totally symmetric under spin exchange. It turns out that the orbital wave function
of the ∆-isobar is also totally symmetric. This seems to contradict the Pauli principle.
The quarks are spin 1/2 fermions and still they seem to have a totally symmetric wave
function. In order to resolve this puzzle a new quantum number for quarks was postulated
by Gell-Mann. Besides orbital quantum numbers and spin quarks also carry “color” —
the generalized charge of the strong interactions. There are three different color states
of a quark, arbitrarily called red, green and blue. Indeed, the color wave function of the
three quarks inside a ∆-isobar (and also of a proton or neutron) is totally anti-symmetric
1
|ai = √ (|rgbi − |rbgi + |gbri − |grbi + |brgi − |bgri).
6
(16.2.5)
This state forms a 1-dimensional representation of the color gauge group SU (3) (a color
singlet). In particular, the three quarks all have different colors. The strong interactions
are not only spin-dependent but, most important, color-dependent. In particular, quarks
are always in a color-singlet state. This leads to one of the most important properties of the
strong interactions — the so-called quark confinement. Despite numerous experimental
efforts, no single quark has ever been observed. They are always confined together in colorsinglet groups of three, forming protons, neutrons, ∆-isobars or other strongly interacting
elementary particles.
182
16.3
CHAPTER 16. SYSTEMS OF IDENTICAL PARTICLES
The Helium Atom
The helium atom consists of three particles: the atomic nucleus and two electrons, interacting through their mutual Coulomb potentials. The Helium nucleus contains Z = 2
protons and two neutrons and thus has electric charge Ze = 2e. The two negatively
charged electrons make the whole atom electrically neutral. The helium nucleus is about
8000 times heavier than an electron and can thus, to a very good approximation, be treated
as fixed in space. The Hamiltonian then takes the form
H = H1 + H2 + V12 .
(16.3.1)
~2
Ze2
∆i −
2M
|~ri |
(16.3.2)
Here
Hi = −
is the Hamilton operator for electron i ∈ {1, 2} at position ~ri in the Coulomb field of
the nucleus located at the origin. The Laplace operator ∆i entering the kinetic energy of
electron i takes second derivatives with respect to the coordinates ~ri . The term
V12 =
e2
|~r1 − ~r2 |
(16.3.3)
represents the Coulomb repulsion between the two electrons. The Hamilton operator H
does not contain any spin-dependent forces because we have neglected tiny relativistic
effects like the spin-orbit interaction. Still, as a consequence of the Pauli principle, we will
find that the spectrum depends on the spin.
The spins s = 1/2 of the two electrons can couple to a total spin S = 0 or S = 1. The
corresponding spin states are anti-symmetric for S = 0, i.e.
|(
11
1
)0, 0i = √ (| ↑↓i − | ↓↑i),
22
2
(16.3.4)
and symmetric for S = 1, i.e.
11
)1, 1i = | ↑↑i,
22
11
1
|( )1, 0i = √ (| ↑↓i + | ↓↑i),
22
2
11
|( )1, −1i = | ↓↓i.
22
|(
(16.3.5)
The S = 0 states are referred to as para-, and the S = 1 states are referred to as orthohelium. The total wave function of the two electrons is given by
|Ψi = Ψ(~r1 , ~r2 )|(
11
)S, Sz i.
22
(16.3.6)
Here the orbital wave function Ψ(~r1 , ~r2 ) solves the time-independent Schrödinger equation
HΨ(~r1 , ~r2 ) = EΨ(~r1 , ~r2 ).
(16.3.7)
16.3. THE HELIUM ATOM
183
In addition, according to the Pauli principle, Ψ(~r1 , ~r2 ) must be symmetric under the
exchange of the coordinates ~r1 and ~r2 for para-helium (S = 0), and anti-symmetric for
ortho-helium (S = 1). Symmetric and anti-symmetric wave functions will in general have
different energies, such that a spin-dependent spectrum emerges although the Hamiltonian
does not contain explicitly spin-dependent forces.
Solving the Schrödinger equation for the helium atom is much more difficult than for
hydrogen and can, in fact, not be done in closed form. Instead, one can use numerical
methods, perturbation theory, or a variational approach. Here we perform an approximate
calculation treating V12 as a small perturbation. As a first step, we completely neglect
the Coulomb repulsion between the electrons and we put V12 = 0. Then the Hamiltonian
H = H1 +H2 separates into two hydrogen-like problems. The corresponding single-particle
Schrödinger equation
(16.3.8)
Hi Ψni ,li ,mi (~ri ) = Eni Ψni ,li ,mi (~ri )
has been solved earlier. The energy eigenvalues for bound states are given by
En = −
Z 2 e4 M
.
2~2 n2
(16.3.9)
The correctly normalized and (anti-)symmetrized two-particle orbital wave function
then takes the form
1
Ψ(~r1 , ~r2 ) = √ (Ψn1 ,l1 ,m1 (~r1 )Ψn2 ,l2 ,m2 (~r2 ) ± Ψn1 ,l1 ,m1 (~r2 )Ψn2 ,l2 ,m2 (~r1 )).
2
(16.3.10)
The plus-sign corresponds to para-helium (spin state anti-symmetric, orbital wave function
symmetric) and the minus-sign corresponds to ortho-helium (spin state symmetric, orbital
wave function anti-symmetric). Neglecting V12 , the ground state has n1 = n2 = 1, l1 =
l2 = 0 and thus m1 = m2 = 0. In that case, the anti-symmetric orbital wave function
1
Ψ(~r1 , ~r2 ) = √ (Ψ1,0,0 (~r1 )Ψ1,0,0 (~r2 ) − Ψ1,0,0 (~r2 )Ψ1,0,0 (~r1 )) ,
2
(16.3.11)
simply vanishes. Hence, the ground state is para-helium (a non-degenerate spin singlet)
with the symmetric orbital wave function
1
√ (Ψ1,0,0 (~r1 )Ψ1,0,0 (~r2 ) + Ψ1,0,0 (~r2 )Ψ1,0,0 (~r1 ))
2
√
=
2Ψ1,0,0 (~r1 )Ψ1,0,0 (~r2 ).
Ψ(~r1 , ~r2 ) =
(16.3.12)
This wave function seems to be incorrectly normalized to 2. However, the correct normalization condition for two identical electrons takes the form
Z
1
d3 r1 d3 r2 |Ψ(~r1 , ~r2 )|2 = 1.
(16.3.13)
2
The prefactor 1/2 arises due to the indistinguishability of the electrons. The value of the
ground state energy is given by E = 2E1 , which is a factor 2Z 2 = 8 larger than the ground
184
CHAPTER 16. SYSTEMS OF IDENTICAL PARTICLES
state energy of the hydrogen atom. Of course, we should keep in mind that we have not
yet included the electron repulsion term V12 . The first excited states still have n1 = 1,
l1 = m1 = 0 (a 1s state), but now n2 = 2 and hence the total energy is E = E1 + E2 .
The possible values for the angular momentum are l2 = m2 = 0 (a 2s state) and l2 = 1,
m2 = 0, ±1 (one of three 2p states). In this case, both para- and ortho-helium states exist
with the orbital wave functions
1
Ψ(~r1 , ~r2 ) = √ (Ψ1,0,0 (~r1 )Ψ2,l,m (~r2 ) ± Ψ1,0,0 (~r2 )Ψ2,l,m (~r1 )) .
2
(16.3.14)
Altogether there are 16 degenerate excited states: the four spin states combined with the
four possible choices for l2 , m2 .
16.4
Perturbation Theory for the Helium Atom
In the next step we will include the electron-electron Coulomb repulsion V12 using perturbation theory. First, we consider the correction ∆E to the ground state energy E = 2E1 ,
which is given by
Z
∆E = hΨ|V12 |Ψi =
d3 r1 d3 r2
e2
|Ψ1,0,0 (~r1 )|2 |Ψ1,0,0 (~r2 )|2 .
|~r1 − ~r2 |
(16.4.1)
From our study of the hydrogen atom, we know the ground state wave function
Ψ1,0,0 (~r) = √
1
πa3
exp(−|~r|/a),
where a = ~2 /(Ze2 M ) is the Bohr radius. In the next step we write
q
p
|~r1 − ~r2 | = |~r1 |2 + |~r2 |2 − 2~r1 · ~r2 = r12 + r22 − 2r1 r2 cos θ.
(16.4.2)
(16.4.3)
Inserting this and performing all angular integrations except the one over the angle θ
between the vectors ~r1 and ~r2 one obtains
Z
8π 2 e2
exp(−2r1 /a) exp(−2r2 /a)
.
(16.4.4)
dr1 dr2 dθ r12 r22 sin θ p 2
∆E = 2 6
π a
r1 + r22 − 2r1 r2 cos θ
Substituting x = cos θ, dx = − sin θdθ one finds
Z
π
1
Z
1
1
dx p 2
=
2
+ − 2r1 r2 cos θ
r1 + r2 − 2r1 r2 x
0
−1
q
q
1
2
2
2
2
−
r1 + r2 − 2r1 r2 − r1 + r2 + 2r1 r2 =
r1 r2
r1 + r2 − |r1 − r2 |
2/r1 for r1 > r2
=
(16.4.5)
2/r2 for r2 > r1 .
r1 r2
dθ sin θ p
r12
r22
=
16.4. PERTURBATION THEORY FOR THE HELIUM ATOM
185
Hence, we now obtain
Z
16e2 ∞
∆E =
dr1 r1 exp(−2r1 /a)
a6
Z r1 0
Z
×
dr2 r22 exp(−2r2 /a) + r1
=
0
5e2
8a
∞
dr2 r2 exp(−2r2 /a)
r1
=
5Ze4 M
.
