Download On the Reality of the Quantum State

School of Physics & Astronomy
The University of Edinburgh
On the Reality of the Quantum State
The Historical Development of the Interpretation of
Quantum Mechanics
Senior Honours Group Project
K. Campbell, T. Downes, A. Khamseh, C. Lumby, M. Rüfenacht, G. Thompson
Supervisor: A. Shotter, [email protected]
August 28, 2013
“But our present [quantum mechanical] formalism is not purely epistemological; it
is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature — all scrambled up by Heisenberg and Bohr into an
omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we
cannot separate the subjective and objective aspects of the formalism, we cannot
know what we are talking about; it is just that simple.”
- Edwin Thompson Jaynes [1]
Probability in Quantum Theory (1996)
On the Reality of the Quantum State
The Historical Development of the Interpretation of Quantum
Mechanics
Senior Honours Group Project
K. Campbell, T.Downes, A. Khamseh, C.Lumby, M. Rüfenacht, G. Thompson
Abstract
Since its 1927 inception, the Copenhagen interpretation has been the prevailing
interpretation of quantum mechanics. In May 2012 On the Reality of the Quantum State
was published and showed that the quantum state cannot merely represent information.
We present an in-depth review of the development of the Copenhagen Interpretation along
with other competing theories, and go on to argue that this modern paper leads to
advances in our understanding of the interpretation of quantum mechanics.
Contents
1 What is an Interpretation and Why do We Need One?
2 Origins and Foundations
2.1 The Need for New Physics: 1899 - 1905
2.1.1 Blackbody Radiation . . . . . . .
2.1.2 Planck’s Law . . . . . . . . . . .
2.2 Einstein Sheds Some Light . . . . . . . .
2.2.1 Light Quanta . . . . . . . . . . .
2.2.2 The Photoelectric E↵ect . . . . .
2.3 Quantum in the Atom . . . . . . . . . .
2.3.1 The Bohr Model . . . . . . . . .
2.3.2 Einstein Connects the Dots . . .
2.4 Correspondence Principle . . . . . . . .
2.5 Wave-Particle Duality . . . . . . . . . .
2.5.1 The Compton E↵ect . . . . . . .
2.5.2 De Broglie’s Matter Waves . . .
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3 Quantum Mechanics in its Infancy
3.1 Birth of the Formalism . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Matrix Mechanics . . . . . . . . . . . . . . . . . . . . .
3.1.2 The Guiding Principle in the New Quantum Mechanics
3.1.3 Wave Mechanics . . . . . . . . . . . . . . . . . . . . . .
3.2 Early Interpretations of the New Quantum Mechanics . . . . .
3.2.1 Schrödinger’s Electromagnetic Interpretation . . . . . .
3.2.2 Hydrodynamic Interpretations . . . . . . . . . . . . . .
3.2.3 De Broglie’s Double Solution Interpretation . . . . . . .
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4 The Development of the Copenhagen Interpretation
4.1 The Conceptual situation . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Born’s Probabilistic Interpretation . . . . . . . . . . . . . . .
4.1.2 Schrödinger’s Visit to Copenhagen . . . . . . . . . . . . . . .
4.2 The Indeterminacy Relations . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The Conceptual Context . . . . . . . . . . . . . . . . . . . . .
4.2.2 On the Intuitive Content of Quantum-Theoretical Mechanics
4.2.3 Later Developments . . . . . . . . . . . . . . . . . . . . . . .
4.2.4 Philosophical Implications . . . . . . . . . . . . . . . . . . . .
4.3 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The Como Lecture . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Analysis and Initial Responses . . . . . . . . . . . . . . . . .
4.4 The Copenhagen Interpretation of Quantum Mechanics . . . . . . .
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5 The Bohr-Einstein Debates
5.1 Round 1: Solvay 1927 . . . . . . . . . . . . .
5.1.1 Setting the Scene . . . . . . . . . . . .
5.1.2 Einstein and the Uncertainty Principle
5.1.3 Einstein Strikes Again . . . . . . . . .
5.1.4 Einstein’s Ensembles . . . . . . . . . .
5.2 Round 2: Solvay 1930 . . . . . . . . . . . . .
5.3 Round 3: EPR . . . . . . . . . . . . . . . . .
5.3.1 Einstein Moves to Princeton . . . . . .
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5.4
5.5
5.3.2 EPR . . . . . . . . . . . . . . . . . . . . .
5.3.3 Response to EPR . . . . . . . . . . . . . .
Further Developments on EPR, Non-Locality and
The Measurement Problem . . . . . . . . . . . .
6 Di↵erent Interpretations
6.1 The Collapse of the Wave Function, Von
6.2 Bohm’s Interpretation and Bell . . . . .
6.3 Many-Worlds Interpretations . . . . . .
6.3.1 Introduction . . . . . . . . . . .
6.3.2 Relative State Formulation . . .
6.3.3 Many-Worlds Theory . . . . . .
6.3.4 Bare Theory . . . . . . . . . . .
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Bell
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Neumann, and Wigner Interpretations
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7 Recent Developments and Applications
7.1 Quantum Computing . . . . . . . . . . . . . . . . . . . . . .
7.2 Extensions and Variations of the Copenhagen Interpretation
7.3 Alternative Interpretations . . . . . . . . . . . . . . . . . .
7.4 Unscrambling the Foundations of the Theory . . . . . . . .
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8 Conclusion
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Appendices
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A Quantum State Superposition
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B Einstein’s Derivation of Planck’s Formula from the Bohr Model
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C Interpretations of the Correspondence Principle
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D The Formal Equivalence of the Wave and Matrix Formalisms
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E Bohr-Schrödinger Debate 1926
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F The Problem with Ensembles
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G Schrödinger’s Cat
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H Quantum Mechanical Formalism
65
1
What is an Interpretation and Why do We Need One?
The aim of the report is to give a historical review of the developments of quantum mechanics
with a specific focus on the interpretational aspects of the theory. To address this it is important
to understand what an interpretation is and why it is needed.
In his book An Interpretive Introduction to Quantum Field Theory [2], Paul Teller defines an
interpretation to be
“A relevant similarity relation hypothesized to hold between a model and the aspects of actual
things that the model is intended to characterize.”
A di↵erent point of view on what constitutes an interpretation is given by Mittelstaedt [3]
as:
“Any interpretation of quantum theory should provide interrelations between the theoretical
expressions of the theory and possible experimental outcomes. In particular, any interpretation
of quantum mechanics has to clarify which are the theoretical terms that correspond to
measurable quantities and whether there are limitations of the measurability.”
Such a limitation, for example, could be the simultaneous measurement of two observables that
are said to be incompatible.
Another important topic of an interpretation is the understanding of the Schrödinger wave
function. The wave function is central to both theoretical and applied quantum mechanics and
we believe that any proper interpretation should specify its role and relation to measurement.
More generally, we are of the opinion that an interpretation is about understanding quantum
theory from a viewpoint of knowledge, fundamental principles, and underlying operational mechanics1 .
When discussing why we need an interpretation of quantum mechanics, it is worth drawing
parallels to classical mechanics. In classical formulations of mechanics we are dealing with
macroscopic problems that are familiar to us. This familiarity has lead to a somewhat more
relaxed boundary between formalism and interpretation. However, in quantum mechanics appealing to experience is not an option and as such there is an urgent need for a consistent
interpretational scheme.
An important interpretational concept in quantum theory is that of the measurement problem.
The measurement problem is a key aspect in any interpretation of quantum mechanics and
its specification is fundamental to the validification and understanding of quantum mechanics
outside its simple applicational use. To further discuss the matter one needs an understanding
of the idea of superposition2 . We will return to the measurement problem at a later stage and
instead turn our focus towards the very beginning of quantum mechanics.
1
Typical questions raised in relation to knowledge, fundamental principles, and underlying operational mechanics respectively would be: Is the wave function a state of complete knowledge? What is the collapse of the
wave function? How is theory related to measurement? These questions will be dealt with throughout the report.
2
For a full explanation and justification of the concept of superposition, which is crucial to the understanding
of quantum mechanics, refer to appendix A.
1
2
Origins and Foundations
2.1
2.1.1
The Need for New Physics: 1899 - 1905
Blackbody Radiation
The story of quantum mechanics began around the end of the 19th century, a time in which
physics was considered essentially complete. Max Planck, then a young German schoolboy,
was heavily discouraged from entering the field, being told by a Professor “In this field, almost
everything is already discovered, and all that remains is to fill a few holes.” [4] The famous
physicist Lord Kelvin even proclaimed “There is nothing new to be discovered in physics now”.
