Download Physics 2018: Great Ideas in Science: The Physics Module Quantum

Physics 2018: Great Ideas in Science:
The Physics Module
Quantum Mechanics Lecture Notes
Dr. Donald G. Luttermoser
East Tennessee State University
Edition 1.0
Abstract
These class notes are designed for use of the instructor and students of the course Physics 2018:
Great Ideas in Science. This edition was last modified for the Fall 2007 semester.
I.
Quantum Mechanics
A. Great Ideas in Physics.
1. Astronomy, the study of the night time sky, is the oldest of the
sciences.
a) Besides the sky, astronomy is the study of the objects that
make up the solar system (our home system of the Sun),
our home galaxy (the Milky Way), and the Universe as a
whole.
b)
In an attempt to understand what was occurring in the
night sky, humans invented a way to study the sky and
nature in general called physics – the study and matter
and energy and how these two interact with each other.
2. It is debatable as to which ideas in physics and astronomy are
the most important, but such a list should include the following
items:
a) We live on a round planet called Earth – Eratosthenes
(276–195 B.C.), an ancient Greek astronomer, was the
first to accurately determine the diameter of this round
Earth around 200 B.C.
b)
The solar system is heliocentric (i.e., Sun at the center)
not geocentric (i.e., Earth at the center).
i)
Aristotle (384–322 B.C.) assumed the Earth was
motionless and everything in the sky revolved around
us.
ii)
Aristarchus of Samos (310–230 B.C.) reasoned that
the Sun must be at the center.
I–1
iii) In order to explain the periodic “backward” (i.e.,
retrograde) motion of the planets on the sky, Claudius
Ptolemy, who lived around 140 A.D. and a firm
believer in Aristotle’s philosophy, developed a geocentric system that had the planets revolving on
smaller circles (called epicycles) whose “centers”
orbited the Earth (with the larger circular orbits
called deferents ).
c)
iv)
In 1543, Nicholas Copernicus (1473–1543), a Polish astronomer and cleric, published his heliocentric model for the solar system where Earth was
a planet, similar to the other planets, in circular
orbit about the Sun =⇒ the Copernican Revolution.
v)
Johannes Kepler (1571–1630), a German mathematician and astronomer, modified the Copernican
model by having the planets orbit the Sun in elliptical and not circular paths when he formulated
the three laws of planetary motion.
Invention of the scientific method owes much to the
work of Galileo Galilei (1564–1642), an Italian astronomer
and physicist. Galileo is considered to be the father of
experimental physics.
i)
ii)
Determined that objects of different masses fall at
the same rate on the Earth’s surface (which contradicted the teachings of Aristotle).
Came up with the concept of the pendulum clock.
I–2
iii)
iv)
d)
Developed the various concepts of motion.
First to use the telescope to study the cosmos =⇒
discovered the 4 large moons of Jupiter (i.e., the
Galilean moons), that Venus goes through phases
(like our Moon), that the Moon’s surface wasn’t
smooth, and that dark spots appear on the Sun
(i.e., sunspots) from time to time.
Isaac Newton (1642–1727), an English astronomer and
physicist, was perhaps the greatest scientist whoever lived!
The work he did is often referred to as the Newtonian
Revolution.
i)
Invented calculus to describe his physics.
ii)
Developed the laws of motion.
iii)
Developed the law of gravity.
iv)
Invented the reflecting telescope.
v)
Developed many theories in optics and showed
that white light is composed of the rainbow of colors.
e) James Clerk Maxwell (1831–1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland and had two major impacts on physics.
i)
His most significant achievement was developing
a set of equations that showed how electricity and
magnetism are related =⇒ Maxwell’s equations.
These equations merged the electric force and the
I–3
magnetic force into one force called electromagnetism.
ii)
He also developed the Maxwell distribution, a statistical means to describe the number density of
gases used in the kinetic theory of gases.
f ) In 1905, Albert Einstein (1879–1955), a German physicist,
rewrote Newton’s laws of motion in his Theory of Special Relativity. A bye-product of this theory was the
famous equation E = mc2 =⇒ mass can be converted to
energy and energy back to mass.
g) In 1915, Einstein rewrote Newton’s law of gravity in his
General Theory of Relativity.
h)
The quantum revolution began in the early part of the
20th century and has many people responsible for its development. Note that the word quantum means small
individual packet or step.
i)
German physicist Max Planck (1858–1947) derived
a formula describing blackbody radiation based on
radiating atomic oscillators.
ii)
Danish physicist Niels Bohr (1885–1962) developed a quantum model for the hydrogen atom.
iii) German physicist Werner Heisenberg (1901–1976)
invented matrix mechanics, the first formalization
of quantum mechanics in 1925, which he developed
with the help of Max Born and Pascual Jordan.