8~2
(16.4.6)
186
CHAPTER 16. SYSTEMS OF IDENTICAL PARTICLES
Chapter 17
Quantum Mechanics of Elastic
Scattering
In order to investigate the structure of molecules, atoms, atomic nuclei, or individual protons, one must interact with these systems from outside. To study these fundamental
objects, one can probe them with other elementary particles such as electrons, photons,
or neutrinos. For this purpose, one performs scattering experiments. For example, by
bombarding atoms with electrons Rutherford and his collaborators were able to demonstrate that the atomic nucleus is tiny compared to the entire atom. Similarly, electron
scattering experiments have revealed that protons are not truly elementary but consist
of quarks and gluons. In Rutherford’s experiments the atomic nucleus remained intact
and in its ground state, i.e. the electrons were scattered elastically. The experiments that
revealed the quark structure of the proton, on the other hand, were deeply inelastic. In
such a scattering event the proton is broken up into numerous fragments. In this chapter
we consider the quantum mechanics of elastic scattering.
17.1
Differential Cross Section and Scattering Amplitude
Let us consider projectile particles (for example, electrons) moving along the z-direction
with constant momentum ~k = k~ez . They hit a heavy target (for example, an atom) that
is at rest. The target exerts a potential V (~x) (in our example the Coulomb potential) on
the projectile. Here we assume that the target is static and unaffected by the projectile
(elastic scattering), and can be described entirely in terms of the scattering potential V (~x).
In reality the target, of course, also suffers some recoil which we neglect here. It would be
easy to incorporate this by describing the collision in the center of mass frame.
The incoming projectile particles are described by an incident probability current density j which measures the number of incident projectile particles per beam area and time.
The projectile particles are then affected by the scattering potential and in general change
187
188
CHAPTER 17. QUANTUM MECHANICS OF ELASTIC SCATTERING
their direction of motion until they leave the interaction region and disappear to infinity.
The scattering angles (θ, ϕ) describe the direction of the momentum of the projectile after
the collision. When we place a detector covering an angle dΩ in that direction, we measure the current dJ which counts the number of projectile particles per unit time. The
differential cross section is defined as
dσ(θ, ϕ)
dJ 1
=
.
dΩ
dΩ N j
(17.1.1)
Here N is the number of scattering centers (in this case atoms) in the target. The differential cross section thus counts the number of projectiles scattered into the direction (θ, ϕ),
normalized to the incident current and the number of scattering centers. Here we assume
a dilute target. Then each projectile particle scatters only on a single target particle and
the current dJ is proportional to N . For multiple scattering events this would not be the
case. The theory allows us to derive the differential cross section dσ(θ, ϕ)/dΩ from the
scattering potential V (~x). Experiments, on the other hand, directly measure dσ(θ, ϕ)/dΩ
and thus indirectly yield information on V (~x) and thus on the structure of the target. One
can also define the total cross section
Z
dσ(θ, ϕ)
.
(17.1.2)
σ = dΩ
dΩ
We limit the discussion to scattering potentials that vanish at infinity. The energy of
the incident projectile particles is then given by
E=
~2 k 2
.
2M
(17.1.3)
We are looking for a stationary scattering wave function |Ψ(~k)i that solves the timeindependent Schrödinger equation
H|Ψ(~k)i = E|Ψ(~k)i.
(17.1.4)
At asymptotic distances the wave function can be decomposed into the incident plane
wave exp(ikz) and a scattered wave, i.e.
exp(ikr)
h~x|Ψ(~k)i ∼ A exp(ikz) + f (θ, ϕ)
.
(17.1.5)
r
Here r = |~x| is the distance from the target. The factor exp(ikr)/r describes a radial wave
with an angular-dependent amplitude f (θ, ϕ) — the so-called scattering amplitude.
Let us now derive a relation between the scattering amplitude f (θ, ϕ) and the differential cross section dσ(θ, ϕ)/dΩ. The total probability current density is given by
h
i
~k)|~xi)∇h~
~ x|Ψ(~k)i − (∇hΨ(
~
~ x|Ψ(~k)i .
~j(~x) = ~ hΨ(~k)|~xi∇h~
(17.1.6)
2M i
For the incident plane wave A exp(ikz) one obtains
~jin (~x) = ~k |A|2~ez .
M
(17.1.7)
17.2. GREEN FUNCTION FOR THE SCHRÖDINGER EQUATION
189
The probability current density of the scattered wave, on the other hand, is given by
~
exp(ikr)
2
∗ exp(−ikr)
~
~jout (~x) =
|A| f (θ, ϕ)
∇f (θ, ϕ)
2M i
r
r
exp(ikr)
exp(−ikr)
~ (θ, ϕ)∗
f (θ, ϕ)
−
∇f
r
r
~k 2
1
1
=
.
(17.1.8)
|A| |f (θ, ϕ)|2 2 + O
M
r
r3
Hence, we can identify the differential cross section as
dσ(θ, ϕ)
dJ 1
|~jout |r2
= |f (θ, ϕ)|2 .
=
=
~
dΩ
dΩ jN
|jin |
(17.1.9)
This shows that the asymptotic form of the wave function determines the differential cross
section.
17.2
Green Function for the Schrödinger Equation
Let us consider the Schrödinger equation
~2
−
∆ + V (~x) h~x|Ψ(~k)i = Eh~x|Ψ(~k)i
2M
(17.2.1)
for scattering states of positive energy E = ~2 k 2 /2M . First, we replace the potential by
a δ-function
~2
V (~x) =
δ(~x),
(17.2.2)
8M π
and we consider the following equation for the Green function G0 (~x)
[∆ + k 2 ]G0 (~x) = −4πδ(~x).
(17.2.3)
In momentum space this equation takes the form
G̃0 (~k 0 ) =
4π
.
k 02 − k 2
(17.2.4)
Here G̃0 (~k 0 ) is the Fourier transform of G0 (~x), which is obtained by an inverse Fourier
transform
Z
1
d3 k 0 G̃(~k 0 ) exp(i~k 0 · ~x)
G0 (~x) =
(2π)3
Z
Z 1
1 ∞ 0
k 02
=
dk 02
d cos θ exp(ik 0 |~x| cos θ)
π 0
k − k 2 −1
Z ∞
1
k 0 exp(ik 0 |~x|)
exp(ik|~x|)
=
dk 0 0
=
.
(17.2.5)
0
πi|~x| −∞
(k − k)(k + k)
|~x|
In the last step we have closed the integration contour in the complex plane and we have
used the residue theorem.
190
17.3
CHAPTER 17. QUANTUM MECHANICS OF ELASTIC SCATTERING
The Lippmann-Schwinger Equation
The Schrödinger equation is equivalent to the Lippmann-Schwinger equation
Z
1 2M
x − ~x0 |)
3 0 exp(ik|~
~
~
h~x|Ψ(k)i = h~x|ki −
d
x
V (~x0 )h~x0 |Ψ(~k)i.
4π ~2
|~x − ~x0 |
(17.3.1)
Here h~x|~ki = A exp(i~k · ~x) is the incident plane wave. Acting with the Laplacian [∆ + k 2 ]
and using the defining property of the Green function we indeed obtain
~2
[∆ + k 2 ]h~x|Ψ(~k)i =
2M Z
1
exp(ik|~x − ~x0 |)
−
d3 x0 [∆ + k 2 ]
V (~x0 )h~x0 |Ψ(~k)i =
4π
|~x − ~x0 |
Z
d3 x0 δ(~x − ~x0 )V (~x0 )h~x0 |Ψ(~k)i = V (~x)h~x|Ψ(~k)i,
(17.3.2)
which is nothing but the Schrödinger equation.
Let us consider the scattering wave function h~x|Ψ(~k)i at asymptotic distances ~x. The
wave vector pointing in the corresponding direction is then given by ~k 0 = k~x/|~x| and we
can write
p
p
k|~x − ~x0 | = k x2 + x02 − 2~x · ~x0 ≈ k|~x| 1 − 2~x · ~x0 /|~x|2
≈ k|~x| − k~x · ~x0 /|~x| = k|~x| − ~k 0 · ~x0 .
(17.3.3)
At asymptotic distances the Lippmann-Schwinger equation thus takes the form
Z
1 2M exp(ik|~x|)
h~x|Ψ(~k)i ≈ h~x|~ki −
d3 x0 exp(i~k 0 · ~x0 )V (~x0 )h~x0 |Ψ(~k)i.
(17.3.4)
4π ~2
|~x|
Hence, we can identify the scattering amplitude as
Z
2M
0
2 2M
~
f (k ) = −2π 2
d3 x0 exp(i~k 0 · ~x0 )V (~x0 )h~x0 |Ψ(~k)i = −2π 2 2 h~k 0 |V |Ψ(~k)i.
~
~
17.4
(17.3.5)
Abstract Form of the Lippmann-Schwinger Equation
Let us now define a free propagator as
G0 (E) =
1
,
E − p2 /2M
(17.4.1)
where p is the momentum operator. The corresponding matrix element
Z
1
2M
1
0
0
h~x|G0 (E)|~x i = h~x|
|~x i = d3 k 0 h~x|~k 0 i 2 2
h~k 0 |~x0 i
2
E − p /2M
~ k − k 02
Z
~k 0 · (~x − ~x0 )
exp
i
1 2M
=
d3 k 0
= G0 (~x − ~x0 ),
(2π)3 ~2
k 2 − k 02
(17.4.2)
17.4. ABSTRACT FORM OF THE LIPPMANN-SCHWINGER EQUATION
191
is just the Green function defined above. This allows us to write the Lippmann-Schwinger
equation in the more abstract form
|Ψ(~k)i = |~ki + G0 (E)V |Ψ(~k)i.