[5]
One of the few remaining problems demanding research was a theoretical description of the electromagnetic radiation emitted by hot objects. The pressure to solve this problem was motivated
on two fronts - for its theoretical merits and its industrial applications. As early as 1859 Gustav
Kirchho↵, a German Physicist at Heidelberg University, developed the concept of a blackbody an object that absorbs all incident radiation, reflecting none - as a good approximation for all
materials emitting radiation. Kirchho↵ declared that the wavelength range and intensities of
the radiation emitted by the blackbody, known as its emissivity, would be a universal function of
radiation wavelength and body temperature only, independent of the object material and size.
He went on to state that this function was “extremely important to determine” [6], which was
reiterated by Paschen who remarked “the determination of [this theory] is sufficiently important
to endow a professorship” [6]. Physicists were further encouraged by the large grants a↵orded
by German electric companies, who at the time were attempting to produce the brightest light
bulbs possible given minimum energy input, an optimisation problem that would be solved by
a correct blackbody theory.
By the end of 1893 Wien had discovered [7] what is now known as Wien’s law, giving the
black-body emissivity I in terms of the body frequency ⌫ and temperature T by3
I = c1 ⌫ 3 exp( c2 ⌫/T )
This law agrees spectacularly well with experimental data at short wavelengths, but as early
as 1897 experiments by Lummer and Pringsheim demonstrated that it was untenable for large
wavelengths [6]. It is here that Planck, now professor of theoretical physics at the University
of Kiel, focussed his attention on the problem. Planck had a lifelong love of entropy, declaring
early on that he “considered [it] the most important property of any physical system along with
energy” [6], and even completing his PhD thesis in the area.
3
The values of c1 and c2 were at the time of no concern to Wein nor Paschen, but following Planck’s work
were found to be c1 = 2h/c3 and c2 = h/k.
2
2.1
The Need for New Physics: 1899 - 1905
2.1.2
The University of Edinburgh
Planck’s Law
After several years of careful consideration, by 1900 Planck - through a careful examination of
the entropy of electromagnetic radiation trapped in a blackbody cavity, as well as the limiting
cases of Wien’s Law - was able to correct Wien’s equation to
I = c1 ⌫ 3
1
ec2 ⌫/T
1
1
which using the identity ec2 /T
⇡ e c2 /T for large ⌫ then approximates to Wien’s Law. This
1
equation was presented (albeit in a slightly di↵erent form) at a meeting of the German Physical
Society in Berlin on October 19th, 1900. Heinrich Rubens, present at the meeting, spent the
following night comparing his experimental results to Planck’s new law and “found completely
satisfactory agreement in all cases” [6]. This would be the first in a long line of verifications
proving Planck’s Law modelled blackbody radiation perfectly.
However, herein lay a huge problem for Planck’s new law - it was completely devoid of any theoretical justification. Seemingly derived from a modification of its own approximation, Planck
was well aware of necessity of an air-tight derivation of his equation, though never imagined
it would shake the foundations of physics as it was known. In search of a proof Planck soon
realised he would have to abandon the phenomenological thermodynamics he held so dear and
re-examine “the relationship between entropy and probability - the ideas of Boltzmann.” This
was a particularly big step for Planck who up until as recently as 1896 had not believed in an
atomistic view of the universe [6].
Planck started with Boltzmann’s definition of entropy,
S = k ln W
where W is the “corresponding calculated probability” or the weight function as we now know
it - the number of microstates and configurations a system can be in given some microscopic
parameters. In order to use the “distasteful” Boltzmann weight function, Planck recognised the
need to discretise the system and borrowed from Boltzmann’s 1877 paper in which he allows n
molecules to acquire the kinetic energy
0, ✏, 2✏, 3✏, ..., p✏
This was indeed an odd step as it was the first suggestion a continuous variable - kintetic energy
- can take on only discrete values. Planck imagined the walls of the blackbody contained N
distinguishable oscillators, which then shared among them P indistinguishable quanta of energy.
The number of possibilities for this becomes
W =
✓
N +P
P
1
◆
=
(N + P 1)!
(N 1)!P !
Planck then applied Stirling’s approximation N ! ⇡ (N/e)N to get
W =
The University of Edinburgh
(N + P )N +P
NNPP
3
2.2
Einstein Sheds Some Light
The University of Edinburgh
Recognising that given P energy quanta each of energy ✏ we can write the entropy in terms of
the total energy U = P ✏ by

S=k (
U
U
+ 1) ln( + 1)
✏
✏
U
U
ln( )
✏
✏
Planck compared this to the entropy he had calculated given his correction to Wien’s Law:

a0
U
U
S=
( 0 + 1) ln ( 0 + 1)
a a⌫
a⌫
U
U
ln 0
0
a⌫ a⌫
He set k = a0 /a and renamed a0 to h, calling it the elementary quantum of action since “it has
dimensions of a product of energy and time”. [6] Here, out of Planck’s “act of desperation” falls
the now famous equation
✏ = h⌫
relating the energy of the blackbody to the emitted radiation.
Upon presenting his proof to the German Physical Society in December 1900, Planck was praised
for his work and it quickly became widely accepted. Many considered the quantisation of energy
to be a mathematical trick, Planck himself regarding it as a “purely formal assumption” to which
he “really did not give much thought” [7]. There was even some suggestion he planned to take
the limit as n becomes continuous and the spacing between consecutive energy levels goes to
zero [6].
Planck was seemingly unaware of having stumbled upon arguably the most important discovery
in physics. Indeed, he was far more pleased in having discovered two new fundamental constants
- Boltzmann’s constant k and Planck’s constant h - telling his seven year old son he had “made
a discovery as important as that of Newton” [7]. Ironically, Planck spent the rest of his life
resisting quantum theory, trying to “at least make the influence of quantum theory as little as it
could possibly be”. However, despite being ignorant of its implications, Planck had unleashed
the juggernaut of physics that would go on to be developed and refined by hundreds of others
into the most successful scientific theory ever.
2.2
2.2.1
Einstein Sheds Some Light
Light Quanta
In 1905, Albert Einstein was employed in the Swiss patent office in Bern as technical expert,
third-class. Working 8 hours a day, 6 days a week left him “8 hours for fooling around, and
then there’s Sundays” [7]. It was in this free time that the 26 year old, whose name would
one day become synonymous with genius, prepared three papers to be published in Annalen
der Physik, each of which was revolutionary and would have catapulted him to fame in its
own right. The first was his theory of Brownian motion - the apparently random motion of
particles under a microscope - which he used as proof of the existence of atoms. In the second
of these, On the Electrodynamics of Moving Bodies, Einstein introduced special relativity and in
doing so the world’s most famous equation, E = mc2 . It was in his third paper, On a Heuristic
Viewpoint Concerning the Production and Transformation of Light, that Einstein revolutionised
our understanding of quantum theory and it was for this he would eventually win the Nobel
prize in physics.
The University of Edinburgh
4
2.2
Einstein Sheds Some Light
The University of Edinburgh
Einstein started his train of thought by pointing out a contradiction in Planck’s reasoning - if the
oscillators in the blackbody can only oscillate at discrete frequencies then how can the emitted
radiation have a continuous spectrum? While interference e↵ects can produce a larger range
of spectra than those of the oscillators, no e↵ect could produce a truly continuous distribution
of radiation. Einstein followed Planck’s approach of examining entropy, this time focussing on
the entropy of the radiation rather than the blackbody oscillators. From Planck’s formula he
derived an expression for the radiation entropy s by
s

s0 = k. ln (
v ⌘
) h⌫
v0
where v/v0 is the fractional volume and ⌘ is the energy radiant in volume v. Einstein compared
this to Boltzmann’s definition of the entropy, S = kW , and found that this required ⌘/h⌫
to be the number of “energy quanta”. From this he concluded “monochromatic radiation of
low density behaves with respect to thermal phenomena as if it were composed of independent
energy quanta” [6].
Einstein had introduced a stunning proposition - light behaves as a discrete particle, rather than
a wave. Though early ideas of light were atomistic - that it was composed of many di↵erent
particles - in 1821 Augustin-Jean Fresnel proved that polarisation of light could be explained
only if it was a wave. Almost 90 years later Einstein required light must be a particle, perhaps
the first time a truly contradictory phenomena of Quantum Mechanics was discovered. To this
day it still remains somewhat of a mystery.