His uncertainty principle, developed in 1927, states
that the simultaneous determination of two paired
I–4
observable quantities, for example the position and
momentum of a particle, has an unavoidable uncertainty. Together with Bohr, he formulated the
Copenhagen interpretation of quantum mechanics.
iv)
In 1926 German physicist Erwin Schrödinger
(1887–1961) published a paper on wave mechanics
and what is now known as the Schrödinger equation. In this paper he gave a “derivation” of the
wave equation for time independent systems, and
showed that it gave the correct energy eigenvalues
for the hydrogen-like atom.
v)
There are others that we could cite here, but the
above four are the most important.
i) From 1970 through 1973, particle physicists developed
the Standard Model of particle physics which describes
three of the four known fundamental interactions between
the elementary particles that make up all matter.
i)
A large number of physicists were responsible for
its development.
ii)
To date, almost all experimental tests of the three
forces described by the Standard Model have agreed
with its predictions.
iii) Through the Standard Model all of the large variety of so-called “elementary” particles that have
been discovered in particle accelerators can be explained as a composite of any of six quarks and
six leptons.
I–5
B. The Nature of Physics.
1. 2 main branches:
a) Classical Physics:
i)
Classical Mechanics (also called Newtonian
Mechanics).
ii)
Thermodynamics (the study of heat).
iii)
b)
Fluid Mechanics (the study of fluids).
iv)
Electromagnetism (the study of electricity and
magnetism).
v)
Optics (the interaction of light with lenses and
mirrors).
vi)
Wave Mechanics (the study of wave motion).
Modern Physics:
i)
Special Relativity and General Relativity.
ii)
Quantum Mechanics (also called Atomic Physics).
iii)
iv)
v)
Nuclear Physics.
Statistical Mechanics (thermodynamics in terms
of probabilities).
Condensed Matter (once called Solid State Physics).
I–6
2. In classical physics, matter moves (i.e., follows trajectories) as a
result of a force being applied to it.
a) Contact forces: Force exerted through a collision as described by Newton’s 2nd law of motion: F = ma.
b)
Field (or natural) forces: Force exerted on an object
from its location in some natural potential field. There
are 4 field forces in nature:
Interaction
Relative Strength Range
Strong‡
1
10−15 m
Electromagnetic† ‡
10−2
∞
†‡
−6
−17
Weak
10
10
m
−43
Gravitational
10
∞
† - Under high energies, the electromagnetic and
weak forces act as one — the Electroweak force.
‡ - Under even higher energies, all of the natural
forces (except gravity) also may act as one,
as described by the Grand Unified Theory.
3. There are 6 key definitions that are useful in the description of
physics.
a) Concept: An idea or physical quantity used to analyze
nature (e.g., “space,” “length,” “mass,” and “time”
are concepts).
b)
Laws: Mathematical relationships between physical quantities.
c)
Principle: A very general statement on how nature operates (e.g., the principle of relativity, that there is no
absolute frames of reference, is the bases behind the theory of relativity).
I–7
d)
Models: A representation of a physical system (e.g., the
Bohr model atom).
e) Hypothesis: The tentative stages of a model that has
not been confirmed through experiment and/or observation (e.g., Ptolomy’s model solar system).
f ) Theory: Hypotheses that are confirmed through repeated
experiment and/or observation (e.g., Newton’s theory of
gravity). The word “theory” has different meanings in
common English (i.e., it can mean that one is making a
guess at something). However, it has a very precise
meaning in science! Something does not become
a theory in science unless it has been validated
through repeated experiment as described by the
scientific method.
4. At this point, we will differences between the classical view of
physics and the quantum view of physics.
C. The Classical Point of View.
1. A system is a collection of particles that interact among themselves via internal forces and that may interact with the world
outside via external fields.
a) To a classical physicist, a particle is an indivisible mass
point possessing a variety of physical properties that can
be measured.
i)
Intrinsic Properties: These don’t depend on
the particle’s location, don’t evolve with time, and
aren’t influenced by its physical environment (e.g.,
rest mass and charge).
I–8
ii)
Extrinsic Properties: These evolve with time
in response to the forces on the particle (e.g., position and momentum).
b)
These measurable quantities are called observables.
c)
Listing values of the observables of a particle at any time
=⇒ specify its state. (A trajectory is an equivalent way
to specify a particle’s state.)
d)
The state of the system is just the collection of the states
of the particles comprising it.