(17.4.3)
Indeed, by left-multiplication with h~x| one obtains
h~x|Ψ(~k)i = h~x|~ki + h~x|G0 (E)V |Ψ(~k)i
Z
~
= h~x|ki + d3 x0 h~x|G0 (E)|~x0 ih~x0 |V |Ψ(~k)i
Z
~
= h~x|ki + d3 x0 G0 (~x − ~x0 )V (~x0 )h~x0 |Ψ(~k)i.
(17.4.4)
In the next step we define a full interacting propagator
G(E) =
1
1
=
,
2
E−H
E − p /2M − V
(17.4.5)
and we write
G(E) − G0 (E) =
=
−
=
1
E−
p2 /2M
−
1
E − p2 /2M
−V
1
1
(E − p2 /2M )
2
E − p /2M − V
E − p2 /2M
1
1
(E − p2 /2M − V )
2
E − p /2M − V
E − p2 /2M
1
1
V
= G(E)V G0 (E).
E − p2 /2M − V E − p2 /2M
(17.4.6)
Then the abstract form of the Lippmann-Schwinger equation takes the form
|Ψ(~k)i = |~ki + G0 (E)V |Ψ(~k)i
= |~ki + G(E)V |Ψ(~k)i − G(E)V G0 (E)V |Ψ(~k)i
= |~ki + G(E)V |Ψ(~k)i − G(E)V (|Ψ(~k)i − |~ki)
= |~ki + G(E)V |~ki.
Indeed, multiplying this equation with E − H, we obtain
p2
~
~
~
(E − H)Ψ(k)i = (E − H)|ki + V |ki = E −
|~ki = 0,
2M
(17.4.7)
(17.4.8)
which is nothing but the Schrödinger equation.
Iterating the abstract form of the Lippmann-Schwinger equation one obtains
|Ψ(~k)i = |~ki + G0 (E)V |Ψ(~k)i
= |~ki + G0 (E)V |~ki + G0 (E)V G0 (E)V |Ψ(~k)i
=
∞
X
(G0 (E)V )n |~ki.
n=0
(17.4.9)
192
CHAPTER 17. QUANTUM MECHANICS OF ELASTIC SCATTERING
Chapter 18
The Adiabatic Berry Phase
In this chapter we will consider quantum mechanical systems with a Hamiltonian that
depends on some slowly varying external parameters, such that the system undergoes
an adiabatic time evolution. In 1928 Born and Fock derived the adiabatic theorem in
quantum mechanics. According to the theorem, a quantum system then evolves from
an eigenstate of the initial Hamiltonian through the momentary eigenstates of the timedependent Hamiltonian. When the Hamiltonian undergoes a periodic time evolution,
the system ultimately returns to the initial eigenstate, at least up to a complex U (1)
phase, known as the Berry phase which was noticed by Michael Berry in 1983. When
the eigenstate is n-fold degenerate, the Berry phase becomes a non-Abelian U (n) matrix.
There is an abstract Abelian or non-Abelian Berry gauge field in the space of slowly
varying external parameters, whose parallel transport along a closed path in parameter
space yields the Berry phase. Non-Abelian gauge fields even arise in the classical physics
of falling cats.
18.1
Abelian Berry Phase of a Spin
1
2
in a Magnetic Field
~
Let us consider a spin 12 in a time-dependent magnetic field B(t).
The time-dependent
Hamiltonian then takes the form
~
H(t) = µB(t)
· ~σ ,
(18.1.1)
where ~σ denotes the Pauli matrices, and µ is a magnetic moment.
The states of a spin
1
2
can be parameterized as
|~ei = a| ↑i + b| ↓i,
193
|a|2 + |b|2 = 1,
(18.1.2)
194
CHAPTER 18. THE ADIABATIC BERRY PHASE
which gives rise to the projection operator
∗
∗
P (~e) = |~eih~e| = (a| ↑i + b| ↓i)(a h↑ | + b h↓ |) =
1
1
1 + e3 e1 − ie2
=
= (1 + ~e · ~σ ),
2 e1 + ie2 1 − e3
2
|a|2 a∗ b
b∗ a |b|2
(18.1.3)
where we have identified
~e = h~e|~σ |~ei = (a∗ b + b∗ a, −ia∗ b + ib∗ a, |a|2 − |b|2 ).
(18.1.4)
The vector ~e ∈ S 2 associates a spin state with a point on the so-called Bloch sphere.
Thereby we do not distinguish the state |~ei from states exp(iα)|~ei which differ only by an
irrelevant phase, which cancels in the physical projection operator P (~e) = |~eih~e|.
~
~
Identifying ~e(t) = ±B(t)/|
B(t)|
we obtain
1
~
H(t)|~e(t)ih~e(t)| = H(t)P (t) = µB(t)
· ~σ (1 + ~e · ~σ )
2
µ~
µ~
µ~
µ ~
=
B(t) · ~σ + B(t) · ~e(t) = B(t)
· ~σ ± |B(t)|
2
2
2
2
~
= ±µ|B(t)||~e(t)ih~e(t)|.
(18.1.5)
~
~
Hence, the state |~e(t)i = | ± B(t)/|
B(t)|i
is a momentary eigenstate of the Hamiltonian
~
with eigenvalue ±µ|B(t)|.
Let us now consider the time-dependent Schrödinger equation
i∂t |Ψ(t)i = H(t)|Ψ(t)i.
(18.1.6)
For a slowly varying external magnetic field, the adiabatic theorem then suggests the
ansatz
Z t
0
0
~
|Ψ(t)i = exp ∓i
dt µ|B(t )| exp(iγ± (t))|~e(t)i,
(18.1.7)
0
i.e. the system always remains in a momentary eigenstate |~e(t)i, but it also accumulates
a phase. Inserting this ansatz in the time-dependent Schrödinger equation, we obtain
Z t
h
i
~
~ 0 )| exp(iγ± (t))|~e(t)i,
i∂t |Ψ(t)i = ±µ|B(t)|
− ∂t γ± (t) + i∂t exp ∓i
dt0 µ|B(t
0
Z t
0
0
~
~
H(t)|Ψ(t)i = ±µ|B(t)| exp ∓i
dt µ|B(t )| exp(iγ± (t))|~e(t)i,
(18.1.8)
0
such that
|~e(t)i∂t γ± (t) = i∂t |~e(t)i.
(18.1.9)
This equation can be satisfied only when the time-evolution is indeed adiabatic. In that
case, one obtains
∂t γ± (t) = ih~e(t)|∂t |~e(t)i.
(18.1.10)
18.1. ABELIAN BERRY PHASE OF A SPIN
1
2
IN A MAGNETIC FIELD
195
The Berry phase is defined for a cyclic variation of the external magnetic field with
~ + T ) = B(t),
~
the period T , for which B(t
T
Z
Z
T
dt h~e(t)|∂t |~e(t)i.
dt ∂t γ± (t) = i
γ± (T ) =
(18.1.11)
0
0
~
Since |~e(t)i depends on t only through B(t),
we can write
~
~ B |~e(B(t))i,
~
~
·∇
|~e(t)i = |~e(B(t))i
⇒ ∂t |~e(t)i = ∂t B(t)
(18.1.12)
such that
Z
γ± (T ) = i
T
~
~
~ B |~e(B(t))i
~
· h~e(B(t))|
∇
=i
dt ∂t B(t)
Z
~ · h~e(B)|
~ ∇
~ B |~e(B)i.
~
dB
(18.1.13)
C
0
~
This shows that the Berry phase depends only on the curve C along which B(t)
varies
with time, but not on the velocity of the variation, at least as long as it remains adiabatic.
Consequently, the Berry phase is a purely geometric and not a dynamical object. This
suggests to introduce an abstract vector potential, also known as the Berry connection,
~ B)
~ = ih~e(B)|
~ ∇
~ B |~e(B)i,
~
A(
(18.1.14)
which is real-valued despite the factor i. The Berry phase is then identified as a Wilson
loop of the abstract Abelian Berry gauge field. It should be pointed out that this gauge
field does not exist in coordinate space, but rather in the space of external parameters of
~
a quantum system, in this case, in the space of all possible external magnetic fields B.
Let us now make a different choice for the arbitrary phase of a momentary eigenstate
~ 0 = exp(iα(B))|~
~ e(B)i.
~
|~e(B)i
(18.1.15)
This implies an Abelian gauge transformation of the vector potential
~ B)
~ 0 = A(
~ B)
~ −∇
~ B α(B).
~
A(
(18.1.16)
The Berry phase is gauge invariant, i.e.
Z
Z
0
0
~ · A(
~ B)
~ = dB
~ · [A(
~ B)
~ −∇
~ B α(B)]
~ = γ± (T ).
γ± (T ) = dB
C
(18.1.17)
C
Let us now associate an abstract Berry field strength with the Abelian vector potential
~ =∇
~ B × A(
~ B),
~
F~ (B)
(18.1.18)
which obviously is gauge invariant. Using Stoke’s theorem, the Berry phase can then be
expressed as
Z
Z
Z
~
~
~
~
~
~
~
γ± (T ) = dB · A(B) =
d~s · ∇B × A(B) =
d~s · F~ (B),
(18.1.19)
C
S
S
196
CHAPTER 18. THE ADIABATIC BERRY PHASE
i.e. it represents the flux of the Berry field strength through a surface S bounded by the
closed curve C.
Let us evaluate the Berry gauge field for the spin
1
2
in an external magnetic field
~ = |B|(sin
~
B
θ cos ϕ, sin θ sin ϕ, cos θ).
(18.1.20)
~ = (0, 0, |B|)
~ the ground state is given by |~e(B)i
~ = | ↑i. The ground state for a
For B
~
general orientation of B is obtained by a rotation
~ = exp(iϕSz ) exp(−iθSx )| ↑i.
|~e(B)i
(18.1.21)
Based on this, it is straightforward to work out
~ ϕ |~e(B)i
~ = i cos θ,
h~e(B)|∂
2
~ θ |~e(B)i
~ = h~e(B)|∂
~
~ = 0,
h~e(B)|∂
e(B)i
~ |~
|B|
such that one obtains
~ B)
~ =−
A(
cos θ
~e .