However, very little thought was given to the interpretation of these ideas, and even to Einstein’s
work itself. Of his three published papers, his third on the light-quantum hypothesis found the
least support. In particular Planck, who was a great believer in Einstein’s special relativity,
found the extension of the quantum to radiation highly unsatisfactory. Planck believed that light
was only quantised at points of emission and absorption and could fill the continuous spectrum
of wavelengths as it propagated in between, obeying Maxwell’s equations [7] [6].
2.2.2
The Photoelectric E↵ect
In order to promote his theory further, Einstein turned to another of classical physics’ unsolved
problems - the photoelectric e↵ect. This was first observed by chance in 1887 by Heinrich Hertz,
who noticed the spark between two charged spheres became brighter when one was illuminated
by ultraviolet light. Further observations showed that an electron will be ejected from a metal
if irradiated by light above a certain frequency. Increasing the intensity of the light will only
increase the number of electrons ejected if the light is above the given frequency, with no ejection
occurring below this no matter what intensity. This is completely at odds with classical physics
where the energy of any wave - in this case light - is given by intensity only.
Using Planck’s formula ✏ = h⌫ applied to light also, Einstein could successfully explain the
photoelectric e↵ect, and derived a formula for the maximum velocity of the electron:
mv 2
= h⌫
2
where
is the work function, a constant unique to every material which is the minimum
energy required by the incident photon to eject an electron. In this single, simple equation
Einstein showed that the energy of the ejected electrons is both independent of intensity, and
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5
2.3
Quantum in the Atom
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for frequency ⌫ < /h no electrons will be emitted, perfectly explaining what had puzzled
physicists for the previous 18 years.
By the late 1910s Einstein’s light quantisation was increasingly accepted by the scientific community, culminating in Einstein being awarded the 1921 Nobel prize for his
theory explaining the photoelectric e↵ect [8].
However, having brought early quantum theory into the mainstream, Einstein turned his
attention to his theory of general relativity,
Sommerfield remarking that Einstein was “so
deeply involved in the problems of gravitation
that he turns a deaf ear to all else.” [7] It
would be another 20 years before Einstein returned to the quantum stage, so desperate to
disprove the theory he had once helped cre- Figure 2.1: Ejection of electrons from a metal
ate.
due to incident ultraviolet radiation.
2.3
Quantum in the Atom
Following his famous 1911 discovery of the
atomic nucleus in the Geiger-Marsden experiment [9, pp. 68-69], Ernest Rutherford went on
to describe the structure of the atom. Inspired by the classical success of Newton, Rutherford
proposed a planetary model in which negatively charged electrons orbit a positively charged
nucleus. However, this classical model had two important shortcomings [10, pp. 30-31]. Firstly,
in Newton’s planetary model, planets exhibit stable orbits. Orbiting electrons on the other hand
are accelerating and thus should radiate energy in accordance with Maxwell’s electromagnetic
theory. As electrons radiate energy their orbits would decrease continuously and ultimately lead
to the electrons collapsing into the nucleus. Such a collapse would have a timescale of around
10 8 s. This obviously isn’t the case as we know atoms are stable. Secondly, the frequency of the
radiation matches the orbiting frequency, so as the electron collapses into the nucleus the orbiting frequency and thus the radiation frequency increase continuously. However, experiments
show that radiation spectra for atoms are discrete.
2.3.1
The Bohr Model
In 1912 Niels Bohr started work on atomic models but took a di↵erent approach to that of
Rutherford. Bohr neglected the classical prediction of electron collapse as he did not believe
atoms could be explained by classical physics [9, p. 109]. Instead he based his model partially
on Rutherford’s model, partially on Planck’s quantisation principle, and partially on Einstein’s
photon concept. As in Rutherford’s model the electron orbits the nucleus due to electrostatic
interaction, but for simplicity Bohr only dealt with circular orbits.
He proposed the following [10, p. 31]:
• Only a discrete set of stable orbits are allowed. Such an orbit is called a stationary state.
This leads to quantisation of energy, i.e. an atom can only have certain energies E1 , E2 ,
E3 , etc.
• A stable orbit is one in which the orbital angular momentum of the electron is an integer
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2.3
Quantum in the Atom
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multiple of the reduced Planck constant:
L = n~
(1)
This expression is referred to as Bohr’s quantisation rule of angular momentum.
• While in a stationary state the electron radiates no energy. Instead, energy is only radiated
when the electron moves from one stationary orbit to another. If the electron jumps from
an orbit of energy En to an orbit of energy Em , then the electromagnetic radiation is
carried away by a photon of energy
h⌫ = En
Em
(2)
This implies that an electron may not only radiate energy by moving to a lower orbit, but
it may also absorb energy by jumping to a higher one.
The Bohr model followed nicely from Planck’s earlier work as it embodied both the quantisation
principle as well as making use of Planck’s fundamental constant. Furthermore, the third
postulate was a direct application of Planck’s relation for the energy of a photon of frequency
⌫.
Experimental Evidence
One of the most important features of the Bohr model is its agreement with experimental results.
In particular, it brilliantly describes the hydrogen atom. Here the electrostatic force is that of
the Coulomb force, which (for a circular orbit) can be equated to the centripetal force:
e2
v2
=
m
e
4⇡✏0 r2
r
(3)
where e is the charge of the electron, ✏0 is the electric permittivity, me is the mass of the
electron, and v and r are the speed and radius of the orbiting electron respectively.
Then using equation (1) in the form L = me vr = n~ leads to the following expressions for the
radius and velocity of the electron:
rn =
4⇡✏0 ~2 2
n ⌘ n2 a 0
me e2
n~
vn =
=
me rn
✓
e2
4⇡✏0
◆
1
n~
(4)
(5)
where a0 is the Bohr radius given by
a0 =
4⇡✏0 ~2
= 0.053nm
me e2
(6)
For the ground state case (n = 1) this leads to numerical values of about r = 5 ⇥ 10 11 m and
v = 2.2 ⇥ 106 ms 1 . This gives a diameter of about 10 10 m which is in agreement with atomic
diameters measured from X-ray crystallography [9, p. 110]. For the velocity of the electron
the model predicts speeds much less than that of light.
The model also allows us to derive an expression for the energy of the electron as a function of
n. If we assume that the nucleus (i.e. the proton for the hydrogen atom) is infinitely heavy in
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2.3
Quantum in the Atom
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comparison to the electron then the electron orbits a stationary proton. In this case the energy
is simply the kinetic energy of the electron minus the electrostatic energy:
1
e2
En = me vn2
2
4⇡✏0 rn
where the electrostatic energy is negative for a bound state.
(7)
Substituting in the above expressions for rn and vn and simplifying leads to
En =
e2 1
=
8⇡✏0 rn
me
2~2
✓
e2
4⇡✏0
◆2
1
⌘
n2
R
n2
(8)
where we R is known as the Rydberg constant:
me
R= 2
2~
✓
e2
4⇡✏0
◆2
= 13.6 eV
(9)
Equation (8) is called the Bohr energy for hydrogen. Taking n = 1 corresponds to the ground
state energy of hydrogen, n = 2 is the first excited state, and so on. This result was a huge
achievement - not only did it correspond well with spectroscopic results for the transition lines of
hydrogen, but it was also the first theoretical basis for the empirical Rydberg formula (See [11, p.
70]). The energy levels and transition lines for atomic hydrogen can be see in Figure 2.1.
Figure 2.2: Hydrogen energy levels and transition lines. The quantum number n denotes the
excitation of the electron. Bohr’s model neatly predicted the energy levels for the bound states
and provided theoretical background for the Rydberg formula [10, p. 33].
Shortcomings of the Bohr Model Bohr’s model should work for any single-electron atom
such as He+ and Li2+ so long as one takes into account the increased number of protons in the
nucleus. However, Alfred Fowler proclaimed the theoretical results were not accurate enough for
the case of the spectral lines of He+ [9, pp. 112-113]. Bohr then reviewed his arguments, noting
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2.4
Correspondence Principle
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that in the derivation given above, it has been assumed that the nucleus is infinitely heavy
and hence at rest. Bohr then took into account that both electron and nucleus rotate about a
stationary centre very close to the nucleus. This final correction lead Einstein to conclude that
‘This is an enormous achievement. The theory of Bohr must be correct’ [9, p. 113].
Although Bohr’s model beautifully unified theory and experiment for one-electron atoms, it
failed to describe more complex configurations. It also lacked justification for its basic assumptions: where did the quantisation rule come from and why didn’t a stationary state emit
radiation?