2. According to classical physics, all properties, intrinsic and extrinsic, of a particle could be known to infinite precision =⇒ for
instance, we could measure the precise value of both position and
momentum of a particle at the same time.
3. Classical physics predicts the outcome of a measurement by calculating the trajectory (i.e., the values of its position and momentum for all times after some initial (arbitrary) time t◦ ) of a
particle:
{~r(t), p~(t); t ≥ t◦ } ≡ trajectory,
(I-1)
where the linear momentum is, by definition,
~p(t) ≡ m
d
~r(t) = m ~v (t) ,
dt
(I-2)
with m the mass of the particle.
a) Trajectories are state descriptors of Newtonian physics.
b)
To study the evolution of the state represented by the
trajectory in Eq. (I-1), we use Newton’s Second Law:
X
~ = m ~a ,
F
I–9
(I-3)
P
~ is the sum of all vector forces acting on an obwhere F
ject, m is the mass of an object, and ~a is the acceleration
which results from the applied forces. We also can write
this equation using differential calculus as
m
d2
~r(t) = −∇V (~r, t) ,
dt2
(I-4)
where V (~r, t) is the potential energy of the particle (as a
function of radial distance r and time t) and ∇ is the socalled “del” operator (spatial derivatives in all directions).
This equation reduces to
d2 r
dV (r)
m 2 r̂ = −
r̂ ,
dt
dr
(I-5)
if the potential energy is time independent (note that r̂ is
a unit vector in the radial direction).
c)
To obtain the trajectory for t > t◦ , one only need to know
V (~r, t) and the initial conditions =⇒ the values of ~r
and p~ at the initial time t◦ .
d)
Notice that classical physics tacitly assumes that we can
measure the initial conditions without altering the motion
of the particle =⇒ the scheme of classical physics is based
on precise specification of the position and momentum of
the particle.
4. From the discussion above, it can be seen that classical physics
describes a Determinate Universe =⇒ knowing the initial conditions of the constituents of any system, however complicated,
we can use Newton’s Laws to predict the future.
5. If the Universe is determinate, then for every effect there is a
cause =⇒ the principle of causality.
I–10
D. The Quantum Point of View.
1. The concept of a particle doesn’t exist in the quantum world
— so-called particles behave both as a particle and a wave =⇒
wave-particle duality.
a) The properties of quantum particles are not, in general,
well-defined until they are measured.
b)
Unlike the classical state, the quantum state is a conglomeration of several possible outcomes of measurements of
physical properties.
c)
Quantum physics can tell you only the probability that
you will obtain one or another property.
d)
An observer cannot observe a microscopic system without
altering some of its properties =⇒ the interaction is unavoidable : The effect of the observer cannot be reduced to
zero, in principle or in practice.
2. This is not just a matter of experimental uncertainties, nature
itself will not allow position and momentum to be resolved to
infinite precision (see Figure I-1) =⇒ Heisenberg Uncertainty
Principle (HUP):
∆x(t◦ ) ∆px(t◦ ) ≥
h̄
1 h
= ,
2 2π
2
(I-6)
where h = 6.62620 × 10−27 erg-sec = 6.626 × 10−34 J-sec is
Planck’s Constant.
a) ∆x(t◦ ) is the minimum uncertainty in the measurement
of the position in the x-direction at time t◦ .
b)
∆px (t◦ ) is the minimum uncertainty in the measurement
of the momentum in the x-direction at time t◦ .
I–11
∆x
<x>
∆p
x
<p>
p
Figure I–1: The results of measurement of the x components of the position and momentum of a
large number of identical quantum particles. Each plot shows the number of experiments that yield
the values on the abscissa. Results for each component are seen to fluctuate about a central peak,
the mean value hxi and hpi.
c)
Similar constraints apply to the pairs of uncertainties ∆y(t◦),
∆py (t◦) and ∆z(t◦ ), ∆pz (t◦).
d)
Position and momentum are fundamentally incompatible
observables =⇒ the Universe is inherently uncertain!
e) The HUP strikes at the very heart of classical physics: the
trajectory =⇒ obviously, if we cannot know the position
and momentum of a particle at t◦ , we cannot specify the
initial conditions of the particle and hence cannot calculate the trajectory.
f ) Once we throw out trajectories, we can no longer use Newton’s Laws, new physics must be invented!
I–12
Example I–1. Derive the energy-time uncertainty relation
from the Heisenberg (position-momentum) Uncertainty Relation.