~ sin θ ϕ
2|B|
(18.1.22)
(18.1.23)
The corresponding field strength is the one of a “magnetic monopole” in parameter space
!
~
cos
θ
1 B
1
~ =∇
~ B × A(
~ B)
~ =−
∂θ
~eB =
.
(18.1.24)
F~ (B)
~ sin θ
~
~ 3
2 |B|
|B|
2|B|
18.2
Non-Abelian Berry Phase
Non-Abelian Berry phases arise when one considers the adiabatic evolution of a set of n
degenerate states, which remain degenerate while some external parameters are varied.
After a slow periodic variation of the external parameters, the initial state may then not
turn back to itself, but may turn into another member of the set of degenerate states.
The Berry phase, which rotates the initial into the final state, then takes the form of a
non-Abelian U (n) matrix.
As a concrete example, we consider a nuclear spin resonance experiment in which a
~ proceeds via
probe rotates in a magnetic field. In this case, the interaction of the spin S
the nuclear quadrupole moment and is given by
~
~ 2.
H = ω(B(t)
· S)
(18.2.1)
~ = (0, 0, |B|),
~
When B
the energy eigenstates are |mi with m = −S, −S + 1, ..., S with
2 m2 . In particular, the states |mi and | − mi are degenerate.
~
eigenvalues Em = ω|B(t)|
Again, we obtain the momentary eigenstates by a rotation
~ = exp(iϕSz ) exp(−iθSx )|m(B)i.
~
|m(B)i
(18.2.2)
18.3. SO(3) GAUGE FIELDS IN FALLING CATS
197
In this case, the Berry gauge field is non-Abelian
~ ++ (B)
~ = ihm(B)|
~ ∇
~ B |m(B)i
~ = − m cos θ ~eϕ ,
A
~ sin θ
2 |B|
~ −− (B)
~ = ih−m(B)|
~ ∇
~ B | − m(B)i
~ = m cos θ ~eϕ ,
A
~ sin θ
2 |B|
~ +− (B)
~ = ihm(B)|
~ ∇
~ B | − m(B)i,
~
~ −+ (B)
~ = ih−m(B)|
~ ∇
~ B |m(B)i.
~
A
A
(18.2.3)
~ +− (B)
~ and A
~ −+ (B)
~ vanish, unless m = ± 1 .
A straightforward calculation reveals that A
2
In that case, one obtains
r
1
1
~ ±∓ (B)
~ =
S(S + 1) + (±i~eϕ + ~eθ ) ,
A
(18.2.4)
~
4
2|B|
such that the non-Abelian SU (2) Berry vector potential takes the form
!
r
r
1
1
cos
θ
1
~ B)
~ =
S(S + 1) + ~eθ σ1 + S(S + 1) + ~eϕ σ2 −
~eϕ σ3 .
A(
~
4
4
sin θ
2|B|
(18.2.5)
Under a unitary change U ∈ SU (2) of the two basis states
~ 0 = |n(B)iU
~
~ † ,
|m(B)i
(B)
nm
(18.2.6)
the Berry gauge field transforms as one would expect for a non-Abelian gauge field
~ B)
~ 0 = i0 hn(B)|
~ ∇
~ B |m(B)i
~ 0 = iU (B)hn(
~
~ ∇
~ B |m(B)iU
~
~ †
A(
B)|
(B)
~ A(
~ B)U
~ (B)
~ † + iU (B)
~ ∇
~ B U (B)
~ †.
= U (B)
(18.2.7)
The corresponding non-Abelian field strength is then given by
~ = ∇
~ B × A(
~ B)
~ + iA(
~ B)
~ × A(
~ B)
~
F~ (B)
"
#
r
~
B
1
cos θ
1
1
S(S + 1) + ~ez σ2 − S(S + 1) +
~ez σ3 .
=
σ +
~ 3 3 2|B|
~ 2 sin θ
4
4
2|B|
(18.2.8)
18.3
SO(3) Gauge Fields in Falling Cats
Cats have the ability to land on their feet, even if they are dropped head down from some
height. They twist their body and use their tail, such that their body undergoes a net
rotation. Let us try to understand this phenomenon in mathematical terms. Somewhat
surprisingly, we will encounter a non-Abelian gauge field in the space of all shapes of the
cat’s body, whose non-Abelian Berry phase gives the net rotation angle of the cat.
Let us discretize the cat by a set of point masses mi at positions ~xi , and let us imagine
that the cat has control over the shape of its body, by influencing the relative orientation
198
CHAPTER 18. THE ADIABATIC BERRY PHASE
of the point masses. While the cat is in free fall, it does not feel gravity, at least until
it hits the ground. During the fall, we can simply go to the accelerated center of mass
frame of the cat, and then describe the time-dependent shape of the cat in the absence of
gravity. The key to the understanding of this problem is angular momentum conservation.
The angular momentum of the cat is simply given by
~ =
L
X
mi ~xi ×
i
d~xi
.
dt
(18.3.1)
When the cat is originally released at rest, the angular momentum vanishes and will remain
~ = 0 until the cat hits the ground, as a consequence of angular momentum conservation.
L
Let us now define a possible shape of the cat as a particular configuration of the points
~xi , with configurations being identified if they differ just by an SO(3) spatial rotation.
Any possible shape can be characterized by a reference configuration ~yi , which can then
be realized in all possible orientations
~xi = O~yi ,
OT O = OOT = 1,
detO = 1.
(18.3.2)
Here O ∈ SO(3) is an orthogonal rotation matrix that rotates the reference configuration
~yi into the general orientation ~xi , keeping the shape fixed. Let us now assume that, by
controlling its body, the cat can send the point masses inside its body through any timedependent sequence of shapes, defined by time-dependent reference configurations ~yi (t)
that the cat can choose at will. The question then is how the reference configuration is
rotated by a time-dependent orthogonal rotation O(t) into the actual position of the cat
~xi (t). This simply follows from angular momentum conservation
~ =
L
X
mi ~xi (t) ×
i
d~xi (t)
dt
d
[O(t)~yi (t)]
dt
i
X
dO(t)
d~yi (t)
=
mi O(t)~yi (t) ×
~yi (t) + O(t)
= 0,
dt
dt
=
X
mi O(t)~yi (t) ×
(18.3.3)
i
which thus implies
X
dO(t)
d~yi (t)
~yi (t) = −O(t)
mi ~yi (t) ×
⇒
dt
dt
i
i
X
X
dO(t)
d~yi (t)
~ (t),
mi ~yi (t) × O(t)T
~yi (t) = −
mi ~yi (t) ×
= −M
(18.3.4)
dt
dt
O(t)
X
mi ~yi (t) × O(t)T
i
i
~ (t) is the non-conserved “angular momentum” of the reference configuration. We
where M
now introduce the anti-Hermitean non-Abelian SO(3) vector potential
A(t) = O(t)T
dO(t)
= iAa (t)T a ,
dt
a
Tbc
= −iεabc .
(18.3.5)
18.3. SO(3) GAUGE FIELDS IN FALLING CATS
199
Expressed in components, eq.(18.3.4) takes the form
X
mi εabc yib (t)Ad (t)εdce yie (t) = −M a (t) ⇒
i
X
mi [yie (t)yie (t)δab − yia (t)yib (t)]Ab (t) = M a (t).
(18.3.6)
i
Introducing the moment of inertia tensor
X
Iab (t) =
mi [yie (t)yie (t)δab − yia (t)yib (t)],
(18.3.7)
i
we finally obtain
~ = I(t)−1 M
~ (t).
Iab (t)Ab (t) = M a (t) ⇒ A(t)
(18.3.8)
Of course, the reference configuration ~yi for a given shape of the cat can be chosen
arbitrarily. Let us investigate how the vector potential A(t) changes when the reference
configuration is rotated to a new configuration
~yi (t)0 = Ω(t)~yi (t),
Ω ∈ SO(3).
(18.3.9)
After such a rotation, the new transformation O0 (t) that rotates ~yi (t)0 into ~xi (t), is given
by
~xi (t) = O(t)0 ~yi (t)0 = O(t)0 Ω(t)~yi (t) → O(t)0 = O(t)Ω(t)T .
(18.3.10)
The corresponding vector potential then takes the form
0
dO(t)
dΩ(t)T
T dO(t)
A(t)0 = O(t)0
= Ω(t)O(t)T
Ω(t)T + O(t)
dt
dt
dt
d
Ω(t)T .
= Ω(t) A(t) +
dt
(18.3.11)
Hence, the change of reference configuration for a given shape amounts to an SO(3) gauge
transformation. This is not surprising because the different shapes play the role of gauge
equivalence classes.
Finally, let us calculate the total rotation of the cat during a sequence of shape changes
after which the cat returns to its initial shape. Using
dO(t)
,
dt
O(t)A(t) =
we obtain
Z
O(T ) = P exp
T
dt A(t) ,
(18.3.12)
(18.3.13)
0
where P denotes path ordering. Hence, when the cat returns from its initial shape to the
same final shape after a time T , its net rotation can be computed as a closed Wilson loop
in an SO(3) gauge field, very much like a non-Abelian Berry phase in quantum mechanics.
Of course, all this does not explain why the cat is actually able to perform the difficult
task of landing on her feet. While it is unlikely that it has an SO(3) Wilson loop computer
hard-wired in its brain, at least our brain is capable of describing the cat’s motion using
the abstract mathematical concept of non-Abelian gauge fields.