2.3.2
Einstein Connects the Dots
Having been preoccupied with relativity for several years, Einstein briefly returned to the quantum stage. In 1916-1917 he wrote three papers investigating transitions between stationary
states. Using models of matter in interaction with radiation he invoked the Maxwell-Boltzmann
distribution and Bohr’s expression for the frequency of the emitted photon (eq (2)), and managed to recover Planck’s formula [11, pp. 80-82] [9, pp. 118-119].
Einstein’s derivation 4 has three interesting aspects. Firstly, it produces a link between blackbody radiation and Bohr’s atomic model. Secondly, Bohr’s model assumes transitions happen
at random5 , which means Einstein is abandoning determinism. Thirdly, Einstein’s derivation
reflects that photons possess both momentum and energy. Classically, the momentum of the
photon is indeterminable since it has mass zero. As we shall see soon, de Broglie later managed
to describe the relation between a photons momentum and its energy quite successfully.
2.4
Correspondence Principle
The correspondence principle was a principle developed by Niels Bohr, which turned out to
become an important link between old quantum theory and the later new quantum theory.
Although Bohr had occasionally used this principle since his 1913 model of hydrogen, it wasn’t
methodically used until 1918 and not formally developed or named until 1920 [12] [9, pp. 119122]. In his writings, Bohr was generally vague and never gave any exact definition of the
correspondence principle.
Essentially the correspondence principle may be said to state that quantum theory must at all
times converge towards a classical description when approaching the classical limit. An example
of the correspondence principle is that of the hydrogen atom. Recall from figure 2.1 that in
the classical (large n) limit the energy levels gets so close to each other that they resemble
a continuous spectrum. This is simply the classical picture just as the principle demands.
However, the correspondence principle contains a few more subtleties that are not particularly
important. An in-depth discussion of the correspondence principle can be found in Appendix
C.
However, the correspondence principle wasn’t accepted straight away. It was a polarising subject
within the contemporary physics community and many had an opinion on it. These were also
strongly connected to Bohr’s atomic model. Bohr’s model dealt with circular electron orbits
and only predicted the need of a single (principal) quantum number, n. Then in 1916, Arnold
Sommerfeld went on to extend Bohr’s model by considering elliptical orbits. In doing so he
4
For a complete description of the derivation, see Appendix B.
Random in the sense that the model doesn’t explain the mechanism behind the electron jumping from one
stationary state to another.
5
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2.5
Wave-Particle Duality
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managed to predict the need for an additional two quantum numbers: the inner (or magnetic)
quantum number, m` (1916), and the azimuthal (or orbital) quantum number, ` (1919).
In 1920, Bohr applied his correspondence principle and succeeded in predicting exactly same
quantum numbers by considering the limit where quantum physics meets classical physics.
Although the correspondence principle had proved itself useful, there were some very mixed
opinions on it. In particular Sommerfeld, who had just witnessed his rigorously derived results
being conjured out of thin air, was sceptical: “Bohr has discovered in his principle of correspondence a magic wand (which he himself calls a formal principle), which allows us immediately to
make use of the results of the classical wave theory in the quantum theory.” [12]. Other sceptics included Wolfgang Pauli and Werner Heisenberg, who were at that time students under
Sommerfeld.
When considering why the correspondence principle received a hostile reception it is worth noting that Bohr presented it in a confusing manner. As mentioned earlier, Bohr used the principle
on and o↵ and never defined it exactly. Even today there is no clear definition of the principle, nor do we know if Bohr preferred either of the various interpretations of it (see Appendix
C). In fact, a handful of people, including Max Born and Léon Rosenfeld, misunderstood it
altogether [12].
Despite the criticism, Bohr firmly believed in the principle. The idea that quantum theory must
correspond to classical results in the large n limit went on to play an important role in his 1927
Copenhagen interpretation, as we shall see later.
2.5
Wave-Particle Duality
At the beginning of the 1920s a large number of experimental and theoretical results related to
quantum mechanics were published. These developments played an important role in leading
towards a more advanced description of quantum mechanics. The first important experimental
result was that of the Compton e↵ect.
2.5.1
The Compton E↵ect
In 1923 Arthur Compton conducted an X-ray scattering experiment which lead to an astonishing
result. In his experiment Compton fired a beam of monochromatic X-rays towards a metal
foil target and observed the intensity of the scattered radiation. In doing so he detected two
di↵erent intensity peaks: One beam had scattered and emerged with the same wavelength as the
incident radiation, while a second beam had emerged with an increased wavelength. To explain
this phenomenon, Compton considered individual X-ray photons as particles colliding with the
outermost electrons of the metal foil target. In such a collision the photon would lose some
energy to the electron. By applying Planck’s relation E = h⌫, and using the expression for the
photon frequency = ⌫c , he correctly obtained an increase in wavelength following a decrease
in energy. Utilising this principle, Compton found a strong correspondence between his results
and the proposed theory. By clearly showing a relation between electromagnetic radiation and
particle-like behaviour, the Compton e↵ect played an important role in the overall acceptance
of what was to be known as wave-particle duality.
2.5.2
De Broglie’s Matter Waves
In 1923-24 Louis de Broglie sought to reconcile the nature of light quanta with that of matter
particles and produced three papers on the subject. However, initially none of them were taken
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2.5
Wave-Particle Duality
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into serious consideration [11, pp. 86-87]. In his 1924 doctoral thesis de Broglie discussed the
matter once again, this time catching the eye of Einstein, who was fond of his work [9, p.
125]. The dissertation considered Einstein’s equations for photon energy and momentum and
proposed that these could be extended to hold for both radiation and matter. In this context,
these equations are known as de Broglie relations:
⌫=
E
,
h
=
h
p
(10)
De Broglie explained these equations by appealing to wave-particle duality, namely that the
motion of a particle can be deduced from an associated matter wave. In this view, the first
relation can be understood as relating total energy E of both matter and radiation to the
frequency ⌫ of the associated wave. Similarly, the second relation can be regarded as giving a
relationship between the momentum p and the wavelength of the associated wave. Of special
importance is the wavelength, the so-called de Broglige wavelength. The de Broglie wavelength
can be regarded as a measure of whether the dynamics of a particle is governed by classical- or
quantum mechanics. If the de Broglie wavelength is much less than the size of the characteristic
dimension of the measuring apparatus, then particle mechanics must be applied. Conversely, if
the de Broglie wavelength is of comparable or larger size than the characteristic dimension, then
wave mechanics should be used. This insures that the classical limit is reached when dealing
with objects of everyday size.
For completeness it should be mentioned that de Broglie didn’t just state these relations without
further considerations. For instance, by considering a harmonic wave associated with a particle
of speed v, he managed to show that if the group velocity of the associated wave matches the
velocity of the particle then there must be dispersion. He also successfully showed that the first
relation implies the second. For more details see e.g. [11, pp. 86-93].
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3
Quantum Mechanics in its Infancy
3.1
Birth of the Formalism
3.1.1
Matrix Mechanics
Given that no real unified structure existed for quantum mechanics before 1925 it is difficult to
speak of an explicit interpretation going into this period. The earliest completely self-consistent
formalism was developed by a young Werner Heisenberg while recovering from severe a bout of
hay fever on the island of Heligoland. It was there in an attempt to solve the anharmonic oscillator that he conceived the idea of representing physical quantities by sets of time-independent
complex numbers. Soon after this discovery Max Born soon recognized the “sets that Heisenberg
had used to solve the atomic system were precisely those mathematical entities whose algebraic
properties had been studied by mathematicians ever since Cayley published his memoir on the
theory of matrices. Very quickly Heisenberg’s approach was developed by Born, Jordan and
himself into what is currently known as matrix mechanics, the earliest consistent formalism of
quantum phenomena.
This approach to physics was unfamiliar to the older and more established physicists of the day.
It o↵ered nothing from the point of view of building an intuitive picture of atomic processes.
The theory was physics pared down to its most abstract, a purely mathematical tool. In terms of
the interpretations of quantum mechanics, this formalism added little in its own right. However
it provided a basis for many of the most pervasive and important interpretational schemes that
make up what is now known as the Copenhagen interpretation. At the point of its development
even its creators were yet to develop the conceptual tools with which to build a fully consistent
understanding of their new theory.
3.1.2
The Guiding Principle in the New Quantum Mechanics
Heisenberg’s motivation in producing such an abstract formalism could be said to be a product of
the ‘Copenhagen atmosphere’, namely the view that the classical concepts that had dominated
physics may be insufficient in describing microscopic phenomena. Niels Bohr concluded:
“We must in fact even be prepared to find behaviour that is alien to the application of
ordinary space-time pictures.”