Solution:
A particle moves a distance ∆x in a time interval ∆t. These are
related via the velocity equation
∆x =
p
∆t .
m
Plugging this into Eq. (I-4) gives
∆x ∆p =
p
h̄
∆t ∆p ≥ .
m
2
Special relativity gives the energy of a particle is related to its
momentum by
E 2 = p2 c2 + m2◦ c4 ,
where m◦ is the rest mass of the particle. Taking the derivative
of this equation with respect to momentum gives
2E
dE
= 2pc2 .
dp
Replacing the infinitesimal differentials with small changes in
both E and p gives
E
p ∆p = 2 ∆E .
c
Substituting above gives
h̄
E
.
∆E
∆t
≥
mc2
2
Finally, using Einstein’s well known equation E = mc2 , we see
that
h̄
∆E ∆t ≥ .
(I-7)
2
I–13
3. Since Newtonian and Maxwellian physics describe the macroscopic world so well, physicists developing quantum mechanics
demanded that when applied to macroscopic systems, the new
physics must reduce to the old physics =⇒ this Correspondence Principle was coined by Niels Bohr.
4. Due to quantum mechanics probabilistic nature, only statistical information about aggregates of identical systems can be obtained. Quantum mechanics can tell us nothing about the behavior of individual systems. Moreover, the statistical information
provided by quantum theory is limited to the results of measurements =⇒ thou shall not make any statements that can never be
verified.
E. Blackbody Radiation
1. In the early part of the 20th century, Max Planck asked the question: What is the spectrum of electromagnetic (EM) radiation inside a heated cavity ? More specifically, how does this spectrum
depend on the temperature T of the cavity, on its shape, size, and
chemical makeup, and on the frequency ν of the EM radiation in
it?
a) Earlier in the mid-19th century, Kirchhoff found that the
energy inside such a cavity is independent of the physical
characteristics of the cavity (i.e., size and shape), only ν
and T were important.
b)
Planck was interested in the energy density in the cavity
and sought an expression for the radiative energy density per unit volume ρ(ν, T ) and this density in the
frequency range ν to ν + dν: ρ(ν, T ) dν.
c)
Kirchhoff called his model of a heated cavity in thermal
equilibrium a “black-body radiator.” A blackbody is
I–14
simply anything that absorbs all radiation incident upon
it. Thus a blackbody radiator neither reflects nor transmits energy; it just absorbs or emits it.
2. Wien had already experimentally ascertained that the radiative
energy density of a blackbody was proportional to ν 3 and, from
R
the work of Stefan, that the integrated energy density 0∞ ρ(ν, T ) dν
is proportional to T 4 .
a) Planck realized that ρ(ν, T ) could not solely depend upon
ν 3 since this would imply that the energy density would
blow up at small frequencies (i.e., long wavelengths).
b)
Planck focused on the exchange of energy between the
radiation field and the walls of the cavity.
i)
He developed a simple model of this process by
imagining that the molecules of the cavity walls are
resonators — electrical charges undergoing simple
harmonic motion.
ii)
As a consequence of their oscillations, these charges
emit EM radiation at their oscillation frequency,
which at thermal equilibrium, equals the frequency
ν of the radiation field.
iii) According to classical electromagnetic theory, energy exchange between the resonators and the energy field is a continuous process =⇒ the oscillators
can exchange any amount of energy with the field,
provided that the energy is conserved in the process.
I–15
c)
Planck deduced an empirical formula for the radiative energy density:
Aν 3
ρ(ν, T ) = Bν/T
.
(I-8)
e
−1
i) A and B are constants that were to be determined
by fitting experimental data.
ii)
The functional form of Eq. (I-8) agreed beautifully with observations.
iii) In the limit of ν → ∞ and T → 0, Eq. (I-8)
reduces to Wien’s law.
iv)
However, when Planck developed this functional
form for blackbody radiation, be didn’t have a clue
as to how to prove it theoretically.
v)
Planck made a second assault on the energy density by adopting a statistical method based upon
the concept of entropy as interpreted probabilistically by Boltzmann. He also assumed in this treatment that only discrete amounts of energy can be
absorbed or emitted by the resonators that comprise
the walls of the blackbody.
vi)
He called these discrete amounts of energy quanta.
To each quantum, Einstein took Planck’s idea and
assigned an energy equal to an integral multiple of
hν, where h is now referred to as Planck’s constant.