200
18.4
CHAPTER 18. THE ADIABATIC BERRY PHASE
Final Remarks
One message of all this is that gauge fields may exist in many places. They arise naturally
when we use redundant variables to describe Nature, be they fundamental quantum fields
in the Standard Model of particle physics, ambiguous phases of quantum mechanical wave
functions, or standard orientations of falling cats. From this point of view, gauge fields
are clearly a human invention resulting from our choice of redundant variables. One may
speculate whether Nature herself also uses redundancies in order to realize the phenomena
that we describe with gauge theories. I personally like to think that this may not be
the case. Trying to understand what Nature does (perhaps at the Planck scale) in order
to generate effective gauge theories at low energies is interesting and may perhaps even
reveal deep insights into the emergence of space-time at short distances. While all this
is highly speculative, there is no doubt that, endowed with great curiosity, theoretical
physicists will continue to use their mathematical capabilities to push the boundaries of
current knowledge further into the unknown. This course may be viewed as an invitation
to participate in this most exciting and potentially quite satisfying enterprise.
Appendix A
Physical Units and Fundamental
Constants
In this text we encounter several units. For example, when we discuss quantum mechanical problems that involve electrodynamics we use the so-called Gaussian system which
is natural from a theoretical physicist’s point of view. An alternative often preferred in
technical applications is the MKS system. In any case, units represent man-made conventions, influenced by the historical development of science. Interestingly, there are also
natural units which express physical quantities in terms of fundamental constants of Nature: Newton’s gravitational constant G, the velocity of light c, and Planck’s quantum h.
In this appendix, we consider the issue of physical units from a general point of view. We
also address the strength of electromagnetism and the feebleness of gravity.
A.1
Units of Time
Time is measured by counting periodic phenomena. The most common periodic phenomenon in our everyday life is the day and, related to that, the year. Hence, it is no
surprise that the first precise clock used by humans was the solar system. In our life
span, if we stay healthy, we circle around the sun about 80 times, every circle defining one
year. During one year, we turn around the earth’s axis 365 times, every turn defining one
day. When we build, for example, a pendulum clock, it helps us to divide the day into
24 × 60 × 60 = 86400 seconds. The second (about the duration of one heart beat) is perhaps the shortest time interval that people care about in their everyday life. However, as
physicists we do not stop there, because we need to be able to measure much shorter time
intervals, in particular, as we investigate physics of fundamental objects such as atoms or
individual elementary particles. Instead of using the solar system as a gigantic mechanical
clock, the most accurate modern clock is a tiny quantum mechanical analog of the solar
system — an individual cesium atom. Instead of defining 1 sec as one turn around earth’s
axis divided into 86400 parts, the modern definition of 1 sec corresponds to 9.192.631.770
201
202
APPENDIX A. PHYSICAL UNITS AND FUNDAMENTAL CONSTANTS
periods of a particular microwave transition of the cesium atom. The atomic cesium clock
is very accurate and defines a reproducible standard of time. Unlike the solar system, this
standard could in principle be established anywhere in the Universe. Cesium atoms are
fundamental objects which work in the same way everywhere at all times.
A.2
Units of Length
The lengths we care about most in our everyday life are of the order of the size of our
body. It is therefore not surprising that, in order to define a standard, some stick —
defined to be 1 meter — was deposited near Paris a long time ago. Obviously, this is
a completely arbitrary man-made unit. A physicist elsewhere in the Universe would not
want to subscribe to that convention. A trip to Paris just to measure a length would be
too inconvenient. How can we define a natural standard of length that would be easy to
establish anywhere at any time? For example, one could say that the size of our cesium
atom sets such a standard. However, this is not how this is handled. Einstein has taught
us that the velocity of light c in vacuum is an absolute constant of Nature, independent
of the motion of an observer. Instead of referring to the stick in Paris, one now defines
the meter through c and the second as
c = 2.99792456 × 108 m sec−1 ⇒ 1m = 3.333564097 × 10−7 c sec.
(A.2.1)
In other words, the measurement of a distance is reduced to the measurement of the time
it takes a light signal to pass that distance. Since Einstein’s theory of relativity tells us
that light travels with the same speed everywhere at all times, we have thus established a
standard of length that could easily be used by physicists anywhere in the Universe.
A.3
Units of Mass
Together with the meter stick, a certain amount of platinum-iridium alloy was deposited
near Paris a long time ago. The corresponding mass was defined to be one kilogram.
Obviously, this definition is as arbitrary as that of the meter. Since the original kilogram
has been moved around too often over the past 100 years or so, it has lost some weight
and no longer satisfies modern requirements for a standard of mass. One might think
that it would be best to declare, for example, the mass of a single cesium atom as an
easily reproducible standard of mass. While this is true in principle, it is inconvenient
in practical experimental situations. Accurately measuring the mass of a single atom is
highly non-trivial. Instead, it was decided to produce a more stable kilogram that will
remain constant for the next 100 years or so. Maintaining a standard is important business
for experimentalists, but a theorist doesn’t worry too much about the arbitrarily chosen
amount of matter deposited near Paris.
A.4. UNITS OF ENERGY
A.4
203
Units of Energy
Energy is a quantity derived from time, length, and mass. For example, Einstein’s famous
relation E = M c2 reminds us that energy has the dimension of mass × length2 × time−2 .
The relation also underscores that mass is a form of energy. In particular, in elementary
particle physics it is often convenient to specify the rest energy M c2 instead of the mass
M of a particle. For example, the mass of the proton is
Mp = 1.67266 × 10−27 kg,
(A.4.1)
which corresponds to the rest energy
Mp c2 = 0.940 GeV = 9.40 × 108 eV.
(A.4.2)
An electron Volt (eV) is the amount of energy that a charged particle with elementary
charge e (for example, a proton) picks up when it passes a potential difference of one Volt.
A.5
Natural Planck Units
Irrespective of practical considerations, it is interesting to think about standards that are
natural from a theoretical physicist’s point of view. There are three fundamental constants
of Nature that can help us in this respect. First, in order to establish a standard of length,
we have already used the velocity of light c which plays a central role in the theory of
relativity. Quantum mechanics provides us with another fundamental constant — Planck’s
action quantum (divided by 2π)
~=
h
= 1.0546 × 10−34 kg m2 sec−1 .
2π
(A.5.1)
A theoretical physicist is not terribly excited about knowing the value of ~ in units of
kilograms, meters, and seconds, because these are arbitrarily chosen man-made units.
Instead, it would be natural to use ~ itself as a basic unit with the dimension of an
action, i.e. energy × time. A third dimensionful fundamental constant which suggests
itself through general relativity (or even just through Newtonian gravity) is Newton’s
gravitational constant
G = 6.6720 × 10−11 kg−1 m3 sec−2 .
(A.5.2)
Using c, ~, and G we can define natural units also known as Planck units. First there are
the Planck time
r
G~
tP lanck =
= 5.3904 × 10−44 sec,
(A.5.3)
c5
and the Planck length
r
lP lanck =
G~
= 1.6160 × 10−35 m,
c3
(A.5.4)
204
APPENDIX A. PHYSICAL UNITS AND FUNDAMENTAL CONSTANTS
which represent the shortest times and distances relevant in physics. Today we are very far
from exploring such short length- and time-scales experimentally. One may even expect
that our classical concepts of space and time will break down at the Planck scale. One
may speculate that, at the Planck scale, space and time become discrete, and that lP lanck
and tP lanck may represent the shortest elementary quantized units of space and time. We
can also define the Planck mass
r
~c
MP lanck =
= 2.1768 × 10−8 kg,
(A.5.5)
G
which is expected to be the highest mass scale relevant to elementary particle physics.
Planck units would not be very practical in our everyday life. For example, a heart
beat lasts about 1043 tP lanck , the size of our body is about 1035 lP lanck , and we weigh on
the order of 1010 MP lanck . Still, these are the natural units that Nature suggests to us
and it is interesting to ask why we exist at scales so far removed from the Planck scale.
For example, we may ask why the Planck mass corresponds to about 10−8 kg. In some
sense this is just a historical question. The amount of matter deposited near Paris to
define the kilogram obviously was an arbitrary man-made unit. However, if we assume
that the kilogram was chosen because it is a reasonable fraction of our own weight, our
question may be rephrased as a biological one: Why do intelligent creatures weigh about
1010 MP lanck ? We can turn the question into a physics problem when we think about
the physical origin of mass. Indeed (up to tiny corrections) the mass of the matter that
surrounds us is contained in atomic nuclei which consist of protons and neutrons with
masses
Mp = 1.67266 × 10−27 kg = 7.6840 × 10−20 MP lanck ,
Mn = 1.67496 × 10−27 kg = 7.6946 × 10−20 MP lanck .
(A.5.6)
Why are protons and neutrons so light compared to the Planck mass? This physics question has actually been understood at least qualitatively using the property of asymptotic
freedom of quantum chromodynamics (QCD) — the quantum field theory of quarks and
gluons whose interaction energy explains the masses of protons and neutrons. The Nobel
prize of the year 2004 was awarded to David Gross, David Politzer, and Frank Wilczek
for the understanding of asymptotic freedom.
A.6
The Strength of Electromagnetism
The strength of electromagnetic interactions is determined by the quantized charge unit e
(the electric charge of a proton). In natural Planck units it gives rise to the experimentally
determined fine-structure constant
e2
1
=
.