Heisenberg’s formalism had neatly sidestepped the interpretational difficulties that arose from
the apparent discontinuities of atomic emission phenomena by making no requirement on visualisability. However the general scientific philosophy at the time was firmly classical and this
theory proved hard to grasp for many within the scientific community of the time. Heisenberg’s
guiding principle to the development of his theory was summed up by Max Born three decades
after matrix mechanics had been fully realised.
“The principle states that concepts and representations that do not correspond to physically
observable facts are not to be used in theoretical description.
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3.1
Birth of the Formalism
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This guiding philosophical principle stated in his remark was one that pervaded in positivistic
circles. Positivism was a doctrine contending that sense perceptions are the only admissible
basis of human knowledge and precise thought. It was on this basis that Heisenberg banished
ideas of definite electron orbit radii and periods of rotation simply because these quantities were
not observable.
It is interesting to note at this stage that Einstein - one of the main opponents to the new
theory - used such a principle when eliminating the concepts of absolute velocity and of absolute
simultaneity of events at di↵erent places when forming his ideas of relativity. However, he was
unsettled with the adopted Copenhagen interpretation that had such principles at the heart
of its formation. In an exchange between Einstein and Heisenberg following a presentation of
Heisenberg’s matrix mechanics to the Berlin colloquium in April 1926, Einstein noted:
“What you have told us is extremely strange. You assume the existence of electrons within
the atom, and you are probably quite right to do so. But you refuse to consider their orbits,
even though we can observe electron tracks within the cloud chamber. I should very much like
to hear your reasons for doing so.”
The positivistic approach was instrumental in the development and interpretation of matrix
mechanics. This principle however did not have deep roots in Copenhagen6 , where the concepts
of a semi-classical atom appealed to visualisation rather than to purely experimental observables.
This changed with the inception of matrix mechanics and the development of a new Copenhagen
interpretation.
3.1.3
Wave Mechanics
It is clear that the ground was fertile for a more intuitive quantum theory, one that made use
of classical concepts and if at all possible restored visualisability to quantum mechanics.
In late 1926 while on a retreat in the Swiss Alps Erwin Schrödinger developed a wave equation
(Equation 11) from ideas about de Broglie’s matter waves. This equation was the fundamental
postulate for Schrödinger’s brainchild, the new quantum theory known as wave mechanics. At
the end of January 1926, Schrödinger went on to produce arguably his greatest contribution
to quantum mechanics Quantisation as an Eigenvalue Problem [14], in which he detailed the
quantisation of observables in quantum mechanics as a natural consequence of single valuedness
and finiteness a in the solutions to his equation:
@
i~
@t
=
✓
~2 2
r + V (r)
2m
◆
(11)
Schrödinger then went on to show in a further work Über das Verhältnis der Heisenberg-BornJordanschen Quantenmechanik zu der meinen [15] (On the relationship of Heisenberg-BornJordan Quantum Mechanics to Mine) the direct mathematical equivalence of his own wave
mechanics to the matrix mechanics of Heisenberg7 .
6
This philosophical approach is not solely attributed to Heisenberg [13]. The origin of such a“positivistic
approach to science and specifically to the new quantum theory can be argued to have originated with Ernst
Mach. Mach claimed, before Einstein, that Newton and his concepts of ‘absolute time and ‘uniform motion
without reference to any external object had violated his own principle in natural philosophy of sticking to
only observable facts. However, Heisenberg being heavily influenced by Pauli who was the grandson of Ernst
Mach sheds light on the reason for Heisenberg’s philosophical position in regards to employing only observable
quantities in the theoretical description of physical phenomena.
7
See appendix D
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3.2
Early Interpretations of the New Quantum Mechanics
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Schrödinger’s formalism was largely based on semi-classical ideas and in creating it he had
sought to provide a space-time picture for quantum mechanics. The success of wave mechanics
was realised in explaining the Zeeman and Stark patterns in the hydrogen atom and even the
Compton e↵ect [16], the paradigm of particle physics. As a result of this approach, wave
mechanics gained much support within the scientific community and due to its visualisability
as a classical theory of waves it became a rich source of many interpretational schemes8 .
It is worth noting that quantum mechanics was an atomic theory created specifically to deal
with the properties and behaviour of electrons in atoms which has since been applied a wide
range of microscopic systems. Therefore almost all of the early interpretations concerned only
electrons.
3.2
3.2.1
Early Interpretations of the New Quantum Mechanics
Schrödinger’s Electromagnetic Interpretation
Up to the third part of Schrödinger’s communications of Quantisierung als Eigenwertproblem
the wave function had not yet been assigned any physical meaning and was defined only as
a construct of the formalism. Schrödinger theorised that due to the nature of the emissions of
electrons in atomic systems that the wave function must have some electromagnetic meaning.
The result of these musings was a postulate [15] that the electronic charge distribution is given
as follows.
⇢(r) = Re
✓
@ ⇤
@t
◆
(12)
Schrödinger’s e↵orts to connect the new quantum theory to the classical theory were justified
as his aims to interpret the new quantum theory “would have been reached at once by showing that the formalism F if Schrödinger’s wave mechanics could be regarded as being part of,
or at least isomorphic with, the formalism F * of another theory T * which was fully interpreted” [18], in this case classical electrodynamics. By this rationale interpreting the wave
function as having this electromagnetic meaning, Schrödinger could go on to claim that “the intensity and polarisation have now been made completely understandable on the basis of classical
electrodynamics.”.
Despite this apparent assuredness, Schrödinger had not yet assigned any direct physical meaning
to the wave function itself. It was later discovered by Schrödinger himself in his fourth communication Quantisierung als Eigenwertproblem that his postulated expression for the charge
distribution had the peculiar and physically inconsistent property of integrating to zero over
all space. A property that caused Schrödinger to replace the real part of @@t⇤ with the more
familiar | |2 .
It is not entirely clear from the specific interpretations of the individual mathematical constructs
of the formalism what kind of picture of the microscopic world Schrödinger and his adherents
8
Indi↵erent to the attempts of the Copenhagen physicists Schrödinger categorically denied the existence of
discrete energy levels and discontinuities between them. In fact Schrödinger rejected the concept of energy in
microphysics. In a letter written to Max Planck on May 31st, 1926, Schrödinger explained his rejection, “The
concept ‘energy is something that we have derived from macroscopic experience and really only from macroscopic
experience. I do not believe that it can be taken over into micro-mechanics The energetic property of the
individual partial oscillation is its frequency. [17] Schrödinger never changed his view on this point. Three years
before his death, January 4th, 1961, he wrote a paper “Might perhaps Energy be a merely statistical concept.
This rejection allowed him to bi-pass the quantisation of energy and discontinuous changes between states, this
rejection was to cause much debate, between the Copenhagen physicists.
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Early Interpretations of the New Quantum Mechanics
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had. His picture was that of a world composed entirely of classical waves, matter and radiation
described by purely ‘undulatory’ physics. Schrödinger also dismissed the discontinuities or
‘quantum jumps’ of Heisenberg’s matrix mechanics, describing these phenomena as resonances
of waves or on a more formal level the eigenvalues of the wave equation given by the strict
boundary conditions. As to the observed particulate nature of electrons, Schrödinger proposed
that this could be explained as “wave packets” or mobile areas of high charge distribution
composed of many modes of oscillation. This interpretation was not without issues however.
A letter from Hendrik Lorentz to Schrödinger in May 1926 pointed out that the wave packets
would rapidly disperse.
The core concepts of the electromagnetic and undulatory interpretation were solidified and
presented to the scientific community in Schrödinger’s 1927 paper Energieaustausch nach der
Wellenmechanik [19]. The early success of Schrödinger’s ideas about the physical significance
and interpretation of wave mechanics could be attributed to a hangover from what was, according to Heisenberg, the dogmatic realism that had dominated the microscopic physics of the 19th
Century [20]. This approach to the interpretation of quantum mechanics fitted so well into the
philosophical status-quo that it was for a while the dominant picture of the microscopic world
among the most well established figures in physics.
Despite the apparent success of this interpretation, it was not without significant issues. The
realism of the wave function - such an integral part of Schrödinger’s scheme - is inconsistent
as it is a complex function and can change discontinuously in time. Perhaps worst of all, as
soon as one extends the theory to describe n electron systems, the wave function requires 3n
co-ordinates to be fully specified.