I–16
d)
Having made this assumption, Planck easily derived the
radiation law:
ρ(ν, T ) =
8πν 2
hν
,
c3 ehν/kT − 1
(I-9)
where k is the above mentioned Boltzmann’s constant. As
can be seen, Eq. (I-9) agrees with the empirical relation
expressed in Eq. (I-8).
e) The radiative energy density, ρ(ν, T ), is related to the
monochromatic radiative energy flux Bν (T ) (i.e., the
“brightness” of a glowing object) with the relation
ρ(ν, T ) =
4π
Bν (T ) .
c
(I-10)
f ) As such, the monochromatic energy flux (or brightness)
of a blackbody is
Bν (T ) =
2hν 3 /c2
ehν/kT − 1
(I-11)
in frequency space, where Bν is measured in J/s/m2 /Hz/sr
(‘sr’ is the steradian unit) in SI units and erg/s/cm2 /Hz/sr
in the cgs unit system. Since Bν dν = Bλ dλ and ν = c/λ,
we can also write this function in wavelength space as
Bλ (T ) =
2hc2 /λ5
.
ehc/λkT − 1
(I-12)
Both Eqs. (I-11) and (I-12) are called the Planck function (in frequency and wavelength space, respectively).
3. Planck’s radiation law not only solve the problem of blackbody
radiation, it also opened the door to a new understanding of radiation energy in physics =⇒ quantum physics, also called quantum
mechanics.
I–17
F. The Semi-Empirical Model of Hydrogen.
1. Work that lead to an understanding of the spectrum of the hydrogen atom took place at the end of the 19th and beginning of
the 20th century. As such, much of what of the work described
in this and the next few subsections is presented in the cgs unit
system since those are the units that were being used in physics
at the time.
2. Rydberg (1890), Ritz (1908), Planck (1910), and Bohr (1913)
were all responsible for developing the theory of the spectrum of
the H atom. A transition from an upper level m to a lower level
n will radiate a photon at frequency
!
1
1
2
νmn = c RA Z
−
,
(I-13)
n2 m2
where the velocity of light, c = 2.997925 × 1010 cm/s, Z is the
effective charge of the nucleus (ZH = 1, ZHe = 2, etc.), and the
atomic Rydberg constant, RA, is given by
!
me −1
.
(I-14)
RA = R∞ 1 +
MA
a) The Rydberg constant for an infinite mass is
2π 2 me e4
R∞ =
= 109, 737.31 cm−1 ,
(I-15)
3
ch
where e = 4.80325 × 10−10 esu is the electron charge in
cgs units.
b)
In atomic mass units (amu), the electron mass is me =
5.48597 × 10−4 amu whereas the atomic mass, MA , can be
found on a periodic table (see also Table I-1).
c)
Eq. (I-13) can also be expressed in wavelengths (vacuum)
by the following
!
1
1
1
2
= RA Z
−
.
(I-16)
λmn
n2 m2
I–18
Table I–1: Atomic Masses and Rydberg Constants
Atom
Hydrogen, 1H
Helium, 4 He
Carbon, 12 C
Nitrogen, 14 N
Oxygen, 16 O
Neon, 20Ne
Atomic Mass, MA
(amu)
1.007825
4.002603
12.000000
14.003074
15.994915
19.992440
Rydberg Constant, RA
(cm−1 )
109,677.6
109,722.3
109,732.3
109,733.0
109,733.5
109,734.3
3. Lines that originate from the same level in a hydrogen-like atom/ion
are said to belong to the same series. Transitions out of (or into)
the ground state (n = 1) are lines of the Lyman series, n = 2
corresponds to the Balmer series, and n = 3, the Paschen
series.
4. For each series, the transition with the longest wavelength is
called the alpha (α) transition, the next blueward line from α is
the β line followed by the γ line, etc.
a) Lyman α is the 1 ↔ 2 transition, Lyman β is the 1 ↔ 3
transition, Lyman γ is the 1 ↔ 4 transition, etc.
b)
Balmer or Hα is the 2 ↔ 3 transition, Hβ is the 2 ↔ 4
transition, Hγ is the 2 ↔ 5 transition, etc.
5. Lines that go into or come out of the ground state are referred
to as resonance lines.