(A.6.1)
~c
137.036
The strength of electromagnetism is determined by this pure number which is completely
independent of any man-made conventions. It is a very interesting physics question to ask
α=
A.7. THE FEEBLENESS OF GRAVITY
205
why α has this particular value. At the moment, physicists have no clue how to answer
this question. These days it is popular to refer to the anthropic principle. If α would be
different, all of atomic physics and thus all of chemistry would work differently, and life
as we know it might be impossible. According to the anthropic principle, we can only
live in an inhabitable part of a Multiverse (namely in our Universe) with a “life-friendly”
value of α. In the author’s view the anthropic principle is a last resort which should
only be used when all other possible explanations have failed. Since we will always be
confined to our Universe and thus cannot check whether it is part of a bigger Multiverse,
in practice we cannot falsify the anthropic argument. In this sense, it does not belong to
rigorous scientific thinking. This does not mean that one should not think that way, but
it is everybody’s private business. The author prefers to remain optimistic. Instead of
referring to the anthropic principle, he hopes that perhaps some day a particularly smart
reader of this text may understand why α takes the above experimentally measured value.
A.7
The Feebleness of Gravity
In our everyday life we experience gravity as a rather strong force because the enormous
mass of the earth exerts an attractive gravitational force. The force of gravity acts universally on all massive objects. For example, two protons of mass Mp at a distance r exert a
gravitational force of magnitude
GMp2
Fg =
r2
(A.7.1)
on each other. However, protons carry an electric charge e and thus they also exert a
repulsive electrostatic Coulomb force
Fe =
e2
r2
(A.7.2)
on each other. Electromagnetic forces are less noticeable in our everyday life because
protons are usually contained inside the nucleus of electrically neutral atoms, i.e. their
charge e is screened by an equal and opposite charge −e of an electron in the electron
cloud of the atom. Still, as a force between elementary particles, electromagnetism is much
stronger than gravity. The ratio of the electrostatic and gravitational forces between two
protons is
2
MPlanck
e2
Fe
=
=
α
≈ 1036 .
Fg
GMp2
Mp2
(A.7.3)
Hence, as a fundamental force electromagnetism is very much stronger than gravity, just
because the proton mass Mp is a lot smaller than the Planck mass. As we have seen
before, this is a consequence of the property of asymptotic freedom of QCD. Indeed, it
has been pointed out by Wilczek that the feebleness of gravity is a somewhat unexpected
consequence of this fundamental property of the theory of the strong interactions.
206
A.8
APPENDIX A. PHYSICAL UNITS AND FUNDAMENTAL CONSTANTS
Units of Charge
As we have seen, in Planck units the strength of electromagnetism is given by the finestructure constant which is a dimensionless number independent of any man-made conventions. Obviously, the natural unit of charge that Nature suggests to us is the elementary
charge quantum e — the electric charge of a single proton. Of course, in experiments on
macroscopic scales one usually deals with an enormous number of elementary charges at
the same time. Just like using Planck units in everyday life is not very practical, measuring
charge in units of
r
~c
e=
= 1.5189 × 10−14 kg1/2 m3/2 sec−1
(A.8.1)
137.036
can also be inconvenient. For this purpose large amounts of charge have also been used
to define charge units. For example, one electrostatic unit is defined as
1esu = 2.0819 × 109 e = 3.1622 × 10−5 kg1/2 m3/2 sec−1 .
(A.8.2)
This charge definition has the curious property that the Coulomb force
F =
q1 q2
r2
(A.8.3)
between two electrostatic charge units at a distance of 1 centimeter is
1
esu2
= 10−5 kg m sec−2 = 10−5 N = 1dyn.
cm2
(A.8.4)
Again, the origin of this charge unit is at best of historical interest and just represents
another man-made convention.
A.9
Various Units in the MKS System
An alternative to the Gaussian system is the MKS system that is often used in technical
applications. Although it is natural and completely sufficient to assign the above dimension
to charge, in the MKS system charge is measured in a new independent unit called a
Coulomb in honor of Charles-Augustin de Coulomb (1736 - 1806). In MKS units Coulomb’s
law for the force between two charges q1 and q2 at a distance r takes the form
F =
1 q1 q2
.
4πε0 r2
(A.9.1)
The only purpose of the quantity ε0 is to compensate the dimension of the charges. For
example, in MKS units the elementary charge quantum corresponds to e0 with
e0
e= √
.
4πε0
(A.9.2)
A.9. VARIOUS UNITS IN THE MKS SYSTEM
207
The Coulomb is then defined as
1Cb = 6.2414 × 1018 e0 .
(A.9.3)
Comparing with eq.(A.8.2) one obtains
√
1Cb = 10c 4πε0 sec m−1 esu.
(A.9.4)
It so happens that most pioneers of electrodynamics have been honored by naming a
unit after them. To honor Allessandro Volta (1845 - 1897) the unit for potential difference
is called a Volt and is defined as
1V = 1kg m2 sec−2 Cb−1 .
(A.9.5)
To maintain the Volt standard one uses the Josephson effect related to the current flowing
between two superconductors separated by a thin insulating layer — a so-called Josephson
junction. The effect was predicted by Brian David Josephson in 1962, for which he received
the Nobel prize in 1973. A Josephson junction provides a very accurate standard for
potential differences in units of the fundamental magnetic flux quantum h/2e.
In honor of Andre-Marie Ampere (1775 - 1836), the unit of current is called an Ampere
which is defined as
1A = 1Cb sec−1 .
(A.9.6)
Using Ohm’s law, the standard of the Ampere is maintained using the standards of potential difference and resistance.
The unit of resistance is one Ohm in honor of Georg Simon Ohm (1789 - 1854) who
discovered that the current I flowing through a conductor is proportional to the applied
voltage U . The proportionality constant is the resistance R and Ohm’s law thus reads
U = RI. Historically, the Ohm was defined at an international conference on electricity
in 1881 as
1Ω = 1V A−1 = 1kg m2 sec−1 Cb−2 .
(A.9.7)
Once this was decided, the question arose how to maintain the standard of resistance.
In other words, which conductor has a resistance of 1 Ohm? At another international
conference in 1908 people agreed that a good realization of a conductor with a resistance
of one Ohm are 14.4521 g of mercury in a column of length 1.063 m with constant crosssection at the temperature of melting ice. However, later it was realized that this was not
completely consistent with the original definition of the Ohm. Today the standard of the
Ohm is maintained extremely accurately using an interesting quantum phenomenon — the
quantum Hall effect — discovered by Klaus von Klitzing in 1980, for which he received
the Nobel prize in 1985. Von Klitzing observed that at extremely low temperatures the
resistance of a planar conductor in a strong perpendicular magnetic field is quantized in
units of h/e2 , where h is Planck’s quantum and e is the basic unit of electric charge.
The quantum Hall effect provides us with an accurate and reproducible measurement of
resistance and is now used to maintain the standard of the Ohm by using the fact that
1Ω = 3.8740459 × 10−5
h
.
e2
(A.9.8)
208
APPENDIX A. PHYSICAL UNITS AND FUNDAMENTAL CONSTANTS
In honor of Nikola Tesla (1856 - 1943), magnetic fields are measured in Tesla defined
as
1T = 1kg sec−1 Cb−1 .
(A.9.9)
The corresponding unit in the Gaussian system is 1 Gauss, honoring Carl Friedrich Gauss
(1777 - 1855), with 104 Gauss corresponding to 1 Tesla.
Appendix B
Scalars, Vectors, and Tensors
In this appendix we remind ourselves of the concepts of scalars and vectors and we also
discuss general tensors. Then we introduce the Kronecker and Levi-Civita symbols as well
as Einstein’s summation convention.
B.1
Scalars and Vectors
Physical phenomena take place in space. Hence, their mathematical description naturally
involves space points
~x = (x1 , x2 , x3 ).
(B.1.1)
Here x1 , x2 , and x3 are the Euclidean components of the 3-dimensional vector ~x. The
fact that ~x is a vector means that it behaves in a particular way under spatial rotations.
In particular, in a rotated coordinate system, the components x1 , x2 , and x3 change
accordingly. Let us consider a real 3 × 3 orthogonal rotation matrix O with determinant
1. For an orthogonal matrix
OT O = OOT = 1,
(B.1.2)
where T denotes transpose. Under such a spatial rotation the vector ~x transforms into
~x 0 = O~x.
(B.1.3)
~x · ~y = x1 y1 + x2 y2 + x3 y3
(B.1.4)
The scalar product of two vectors
is unaffected by the spatial rotation, i.e.
~x 0 · ~y 0 = ~xOT O~y = ~x1~y = ~x · ~y .
(B.1.5)
Quantities that don’t change under spatial rotations are known as scalars (hence the name
“scalar” product). The length of a vector is the square root of its scalar product with itself,
209
210
APPENDIX B. SCALARS, VECTORS, AND TENSORS
i.e.
|~x| =
q
√
~x · ~x = x21 + x22 + x23 .
(B.1.6)
Obviously, the length of a vector also is a scalar, i.e. it does not change under spatial
rotations.
B.2
Tensors and Einstein’s Summation Convention
A tensor Ti1 i2 ...in of rank n is an object with n indices which transforms appropriately
under spatial rotations, i.e.
Ti01 i2 ...in =
3 X
3 X
3
X
Oi1 j1 Oi2 j2 ...Oin jn Tj1 j2 ...jn .
(B.2.1)
j1 =1 j2 =1 jn =1
For example, vectors are tensors of rank one such that
Ti0
=
3
X
Oij Tj .
(B.2.2)
j=1
This is nothing but T~ 0 = OT~ expressed in component form. Similarly, scalars are tensors
of zero rank.
For example, general relativity involves a lot of tensors as well as numerous sums over
repeated indices. In order to save space, Einstein thus introduced an elegant summation
convention in which repeated indices are automatically summed over. In his notation
eq.(B.2.1) takes the form
Ti01 i2 ...in = Oi1 j1 Oi2 j2 ...Oin jn Tj1 j2 ...jn .
(B.2.3)
The sums over j1 , j2 , ..., jn need not be written explicitly. They are automatically implied
because j1 , j2 , ..., jn are repeated indices which each occur twice on the right-hand side.
In Einstein’s notation the defining relation of a vector, eq.(B.2.2), simply takes the form
Ti0 = Oij Tj .