3.2.2
Hydrodynamic Interpretations
It had not gone unnoticed by the scientific community that Schrödinger’s wave equation for
a free particle bears a striking resemblance to the classical di↵usion equation. Expanding on
these ideas Erwin Madelung used Schrödinger’s equation and the anzatz = ↵ei to derive the
following from the imaginary part of the resultant equation [21].
r·
✓
~ 2
↵ r
m
◆
=
@↵2
@t
(13)
This equation takes the form of a hydrodynamical continuity where ↵2 is interpreted as the
~
density and r m
takes the place of the velocity field. This eqution in conjunction with the
real part of the aforementioned version of the Schrödinger equation make up a scheme in which
quantum systems can be analysed as the behaviour of incompressible fluids. Another proponent
of this interpretation, Arthur Korn, independently developed a similar scheme for the analysis
of quantum mechanics as the motion of a fluid.
Both Madelung and Korn’s schemes, while appealing to those who would try to apply a classical
rationale to quantum mechanics, had some rather significant issues. One example is the fact
that the theory Madelung proposed could not account for the process of spontaneous absorption
of a photon. It is inconsistencies like these are the reason such hydrodynamic schemes have not
contributed to the major modern interpretations of quantum mechanics.
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3.2
Early Interpretations of the New Quantum Mechanics
3.2.3
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De Broglie’s Double Solution Interpretation
Another scheme that was derived from the Schrödinger and Klein-Gordon wave equations was
de Broglie’s double solution interpretation. This scheme was the culmination of several years
of research for de Broglie and exhibited very clearly the classical approach of visualisiblity
in physics. The scheme reached full maturity in a paper published in 1927, La mecanique
ondulatoire et la structure atomique de la matiere et du rayonnement [22]. In this paper de
Broglie detailed a scheme in which the single particle wave function is recast as a product of the
familiar undulatory wave function and an amplitude envelope with a mobile singularity:
= f (r, t) exp (i!(t
(r)))
(14)
where f is the amplitude envelope and is the spatially dependant phase. From this de Broglie
developed a consistent mathematical scheme in which the singularity represents the location of
the particle, while the particle is guided by the wave field which extends over all space. From
this de Broglie was able to deduce the guidance formula:
v=
r
(r)
m
(15)
Not only did de Broglie’s solution preserve the visualisability of the processes, but it appeared
to imply a feature that all other interpretations of the wave mechanical formalism had lacked: a
trajectory. Here the guiding undulatory field was interpreted as having statistical significance, in
agreement with Born’s statistical interpretation, while the singularity was interpreted as being
the position of the particle itself. The appeal of de Broglie’s interpretation to the scientific
thinkers of the time was that the classical trajectory was maintained and the observed wavelike properties were explained elegantly.
When de Broglie attempted to apply his regime to the analysis of atomic systems he discovered
an issue within the theory, i.e. v = 0 independent of position and time. This implies that the
particle is always stationary with respect to the nucleus, a result that was not at all consistent
with observation. De Broglie’s own suggestion to fix this apparent issue was to introduce an
e↵ect analogous to Brownian motion. In his own words:
“A random element of hidden origin has to be admitted. This implies that the particle’s
regular motion, governed by the guidance law, is continuously submitted to random
perturbations, with the result that the particle all the time switches from one guided
trajectory to another. ”
De Broglie went on to suggest that the nature of the origin of this perturbation was that
the particle is in “energetic contact” with a “sub-quantum medium”. If this caveat could
be accepted, de Broglie had taken the wave-particle duality and recast it as a single unified
phenomenon.
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4
The Development of the Copenhagen Interpretation
4.1
The Conceptual situation
By Spring 1926 the two formalisms - matrix mechanics and wave mechanics - had been reduced
to a common underlying mathematical structure, but still lacked a unified interpretation. The
debate on this physical interpretation was opened in the summer of 1926 and would continue
vigorously for more than a year.
4.1.1
Born’s Probabilistic Interpretation
At this point in time, little consensus had been reached upon the interpretation of quantum
mechanics. One particular object of contention was the enigmatic wave function. Schrödinger’s
interpretation cast | |2 as the charge distribution of an electron. However, this scheme had
many issues, one of which was the explanation of the electron’s observed corpuscular behaviour.
Schrödinger was not the only one having difficulty with his new formalism; in Copenhagen,
while Heisenberg had refused to make use of wave mechanics, Bohr was wrestling with the idea
of this mysterious wave that represented a particle. It would take the radical step of a departure
from one of the most dearly held notions of physics, determinism, to help to initiate the slow
march toward a consistent interpretation of quantum mechanics.
In his 1926 paper On the Quantum Mechanics of Collision Processes Max Born proposed a new
and radical interpretation of the wave function . In his paper Born detailed his analysis of the
predictions of quantum mechanics when applied to an electron scattered from a fixed atomic
target. Having had no success using matrix mechanics to determine the asymptotic behaviour
of the incident and scattered electron he turned to Schrödinger’s wave mechanics for the answer.
Wave mechanics was successful in the analysis of the problem and as a result Born stated in his
paper “I am inclined to regard it as the most profound formulation of the quantum laws” [23].
The result for the scattered wave function was given by:
(1)
nE (r)
=
XZ
m
d!
a·r>0
(E)
(E)
nm (a) sin Knm (a
· r)
0
n
(16)
where d! is the element of solid angle in the direction of the unit vector a. From the standpoint of
Schrödinger’s wave mechanics, the interpretation is handled easily. The electron is an undulatory
phenomenon, so to represent its final state as a wave function over all space fits well within this
view of the microscopic world. Indeed the wave function takes the form of a superposition of
(E)
plane wave modes whose spectrum is given by nm (a). The interpretational difficulties arise
when one considers, as Born did, the corpuscular picture. From this point of view the question
of the direction of motion of a given scattered particle finds no answer. Born concluded that
(E)
the only sensible interpretation was that | nm (a)|2 represents a probability distribution that a
particle will be scattered in a direction parallel to a. This is the origin of the modern standard
for the interpretation of the wave function. For the first time in physics a theory had been
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The Conceptual situation
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developed that was not deterministic. The universe, according to Born, was not the clockwork
machine of classical Newtonian physics:
“ I myself am prepared to give up determinism in the world of atoms.”
This idea was by no means original in its content, merely its application. Born himself would
later admit that he had applied ideas originating from Einstein’s earlier work on a guiding field
for light quanta [24]. While this interpretation was extremely important to the development of
the Copenhagen interpretation, it was still markedly di↵erent in certain important details. In
this version of the statistical interpretation of the wave function is not regarded as something
physically real, merely a state of knowledge of the particle or particles in question. A particle
which in Born’s original interpretation was a particle in the classical sense: a point mass with
a simultaneously well defined position and momentum.
Given that the interpretation was originally applied to scattering experiments it is fitting that
this is where it found most success. Perhaps most notable of the achievements of this interpretation were Born’s approximation and its use by Wentzel to reproduce Rutherford’s experimentally
confirmed scattering formula [25]. Taking this new view of the wave function avoided the difficulties faced by Schrödinger’s interpretation. No issues arose when considering the parameter
space upon which the wave function depends: the rapid dispersal of wave-packets is merely a
change in our state of knowledge and the complex-valuedness of the wave function is is no longer
an issue if it is not considered as something physically real.
However, success was not universal for this interpretation, due in part to the fundamentally
classical view of the nature of a particle. One particular stumbling block was the double slit
experiment. In the limit of a luminosity low enough that only a single particle passes through
the apparatus at a time, the interpretation broke down. If the particles are viewed as the
classical point masses mentioned above, then the self interference of the probability distribution
does not make sense with Born’s picture of the wave function; how could a classical particle
interfere with itself? Born’s interpretation essentially did away with the idea of the wave-like
behaviour of matter. Despite these difficulties the core concept of the statistical interpretation
of the wave function were sufficient to warrant a nobel prize for Born in 1954.
4.1.2
Schrödinger’s Visit to Copenhagen
In July 1926 Schrödinger was invited to Munich by Wilhelm Wien to report on his theory9 .
After receiving a disheartening letter from Heisenberg explaining his failed encounter with
Schrödinger, Bohr invited Schrödinger to come to Copenhagen and deliver a lecture to the
Danish Physical Society on wave mechanics. Schrödinger accepted his o↵er and announced his
arrival on October 1st. The stage was set for the first round of the debate that would shape the
future development of quantum theory and in particular the development of the Copenhagen
Interpretation.
Schrödinger was eagerly received, as Heisenberg recalled: “Bohr’s discussions with Schrödinger
began at the railway station and were continued daily from early morning until late at night.