6. For one e− atoms (i.e., hydrogen-like: H I, He II, C VI, Fe XXVI,
etc. =⇒ in astrophysics, ionization stages are labeled with Roman
numerals: I = neutral, II = singly ionized, etc.), the principal (n)
levels have energies of
2 π 2 m e4 Z 2
En = −
,
n2 h2
I–19
(I-17)
109678
13.60
100000
12.40
cm-1
eV
Brackett
80000
9.92
Energy
Paschen
Wave Number
60000
Pfund
Humphreys
Balmer
7.44
Lyman
H
40000
4.96
Hydrogen Z = 1
20000
2.48
0
0.00
Figure I–2: A partial Grotrian diagram of neutral hydrogen. The lowest 7 levels are shown with
various transitions labeled.
where Z = charge of the nucleus.
a) Negative energies =⇒ bound states
Positive energies =⇒ free states
Ionization limit (n → ∞) in Eq. (I-17) has E = 0.
b)
In astronomical spectroscopy, the ground state is defined
as zero potential (i.e., E1 = 0) and atomic states are
displayed in terms of energy level diagrams (see Figure
I–20
Efield
wavecrest
z
y
x
Bfield
direction
of wave
propagation
λ
Figure I–3: An electromagnetic wave.
I-2), where the energy levels are determined by
En = 13.6 Z
2
1
1− 2
n
!
eV .
(I-18)
n → ∞ defines the ionization potential (IP) of the
atom (or ion), so that for H: IP = 13.6 eV, for He II: IP
= 54.4 eV, etc.
c)
NOTE: 1 eV = 1.602 × 10−19 J = 1.602x10−12 erg =
8066 cm−1 = 12,398 Å = 11,605 K.
d)
The lowest energy state (E = 0) is called the ground
state. States above the ground are said to be excited.
G. Emission and Absorption of Radiation.
1. Electromagnetic Waves.
a) An electromagnetic (EM) wave consists of a transverse,
and mutually perpendicular, oscillating electric and magnetic fields (see Figure I-3).
b)
An atom, in the presence of a passing EM wave, responds
primarily to the electric component of the EM wave.
I–21
c)
i)
If the wave is long as compared to the size of the
atom, the spatial variation of the electric field can
be ignored during the interaction.
ii)
This is the same thing as saying that the period of
oscillation is long as compared to the time it takes
the charge to move around (or within) the atom.
As a result, the atom is essentially exposed to a purely
sinusoidal oscillating electric field, E, of the form
E = E◦ cos(ωt) ẑ ,
(I-19)
where here the electric wave oscillates about the z axis
with an amplitude of E◦ with an angular frequency ω =
2πν.
d)
The potential, φ, is related to the E field by
E = −∇φ ,
(I-20)
hence the potential must be a sinusoidal function as well.
e) The potential of an EM wave passing a bound electron
can perturb the potential energy Ve of said electron via
the potential energy equation from classical EM theory:
1
(I-21)
Ve = q φ ,
2
where q is the charge of the electron. This oscillating
perturbation then can cause the bound electron to change
its state.
2. Absorption, Stimulated Emission, and Spontaneous Emission.
a) We shall see later in the course that bound electrons in an
atom are only found in certain energy states or levels.
Each of these states are described by wave functions.
I–22
i)
The form of an electron wave function is solved
with the partial differential equation called the
Schrödinger equation (see §I.I).
ii)
The solution of this equation depends upon the
potential energy of the given state.
b)
Bound electrons will jump from one state to another based
upon the probability of the transition occurring. This
probability is calculated from the wave function of the
particle/state.
c)
Photon perturbations also can cause electrons to de-excite
in an atom (called stimulated emission).
d)
From the HUP (∆E ∆t ≥ h̄/2), electrons also can deexcite spontaneously (i.e., spontaneous emission).
i)
∆t represents the half-life of the time an electron
stays excited before spontaneously decaying back
to a lower energy state.
ii)
∆E in HUP represents the “half-width” of the
thickness of the energy probability distribution of a
given state. For this natural broadening, this is typically nothing more than a Gaussian (i.e., normal)
distribution. Note that ∆E = 0 for the ground
state of an atom (or molecule) since an electron
stays there indefinitely until perturbed by a passing photon.
H. Matter and Energy: Particles or Waves?
1. In 1905, Einstein proposed that the energy in an EM field is not
spread out over a spherical wavefront, as Maxwell had assumed,
I–23
but instead is localized in indivisible clumps — in quanta.
a) Each quantum of frequency ν travels through space at the
speed of light c, carrying a discrete amount of energy hν
and momentum hν/c.
b)
Thus Einstein formulated the particle view of light.
c)
G.N. Lewis subsequently dubbed Einstein’s and Planck’s
quantum of radiation energy a photon, the name we use
today.
d)
In Einstein’s view, not only is the radiation found in
clumps, but the radiation field itself is quantized !
e) Einstein went on to use this photon model to describe
the photoelectric effect — the ejection of electrons from a
metal, such as sodium, when light impinges on it. Einstein
won a Nobel Prize for his theory of the photoelectric effect.
f ) Millikan reported a precise verification of Einstein’s equation of Planck’s quantized energy idea, E = hν, and the
first measurement of the Planck constant, hence further
showing the validity of the particle-like nature of light.
g) In 1923, Compton published results of his X-ray scattering experiments, and drove the last nail in the coffin of the
wave theory of light. Wavelength shifts were observed as
the X-rays scattered of a thin carbon film which were inconsistent with Maxwell’s theory. However, the scattering
was easily explained in the particle theory of light.