(B.2.4)
Similarly, the scalar product of two vectors, eq.(B.1.4), is rewritten as
~x · ~y =
3
X
x i yi = x i yi ,
(B.2.5)
i=1
while the norm of a vector is given by
|~x| =
√
xi xi .
(B.2.6)
B.3. VECTOR CROSS PRODUCT AND LEVI-CIVITA SYMBOL
211
It is convenient to introduce the Kronecker symbol δij which is 1 for i = j and 0 otherwise. The Kronecker symbol represents the unit-matrix. For example, using Einstein’s
summation convention the orthogonality condition eq.(B.1.2) takes the form
T
Oij Ojk
= Oij Okj = δik .
(B.2.7)
Again, the repeated index j is automatically summed over, thus leading to the multiplication of the matrices O and OT . In three dimensions the trace of the unit-matrix is given
by
δii = 3.
(B.2.8)
The definition of δij immediately implies
Tij δjk = Tik .
(B.2.9)
Furthermore, the Kronecker symbol is invariant under spatial rotations, i.e.
δi01 i2 = Oi1 j1 Oi2 j2 δj1 j2 = Oi1 j1 Oi2 j1 = δi1 i2
B.3
(B.2.10)
Vector Cross Product and Levi-Civita Symbol
Two 3-dimensional vectors ~x and ~y can be combined to another vector ~z by forming the
vector cross product
~z = ~x × ~y = (x1 y2 − x2 y1 , x2 y3 − x3 y2 , x3 y1 − x1 y3 ).
(B.3.1)
The norm of the cross product results from
|~z|2 = (~x × ~y ) · (~x × ~y ) = |~x|2 |~y |2 − (~x · ~y )2 ,
(B.3.2)
while the double cross product of three vectors is given by
~x × (~y × ~z) = (~x · ~z)~y − (~x · ~y )~z.
(B.3.3)
It is very useful to introduce the totally anti-symmetric Levi-Civita symbol εijk whose
non-zero elements are given by
ε123 = ε231 = ε312 = 1, ε321 = ε213 = ε132 = −1.
(B.3.4)
All other elements are zero. The Levi-Civita symbol is anti-symmetric under pair permutations of its indices, e.g.
εjik = −εijk ,
(B.3.5)
while it is symmetric under cyclic permutations, i.e.
εkij = εijk .
(B.3.6)
212
APPENDIX B. SCALARS, VECTORS, AND TENSORS
Using Einstein’s summation convention the vector cross product of eq.(B.3.1) can
simply be written as
zi = εijk xj yk .
(B.3.7)
Just like the Kronecker symbol, the Levi-Civita symbol is invariant under spatial rotations,
i.e.
(B.3.8)
ε0i1 i2 i3 = Oi1 j1 Oi2 j2 Oi3 j3 εj1 j2 j3 = εi1 i2 i3 ,
which holds because O has the determinant
detO =
1
εi i i εj j j Oi j Oi j Oi j = 1.
3! 1 2 3 1 2 3 1 1 2 2 3 3
(B.3.9)
There is a useful summation formula for the Levi-Civita symbol
εijk εilm = δjl δkm − δjm δkl .
(B.3.10)
Using this relation it is easy to verify eq.(B.3.2) for the norm of a cross product
(~x × ~y ) · (~x × ~y ) = εijk xj yk εilm xl ym = (δjl δkm − δjm δkl )xj yk xl ym
= xj xj yk yk − xj yj xk yk
= |~x|2 |~y |2 − (~x · ~y )2 ,
(B.3.11)
Similarly, eq.(B.3.3) for the double cross product results from
[~x × (~y × ~z)]i = εijk xj [~y × ~z]k = εijk xj εklm yl zm
= (δil δjm − δim δjl )xj yl zm = xj zj yi − xj yj zi
= [(~x · ~z)~y − (~x · ~y )~z]i .
(B.3.12)
It may not be obvious that ~z = ~x × ~y is indeed a vector. Obviously, it has three
components, but do they transform appropriately under spatial rotations? We need to
show that
~z 0 = ~x 0 × ~y 0 = O~x × O~y = O~z.
(B.3.13)
Using eq.(B.3.8) we indeed find
T 0
zi = Oij
zj = Oji εjkl x0k yl0 = Oji εjkl Okm xm Oln yn = εimn xm yn .
(B.3.14)
Appendix C
Vector Analysis and Integration
Theorems
Before we can study quantum mechanics we need to equip ourselves with some necessary
mathematical tools including vector analysis and integration theorems. First, we will
consider scalar and vector fields as well as their derivatives. Then we will introduce the
theorems of Gauss and Stokes.
C.1
Fields
The concept of fields is central to many areas of physics, in particular, in electrodynamics
and all other field theories including general relativity and the standard model of elementary particle physics. A field describes physical degrees of freedom attached to points ~x
in space. The simplest example is a real-valued scalar field Φ(~x). Here Φ(~x) is a real
number attached to each point ~x in space. In other words, Φ(~x) is a real-valued function
of the three spatial coordinates. In general, fields depend on both space and time. For
the moment we limit ourselves to static (i.e. time-independent) fields. When we say that
Φ(~x) is a scalar field, we mean that its value is unaffected by a spatial rotation O, i.e.
Φ(~x 0 )0 = Φ(~x) = Φ(OT ~x 0 ).
(C.1.1)
Of course, the spatial point ~x itself is rotated to the new position ~x 0 = O~x, such that
~x = OT ~x 0 . A physical example for a scalar field is the scalar potential in electrodynamics.
Another important field in electrodynamics is, for example, the vector potential
~ x) = (A1 (~x), A2 (~x), A3 (~x)).
A(~
(C.1.2)
This field itself is a vector, i.e. there is not just a single number attached to each point
~ x) is a collection of three real numbers A1 (~x), A2 (~x), and
in space. Instead the field A(~
213
214
APPENDIX C. VECTOR ANALYSIS AND INTEGRATION THEOREMS
A3 (~x). Again, these three components transform in a specific way under spatial rotations
~ x 0 )0 = OA(~
~ x) = OA(O
~ T ~x 0 ).
A(~
(C.1.3)
~ x) is again a vector
It is easy to show that the product of a scalar and a vector field Φ(~x)A(~
~ x) · B(~
~ x) is a scalar field. Similarly,
field, while the scalar product of two vector fields A(~
~
~
the vector cross product of two vector fields A(~x) × B(~x) is again a vector field.
C.2
Vector Analysis
Since fields are functions of space, it is natural to take spatial derivatives of them. A
most important observation is that the derivatives with respect to the various Euclidean
coordinates form a vector — the so-called “Nabla” operator
d
d
d
~ = (∂1 , ∂2 , ∂3 ) =
∇
,
,
.
(C.2.1)
dx1 dx2 dx3
~ is a vector we again mean that it transforms appropriately under
When we say that ∇
spatial rotations, i.e.
d
d
d
~ 0=
~
∇
,
,
= O∇.
(C.2.2)
dx01 dx02 dx03
This immediately follows from the chain rule relation
X ∂xj d
X
X
d
d
T d
=
=
O
=
Oij
.
ji
0
0
dxi
∂xi dxj
dxj
dxj
j
j
(C.2.3)
j
~ acts on a scalar field as a gradient
The operator ∇
dΦ(~x) dΦ(~x) dΦ(~x)
~
∇Φ(~x) = (∂1 Φ(~x), ∂2 Φ(~x), ∂3 Φ(~x)) =
,
,
.
dx1
dx2
dx3
(C.2.4)
~ x). The operator ∇
~
In this way the scalar field Φ(~x) is turned into the vector field ∇Φ(~
can also act on a vector field. First, one can form the scalar product
~ · A(~
~ x) = ∂1 A1 (~x) + ∂2 A2 (~x) + ∂3 A3 (~x) = dA1 (~x) + dA2 (~x) + dA3 (~x) .
∇
dx1
dx2
dx3
(C.2.5)
~ acts as a divergence and turns the vector field A(~
~ x) into the scalar
Then the operator ∇
~ · A(~
~ x). One can also form the vector cross product
field ∇
~ × A(~
~ x) =
∇
(∂1 A2 (~x) − ∂2 A1 (~x), ∂2 A3 (~x) − ∂3 A2 (~x), ∂3 A1 (~x) − ∂1 A3 (~x)) =
dA2 (~x) dA1 (~x) dA3 (~x) dA2 (~x) dA1 (~x) dA3 (~x)
−
,
−
,
−
.
dx1
dx2
dx2
dx3
dx3
dx1
(C.2.6)
C.3. INTEGRATION THEOREMS
215
~ acts as a curl and turns the vector field A(~
~ x) into the vector field ∇
~ × A(~
~ x).
In this case, ∇
It is easy to show that
~ × ∇Φ(~
~ x) = 0, ∇
~ ·∇
~ × A(~
~ x) = 0.
∇
(C.2.7)
Still, one can form a non-zero second derivative
d2
d2
d2
+ 2 + 2,
2
dx1 dx2 dx3
~ ·∇
~ = ∂12 + ∂22 + ∂32 =
∆=∇
(C.2.8)
which is known as the Laplace operator.
C.3
Integration Theorems
We are all familiar with the simple theorem for the 1-dimensional integration of the derivative of a function f (x)
Z b
df (x)
= f (a) − f (b).
(C.3.1)
dx
dx
a
Here the points a and b define the boundary of the integration region (the interval [a, b]).
This simple theorem has generalizations in higher dimensions associated with the names
of Gauss and Stokes. Here we focus on three dimensions. Instead of proving Gauss’ and
Stokes’ theorems we simply state them and refer to the mathematical literature for proofs.
Approximating differentiation by finite differences and integration by Riemann sums, it is
straightforward to prove both theorems.