Schrödinger stayed at Bohr’s house so that nothing would interrupt the conversations.” [27,
p. 73] No current notes on the content of that Copenhagen dialogue exist, though Heisenberg
recalled Schrödinger attacking the assumption of essential discontinuities. He was dissatisfied
9
The experimental physicists in Munich headed by Wien, were enthusiastic about the possibility that now
perhaps this whole “quantum mystery of atomic physics” might be dealt with, and one would be able to return
to the classical concept of honest fields, such as one had learned from Maxwell’s [electromagnetic theory] [26].
Heisenberg’s presence at Schrödinger’s presentation raised concerns over his adopted interpretation. He claimed
inconsistencies in Schrödinger’s theory such as the inability to explain Plank’s radiation law. However ,his
objections were not welcomed by the other physicists present10
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with the lack of explanation provided in the description of “discontinuous jumps” from one
stationary state to the next exclaiming “Is this jump supposed to be gradual or sudden?” and
“what laws govern its motion during the jump”.
Bohr acknowledged Schrödinger’s objections but explained that it “does not prove there are
no quantum jumps. It only proves that we cannot imagine,” appealing to the positivistic
approach of sticking merely to physical observables. Bohr made his position clear stating that
“the representational concepts with which we describe events in daily life and experiments
in classical physics are inadequate when it comes to describing quantum jumps. Nor should
we be surprised to find it so, seeing that the processes involved are not the objects of direct
experience.”
Indi↵erent to Bohr’s view, Schrödinger made his position clear “the comprehensibility of the
external processes in nature is an axiom” - the facts of experience can not contradict each other.
He rejected the “premature” abandonment of the “General conceptions of space and time and
the connection of the interaction of neighbouring space- time points” a focus that had been so
prominent in the formation of general relativity. However, the Copenhagen physicists argued
that from the alleged non visualizability of the atomic phenomena “all the apparently perceptible
pictures should be taken in reality only symbolically.” [28] Schrödinger deeply disagreed with
these ideas and their resulting consequences, such as that the amplitudes of eigenvibrations
might only be interpreted as statistical statements about many atoms, but not as statements
about individual atoms. A proposition recently taken by Max Born in his papers on collision
processes mentioned in the previous section.
These discussions11 would inevitably not be conclusive as neither side was able to provide
a coherent interpretation of their equivalent, yet underdeveloped formalisms. By the end of
Schrödinger’s visit the Copenhagen physicists felt as if they were on the right track even though
they recognised how difficult it would be to convince even leading physicists that they must
abandon intuitive models of atomic processes. The Bohr-Schrödinger debate stimulated energetic discussions which endured long after Schrödinger left Copenhagen. Later, to Heisenberg’s
distaste, Bohr considered wave mechanics “so wonderfully suited to bring out the true correspondence between the quantum theory and the classical ideas.” [29] This particular correspondence
was explored by Bohr and his collaborators in the months preceding Schrödinger’s visit and
greatly aided the formation of the Copenhagen interpretation of quantum mechanics.
4.2
The Indeterminacy Relations
The indeterminacy relations were a huge step towards the self-consistency of the Copenhagen
interpretation. They contained the key to reconciling the extremely abstract nature of the
formalism with observed phenomena. Along with Bohr’s ideas of Complimentarity, their discovery meant that classical notions of position, velocity and energy had to be revised for their
application to the microscopic scale.
4.2.1
The Conceptual Context
Schrödinger’s visit to the Copenhagen institute had been the catalyst for a push by Bohr
and Heisenberg to resolve the difficulties in interpreting quantum mechanics. The preferred
topic of discussion between the two was those phenomena which were the distillation of the
inconsistencies that led many away from the hard-line empiricist school of thought that had
underpinned the development of matrix mechanics. One such phenomenon was the observed
11
See appendix E
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trajectory of the electron in cloud chambers. For a theory that did not even contain a notion
of a trajectory this problem would seem insurmountable.
Concurrent with these events was a close correspondence between Heisenberg and Pauli on the
analysis of collision phenomena and excitations of a quantum rotator. In a letter to Heisenberg
dated 19 October 1926, Pauli detailed his preliminary work with the analysis of these particular
systems. In the second section of the letter in which the quantum rotator was looked at in
detail Pauli noted what is a very typical manifestation of the uncertainty principle. When he
looked at the e↵ect of the perturbation upon the canonical momenta of the system {p}, the
e↵ect of the perturbation could only be specified if all the degrees of freedom {q} were averaged
over.
“One can see things with the p-eye and one can see it with the q-eye, but if one opens both
eyes together, then one goes astray.”
This was what Pauli referred to as the “dark spot” in his calculations. However this “dark spot”
was to stimulate Heisenberg to begin to question the very nature of position and momentum in
quantum mechanics.
Another important contribution to Heisenberg’s approach to the issues of the interpretation of
quantum mechanics came from Einstein. During a conversation with Heisenberg, who had just
presented a talk on the new quantum mechanics, Einstein questioned Heisenberg’s philosophical
approach to the fomulation of his theory. Einstein, contrary to the philosophy that had been
the background for special and general relativity, rejected the empiricist approach of including
only the “observable magnitudes” into the theory. Heisenberg was told by Einstein:
“It is the theory which decides what we can observe.”
Ultimately it was the question of the observed trajectory of the electron that was the catalyst
for the indeterminacy relations. Heisenberg’s discussion’s with Bohr were a source of much
friction between the two and little progress was made in understanding this simplest of observed
phenomena within a quantum theoretical framework. Heisenberg appealed to the formalism to
yield answers while Bohr worked for a more complete conceptual understanding of the theory
as it stood. In February 1927 Bohr left Copenhagen on a four week skiing trip to Norway; the
stage was set for Heisenberg to push forward with ideas about trajectories and crucially the
nature of observation.
4.2.2
On the Intuitive Content of Quantum-Theoretical Mechanics
It was during Bohr’s absence that Heisenberg gained an insight into the reason that no progress
had been made in fitting the observed trajectory of an electron in a cloud chamber with the
formalism of quantum mechanics. By Heisenberg’s own account the breakthrough came when
he was walking in the park, thinking about Einstein’s statement about the nature of observation: “It is the theory which decides what we can observe.” [30]. It is was long after this,
on 23 February 1927, that a nearly fully developed interpretation of the idea of position and
momentum in the quantum realm was the subject of a letter from Heisenberg to Pauli. The
letter included an estimate for the average error in determining a quantity p, namely p, in
terms of the error on the canonically conjugate quantity q in the familiar form:
p⇡
~
q
(17)
In March 1927 Heisenberg submitted the paper On the Intuitive Content of Quantum Theoretical
Kinematics and Mechanics [31]. It was split into four parts:
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1. The reinterpretation of the quantities position, momentum and trajectory in the context
of uncertainty.
2. A derivation of the indeterminacy relations from Dirac-Jordan transformation theory.
3. A discussion of the correspondence applied to the indeterminacy relations
4. A discussion of relevant Gedanken-experiments.
In the first part of the paper, Heisenberg consolidated the ideas that he had proposed to Pauli
in his letter of 23 February 1927. The concept of the position of the electron must be replaced
by the the approximate measured position and its associated uncertainty, similarly in the case
of momentum. That is to say, one cannot talk about the position or momentum of an electron
without reference to how it is measured. According to Heisenberg we cannot measure precisely
the position and momentum due to mechanical limitations on the system, though that is not
to say that they are not defined. How then could it be that the track of an electron in a cloud
chamber should be visible? Heisenberg reasoned that the magnitude of the indeterminacy of
simultaneous measurements of position and velocity must be small enough that on the macroscopic scale the e↵ects were negligible. However, following his debates with Bohr, Heisenberg
believed that the answer must lie in the formalism, to this end he set about deriving it.
The second section of the paper was devoted to the formalism. It contained a derivation based
on the idea that an electron with a given approximate position co-ordinate q and associated
uncertainty q can be represented as a Gaussian wave function. Using Dirac and Jordan’s
transformation theory, Heisenberg demonstrated that a constraint must lie on the relationship
between uncertainties of the the position and the canonically conjugate momentum.
If the initial wave function that represents a state of approximate position, defined as above, is
given by the following expression.
(q) = c exp

q2
2( q)2
(18)
Then according to the Dirac-Jordan transformation theory, the corresponding momentum space
wave function (p) is defined as the Fourier transform, i.e.
(p) =
Z
+1
exp
1

ipq
~
(p)dq
(19)
Which yields the following expression for the p-space wave function.