2. However, classical physics is filled with experiments that show
light as a wave phenomenon: diffraction and interference are two
such experiments.
I–24
a) Light takes on whatever characteristic for which
the experiment is testing. The observation gives the
photon its identity!
b)
Light, having both wave and particle characteristics, is
sometime jokingly referred to as a wavicle.
3. As this wave-particle debate continued for photons, a set of experimentalist set out to run known particles (e.g., electrons) through
the same experiments that produce wave-like characteristics for
light.
a) Surprisingly, electrons also showed wave-like characteristics!
b)
When electrons are passed through a double slit, interference patterns arose on the detector that mimics the
results for photons — the slits defracted the electrons.
c)
Electrons were found to have a wavelength of
λ=√
h
,
2mE
(I-22)
where m and E are the mass and energy of the electron,
respectively.
d)
de Broglie came up with the answer — all microscopic
material particles are characterized by a wavelength and
a frequency, just like photons =⇒ matter waves. This
idea led de Broglie, with the help of Einstein, to equations
relating to the equality of matter and radiant energy.
i)
The photon is a relativistic particle of rest mass
m◦ = 0 and its momentum is defined by
p=
E
.
c
I–25
(I-23)
ii)
The energy of a photon is
E = hν ,
(I-24)
and using this in Eq. (I-23) gives
p=
hν
.
c
(I-25)
iii) For a wave in free space, the wavelength is λ =
c/ν, so Eq. (I-25) becomes
p=
h
.
λ
(I-26)
iv)
For a particle with mass traveling at relativistic
velocities (in a zero potential energy field), its total
energy is
E 2 = p2 c2 + m2◦ c4 .
(I-27)
v)
If its velocity is non-relativistic (v c), then its
kinetic energy is simply
p2
,
T =
2m◦
(I-28)
where T is the kinetic energy, or
T = E − m◦ c2 .
vi)
(I-29)
de Broglie proposed that Eqs. (I-24) and (I-26)
be used for material particles as well as photons.
Thus, for electrons, atoms, photons and all other
quantum particles, the energy and momentum are
related to the frequency and wavelength by
p = h/λ
E = hν
de Broglie-Einstein equations.
I–26
(I-30)
vii) Notice that the de Broglie wavelength equation
λ = h/p implies an inverse relationship between
the total energy E of a particle and its wavelength,
viz.,
hc/E
λ= r
(I-31)
.
m◦ c2 2
1− E
If applied to a photon (by setting the rest mass to
zero), this equation reduces to Eq. (I-24). Hence
the larger the energy of a particle, the smaller is its
wavelength, and vise versa.
e) Trying to understand the meaning of these matter waves
led Schrödinger and Heisenberg to create the physics of
quantum mechanics.
I. The Schrödinger Equation.
1. As previously mentioned, quantum mechanics approaches the
trajectory problem of Newtonian mechanics quite differently. On
a microscopic level, particles do not follow trajectories, but instead are characterized by their wave function, Ψ(x, t), where
x is the 1-dimensional position of the wave function at time t.
(Actually we would need to include all 3-dimensions, x, y, and z,
in the wave function, but there’s no need to complicate this too
much in this class.)
a) The wave function is determined from Schrödinger’s
Equation:
∂Ψ
h̄2 ∂ 2 Ψ
ih̄
=−
+VΨ .
(I-32)
∂t
2m ∂x2
√
i) Here, i = −1 (note that having an “i” in a function or a number makes it a “complex” function or
number),
h
h̄ =
= 1.054573 × 10−34 J s ,
2π
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and ∂ is the symbol for a “partial” differential (which
is covered in Calculus III).
ii)
b)
Whereas Newton’s Second Law, F = ma, is the
most important equation in all of classical physics,
Eq. (I-32) is the most important equation in all of
quantum physics.
Given suitable initial conditions [typically, Ψ(x, 0)], the
Schrödinger equation determines Ψ(x, t) for all future times,
just as, in classical mechanics, Newton’s Second Law determines x(t) for all future times.