Gauss’ integration theorem deals with the volume integral of the divergence of some
~ x)
vector field A(~
Z
Z
3 ~
~
~ x).
d x ∇ · A(~x) =
d2 f~ · A(~
(C.3.2)
V
∂V
Here V is the 3-dimensional integration volume and ∂V is its boundary, a closed 2dimensional surface. The unit-vector normal to the surface is f~.
Similarly, Stokes’ theorem is concerned with the surface integral of the curl of a vector
field
Z
Z
2~ ~
~
~ x).
d f · ∇ × A(~x) =
d~l · A(~
(C.3.3)
S
∂S
Here S is the 2-dimensional surface to be integrated over and ∂S is its boundary, a closed
curve. The unit-vector ~l is tangential to this curve.
216
APPENDIX C. VECTOR ANALYSIS AND INTEGRATION THEOREMS
Appendix D
Differential Operators in Different
Coordinates
Besides ordinary Cartesian coordinates, spherical as well as cylindrical coordinates play
an important role. They can make life a lot easier in situations which are spherically or
cylindrically symmetric.
D.1
Differential Operators in Spherical Coordinates
Spherical coordinates are defined as
~x = r(sin θ cos ϕ, sin θ sin ϕ, cos θ) = r~er ,
~er = (sin θ cos ϕ, sin θ sin ϕ, cos θ),
~eθ = (cos θ cos ϕ, cos θ sin ϕ, − sin θ),
~eϕ = (− sin ϕ, cos ϕ, 0).
(D.1.1)
In spherical coordinates the gradient of a scalar field Φ(r, θ, ϕ) takes the form
~ = ∂r Φ ~er + 1 ∂θ Φ ~eθ + 1 ∂ϕ Φ ~eϕ ,
∇Φ
r
r sin θ
and the Laplacian is given by
1
1
1
∂θ (sin θ∂θ Φ) + 2 2 ∂ϕ2 Φ.
∆Φ = 2 ∂r r2 ∂r Φ + 2
r
r sin θ
r sin θ
A general vector field can be written as
~ x) = Ar (r, θ, ϕ)~er + Aθ (r, θ, ϕ)~eθ + Aϕ (r, θ, ϕ)~eϕ .
A(~
(D.1.2)
(D.1.3)
(D.1.4)
The general expression for the divergence of a vector field in spherical coordinates takes
the form
~ ·A
~ = 1 ∂r r2 Ar + 1 ∂θ (sin θAθ ) + 1 ∂ϕ Aϕ ,
∇
(D.1.5)
2
r
r sin θ
r sin θ
217
218APPENDIX D. DIFFERENTIAL OPERATORS IN DIFFERENT COORDINATES
while the curl is given by
1
1
~
~
∇×A =
∂θ (sin θAϕ ) −
∂ϕ Aθ ~er
r sin θ
r sin θ
1
1
1
1
+
∂ϕ Ar − ∂r (rAϕ ) ~eθ + ∂r (rAθ ) − ∂θ Ar ~eϕ .
r sin θ
r
r
r
(D.1.6)
D.2
Differential Operators in Cylindrical Coordinates
Cylindrical coordinates are defined as
~x = (ρ cos ϕ, ρ sin ϕ, z),
~eρ = (cos ϕ, sin ϕ, 0), ~eϕ = (− sin ϕ, cos ϕ, 0), ~ez = (0, 0, 1).
(D.2.1)
In cylindrical coordinates the gradient of a scalar field Φ(ρ, ϕ, z) takes the form
~ = ∂ρ Φ ~eρ + 1 ∂ϕ Φ ~eϕ + ∂z Φ ~ez ,
∇Φ
ρ
(D.2.2)
and the Laplacian is given by
∆Φ =
1
1
∂ρ (ρ∂ρ Φ) + 2 ∂ϕ2 Φ + ∂z2 Φ.
ρ
ρ
(D.2.3)
A general vector field can be written as
~ x) = Aρ (ρ, ϕ, z)~eρ + Aϕ (ρ, ϕ, z)~eϕ + Az (ρ, ϕ, z)~ez .
A(~
(D.2.4)
The general expression for the divergence of a vector field in cylindrical coordinates takes
the form
~ ·A
~ = 1 ∂ρ (ρAρ ) + 1 ∂ϕ Aϕ + ∂z Az ,
∇
(D.2.5)
ρ
ρ
and the curl is given by
1
∂ϕ Az − ∂z Aϕ ~eρ + [∂z Aρ − ∂ρ Az ] ~eϕ
ρ
1
1
+
∂ρ (ρAϕ ) − ∂ϕ Aρ ~ez .
ρ
ρ
~ ×A
~ =
∇
(D.2.6)
Appendix E
Fourier Transform and Dirac
δ-Function
In this appendix we introduce the Fourier transform as a limit of the Fourier series. We
also introduce the Dirac δ-Function and discuss some of its properties.
E.1
From Fourier Series to Fourier Transform
The Fourier transform is a generalization of a Fourier series which can be used to describe
periodic functions. Let us assume for a moment that Ψ(x) is defined in the interval
[−L/2, L/2] and that it is periodically extended outside that interval. Then we can write
it as a Fourier series
1X
Ψ(x) =
Ψk exp(ikx).
(E.1.1)
L
k
Here Ψk is the amplitude of the mode with wave number k. Due to the periodicity
requirement the wave lengths of the various modes must now be an integer fraction of L,
and therefore the k values are restricted to
k=
2π
m, m ∈ Z.
L
(E.1.2)
The separation of two modes dk = 2π/L becomes infinitesimal in the large volume limit
L → ∞. In this limit the sum over modes turns into an integral
1
Ψ(x) →
2π
Z
∞
dk Ψk exp(ikx).
(E.1.3)
−∞
When we identify Ψ(k) = Ψk we recover the above expression for a Fourier transform.
We can use our knowledge of the Fourier series to derive further relations for Fourier
219
220
APPENDIX E. FOURIER TRANSFORM AND DIRAC δ-FUNCTION
transforms. For example, we know that the amplitudes in a Fourier series are obtained as
Z L/2
Ψk =
dx Ψ(x) exp(−ikx).
(E.1.4)
−L/2
Let us convince ourselves that this is indeed correct by inserting eq.(E.1.1) in this expression
Z L/2
1X
dx
Ψk =
Ψk0 exp(ik 0 x) exp(−ikx)
L
−L/2
k0
Z
X
X
1 L/2
dx exp(i(k 0 − k)x) =
Ψk0 δk,k0 .
(E.1.5)
=
Ψk0
L −L/2
0
0
k
k
Here we have identified the Kronecker δ-function
Z
1 L/2
δk0 ,k =
dx exp(i(k 0 − k)x),
L −L/2
(E.1.6)
which is 1 for k 0 = k and 0 otherwise. Then indeed the above sum returns Ψk . Let us now
take the L → ∞ limit of eq.(E.1.4) in which we get
Z ∞
Ψk →
dx Ψ(x) exp(−ikx).
(E.1.7)
−∞
Again identifying Ψ(k) = Ψk we find the expression for an inverse Fourier transform
Z ∞
Ψ(k) =
dx Ψ(x) exp(−ikx).
(E.1.8)
−∞
Let us check the validity of this expression in the same way we just checked eq.(E.1.4).
We simply insert eq.(3.1.3) into eq.(E.1.8)
Z ∞
Z ∞
1
Ψ(k) =
dx
dk 0 Ψ(k 0 ) exp(ik 0 x) exp(−ikx)
2π
−∞
Z−∞
Z ∞
∞
0
0 1
=
dk Ψ(k )
dx exp(i(k 0 − k)x)
2π −∞
−∞
Z ∞
=
dk 0 Ψ(k 0 )δ(k 0 − k).
(E.1.9)
−∞
Here we have identified
1
δ(k − k) =
2π
0
Z
∞
dx exp(i(k 0 − k)x),
(E.1.10)
−∞
which is known as the Dirac δ-function. It is the continuum analog of the discrete Kronecker δ-function. The Dirac δ-function has the remarkable property that
Z ∞
dk 0 Ψ(k 0 )δ(k 0 − k) = Ψ(k).
(E.1.11)
−∞
This already follows from the fact that the limit L → ∞ led to eq.(E.1.8).
E.2. PROPERTIES OF THE δ-FUNCTION
E.2
221
Properties of the δ-Function
The most important property of the Dirac δ-function is eq.(E.1.11). Other relevant relations are
δ(−k) = δ(k),
(E.2.1)
which follows by the substitution k 0 = −k in the integral
Z ∞
Z −∞
Z ∞
0
0
0
dk Ψ(k)δ(−k) = −
dk Ψ(−k )δ(k ) =
dk 0 Ψ(−k 0 )δ(k 0 ) = Ψ(0). (E.2.2)
−∞
∞
−∞
This shows explicitly that under the integral δ(−k) has the same effect as δ(k). Similarly,
we can show that
1
δ(ak) =
δ(k),
(E.2.3)
|a|
which follows by the substitution k 0 = ak. For a > 0 we have
Z
Z ∞
1 ∞ 0
1
dk Ψ(k 0 /a)δ(k 0 ) = Ψ(0),
dk Ψ(k)δ(ak) =
a −∞
a
−∞
(E.2.4)
while for a < 0
Z ∞
Z
Z
1 −∞ 0
1 ∞ 0
1
0
0
dk Ψ(k)δ(ak) =
dk Ψ(k /a)δ(k ) = −
dk Ψ(k 0 /a)δ(k 0 ) = − Ψ(0),
a
a
a
−∞
∞
−∞
(E.2.5)
which shows explicitly that under the integral δ(ak) has the same effect as δ(k)/|a|. Using
the same method, it is easy to show that
δ(k)Ψ(k) = δ(k)Ψ(0).
(E.2.6)