2. What exactly is the wave function, and what does it do for you
once you got it?
a) Whereas a particle is localized at a point in classical mechanics, a wave function is spread out in space =⇒ it is a
function of x for any given time t.
b)
Born came up with a statistical interpretation of the
wave function, which says that |Ψ(x, t)|2 gives the probability of finding the particle at point x, at time t, or more
precisely,
(
)
probability of finding the particle
2
|Ψ(x, t)| dx =
between x and (x + dx) at time t.
(I-33)
c)
The wave function itself is complex, but |Ψ|2 = Ψ∗Ψ
(where Ψ∗ is the complex conjugate of Ψ) is real and nonnegative — as a probability must be.
d)
For the hypothetical wave function in Figure (I-4), you
would be quite likely to find the particle in the vicinity of
point A, and relatively unlikely to find it near point B.
I–28
{
| Ψ |2
dx
A
B
C
x
Figure I–4: A hypothetical wave function. The particle would be relatively likely to be found near
A, and unlikely to be found near B. The shaded area represents the probability of finding the particle
in the range dx.
3. From the concept of the wave function, it becomes easier to see
how the Heisenberg Uncertainty Principle arises in nature. The
wave function will not allow you to predict with certainty the
outcome of a simple experiment to measure a particle’s position
— all quantum mechanics has to offer is statistical information
about the possible results.
4. As can be seen from this section, to truly understand quantum
mechanics, one must be skilled in handling partial differential
equations and understanding the rules of statistics. One characteristic of wave functions that result from the solution of the
Schrödinger Equation is that particles in negative energy states
(called bound states) can only exist in discrete states described
by quantum numbers:
a) The principal quantum number (n) which is proportional to the total energy of a given bound state and idenI–29
tifies a given “shell” that a bound electron is in.
b)
The orbital angular momentum quantum number
(`) which helps describes the orbital angular momentum
(the classical analogy of an electron “in orbit” about a nucleus) of a given bound state and identifies a given “subshell” within a shell.
c)
The spin angular momentum quantum number (s)
which helps describes the spin angular momentum (the
classical analogy of an electron “spinning” about an axis
just as the Earth spins about an axis). Note that there are
only 2 spin states, “up” and “down” (the classical analogy
of a counterclockwise versus a clockwise spin).
d)
The total angular momentum quantum number (j =
` ± s) which helps describes the total angular momentum.
J. Philosophical Interpretations of Quantum Mechanics.
1. The Realist Position:
a) We view the microscopic world as probabilistic due to the
fact that quantum mechanics is an incomplete theory.
b)
The particle really was at a specific position (say point C
in Figure I-4), yet quantum mechanics was unable to tell
us so.
c)
To the realist, indeterminacy is not a fact of nature, but
a reflection of our ignorance.
d)
If this scenario is, in fact, the correct one, then Ψ is not
the whole story — some additional information (known
as a hidden variable) is needed to provide a complete
description of the particle.
I–30
2. The orthodox position =⇒ the Copenhagen interpretation:
a) The particle isn’t really anywhere in space. The act of
the measurement forces the particle to take a stand —
though how and why we dare not ask!
b)
Observations not only disturb what is to be measured,
they produce it.
c)
Bohr and his followers put forward this interpretation of
quantum mechanics.
d)
It is the most widely accepted position of the interpretation of quantum mechanics in physics.
3. The agnostic position:
a) Refuse to answer! What sense can there be in making
assertions about the status of a particle before a measurement, when the only way of knowing whether you were
right is precisely to conduct the measurement, in which
case what you get is no longer before the measurement.
b)
This has been used as a fall-back position used by many
physicists if one is unable to convince another of the orthodox position.
4. In 1964, John Bell astonished the physics community by showing
that it makes an observable difference if the particle had a precise
(although unknown) position prior to its measurement.
a) This discovery effectively eliminated the realist position.
b)
Bell’s Theorem showed that the orthodox position is the
correct interpretation of quantum mechanics by proving
that any local hidden variable theory is incompatible with
I–31
quantum mechanics (see Bell, J.S. 1964, Physics, 1, 195).
c)
We won’t get into the details of Bell’s Theorem in this
class. Suffice it to say that a particle does not have a
precise position prior to the measurement, any more than
ripples in a pond do =⇒ it is the measurement process
that insists upon one particular number, and thereby in
a sense creates the specific result.
5. The act of the measurement collapses the wave function to a
delta function (e.g., a sharp peak) at some position — Ψ soon
spreads out again after the measurement in accordance to the
Schrödinger equation.
I–32