0
(p) = c exp

p2 ( q)2
2~2
(20)
If one may define the uncertainty in p in the same way as it is defined for q then the the following
relationship is implied.
p=
~
q
(21)
or
p q=~
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The Indeterminacy Relations
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While this derivation is mathematically correct, Heisenberg went on to develop and test the
results by the application of Gedanken-experiments. These are examples of what is known as
“clumsy measurement” experiments in which the act of measurement induces a discontinuous
change and therefore a degree of uncertainty in the motion of the particle. The two examples
that Heisenberg chose to illustrate his principle were the -ray microscope and the Stern-Gerlach
apparatus.
The -ray microscope encapsulated the position momentum indeterminacy relation in a visualisable manner. For the position of an electron to be determined, the electron must be ‘illuminated’
- a photon of light must be scattered from it. The spatial resolution of any microscope is dependent on the wavelength of radiation used to resolve the target. However according to the de
Broglie relation p = h/ this high resolution brings with it a proportionally high perturbative
e↵ect. Through an analysis of this gedanken-experiment Heisenberg concluded the following
relation.
p x⇡h
(23)
The next experiment he was to consider was the use of a Stern-Gerlach apparatus in determining
the energy of the stationary states of and atom. Because the energy imparted to the atoms in
the apparatus must be less than the the energy separating their stationary states E, and the
angular deviation of the beam must be greater than the di↵ractive e↵ects at the aperture of
the apparatus, an upper bound on the force applied and a lower bound on the angular deviation
can be found. For a beam of atoms of width d and momentum p the deviating force F and the
angular deviation are constrained as follows.
F
.
&
E
d
h
pd
(24)
(25)
If the time for which the beam of atoms is being deviated is given by t then the angular
deviation is given by = F t/p. Taken together these relations can be combined to yield the
energy-time indeterminacy relation:
E t
F t
h
&
&
pd
p
pd
(26)
E t&h
(27)
Thus
This new interpretation of the concepts and quantities involved in quantum mechanics seemed
to Heisenberg to complete the picture. The paradoxes that had prompted his analysis of the
meaning of position and momentum had apparently been solved.
When Bohr returned from Norway, just as the manuscript was finished, the friction between the
two resumed almost instantly. Bohr could not accept the core idea underpinning the paper, that
position and momentum are defined in terms of their ability to be measured. Bohr maintained
that it was the application of the classical ideas of conjugate co-ordinates and their conjugate
momenta to quantum phenomena that limited how they could be defined in this realm. Furthermore, the indeterminacy relations alluded to limited ability to measure as a a consequence
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of the limited ability to define. This was around the time at which Bohr was beginning to
solidify his ideas about complementarity in quantum mechanics and it was his application of
those ideas that led him to the above conclusions. Heisenberg was forced to cede the point and
added a post script to his paper containing a promise that the ideas in the paper would be
refined and expanded upon by Bohr from the standpoint of complementarity.
4.2.3
Later Developments
It was discovered in a 1929 paper by Robert Condon that not all non-commuting variables share
the property of resisting simultaneous definition and measurement [32]. The example he used
were the hydrogen eigenfunctions nlm , in the l = 0 state the angular momentum is known with
arbitrary accuracy in all directions i.e. L̂y 100 = L̂z 100 = 0 despite the fact that [L̂z , L̂y ] 6= 0.
Clearly the formulation of the problem required a more general approach, one that could explain
and predict which variables would exhibit these e↵ects and which would not.
In the same year Howard Robertson produced a paper in which he proved a formula that
would give the value of the product of the standard deviations of any two self-adjoint operators
[33]. Robertson’s result was that if A and B are any two such operators, then the following
relationship must hold.
A B
1
|hAB
2
BAi|
(28)
Not only were the derivations and form of the indeterminacy relations elaborated upon, but
the rationale of the thought experiments were also challenged. At the snappily titled ‘Second
international Congress of the International Union for the Philosophy of Science’ in 1954, von
Leichtenstern criticised the validity of the reasoning involved in deriving and indeterminacy
relation from the gamma-ray microscope experiment [34]. His criticism focused on the fact
that for the particle’s momentum to be disturbed by the incoming photon, it must necessarily
exist. That is, the argument is supposed to show the non-existence of the particle’s momentum,
while simultaneously assuming its existence. However, this criticism did not undermine the
principle itself.
It was developments made from the 1930s up to the 1950s that furthered the interpretation of
Heisenberg’s principle. In a Ph.D. thesis written by Ernan McMullin on the interpretations of
heisenberg’s principle [35], four schools of thought were classified. Of these four, two persisted
as the most important to the interpretation of quantum mechanics:
1. The non-statistical or impossibility interpretation in which it is physically impossible to
simultaneously observe both the position and momentum to an arbitrary degree of accuracy.
2. The statistical interpretation in which it is merely the product standard deviations of
simultaneous measurements of conjugate quantities that are subject to a lower bound.
The second interpretation was proposed by Karl Popper and is a result of the ensemble interpretation of quantum mechanics. The first interpretation is the one that contributed to the
Copenhagen interpretation and is perhaps the most common.
4.2.4
Philosophical Implications
It did not go unnoticed by Heisenberg that his indeterminacy relations contradicted causality
- that if all initial conditions and physical laws are known to an arbitrary degree of accuracy,
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then the the future can be predicted exactly. Unlike Born’s statistical interpretation of the wave
function whose conclusion falsified causality, the indeterminacy relations falsify the very premise
of causality - the initial conditions cannot be known. In his paper on the relations, Heisenberg
took the view that in physics the inapplicability and invalidity of a law are synonymous and
therefore one could no longer apply strict causality to the physical world, though this does not
completely rule out any form of causality. Thus it is a pre-requisite for any subscriber to the
Copenhagen interpretation to abandon determinism.
One important consequence of the impossibility interpretation is that it renders meaningless
questions about the structure of fundamental particles. In the classical theory the observer can
look with arbitrary accuracy deep into the microscopic world. However, according to quantum
theory there is no more detail to see. The indeterminacy relations provide epistemological
closure at the smallest scales.
4.3
Complementarity
Prior to April 1925, Bohr assumed whatever followed from quantum theory regarding atomic
radiation would remain identical with what had followed from classical electromagnetic theory
- quantum theory would be encapsulated in the classical theories. From this he could dismiss
Albert Einstein’s light quantum hypothesis. Nonetheless, experimental proof for the existence
for quanta of non-classical radiation had been provided by Walther Bothe and Hans Geiger’s
investigation of the Compton e↵ect. From this the requirement of a correspondence principle
was superseded. Then in early 1927 with Heisenberg’s discovery of the indeterminacy relations Bohr was further justified in developing what came to be known as the Complementarity
Interpretation. 12
No doubt Complementarity was o↵ered as a philosophical principle and replaced the correspondence principle, which had served earlier as an epistemological principle. Furthermore, it
fitted nicely into Bohr’s older philosophical ideas. The discovery of quantum mechanics and
the sharpening of the correspondence principle made it clear that the formal analogy had been
a sign that the quantum formalism was a mathematical generalization of the classical. For
Bohr the idea of correspondence went further than that of maximum mathematical similarity;
his work on the interpretation can be described as an attempt to clarify the epistemological
consequences of the generalization of the classical formalism. This work formed the outlines of
a new view on the foundations of physics. [37]
History tells us of the failed attempts to understand complementarity in atomic physics. That
it is no easy task to figure out precisely what Bohr meant when he spoke about “Complementarity” in atomic physics can be highlighted by Einstein’s attempts: “Despite much e↵ort which
I have expended on it, I have been unable to achieve a sharp formulation of Bohr’s principle
of complementarity!” [38] Complementarity has been described as “not a single, clear-cut unambiguously defined set of ideas but rather a common denominator for a variety of related
viewpoints. Furthermore, it is a viewpoint not necessarily confined to the interpretation of
quantum physics.” [39, p. 87] Bohr’s long-time collaborator Leon Rosenfeld tried to give a more
precise meaning to the concept and content of complementarity. He concluded by simply stating: “Complementarity is no system, no doctrine with ready-made concepts. There is no via
regia to it; no formal definition of it can be ever found in Bohr’s writings, and this worries
12
It is interesting to note that in a correspondence with Kronig, regarding the problems with the old quantum
theory Heisenberg commented “To judge from your letter, a terrible confusion about the radiation theory must
reign in Copenhagen. If I were there, I would, as in the case of Zeeman e↵ects, plead for a dualistic theory:
everything must be described both in terms of the wave theory and in terms of the light-quanta” [36] What
Heisenberg proposed in his perceptive and prophetic remark would take Bohr more than two years to develop.
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people”. [40] Be that as it may, the feature that distinguishes Bohr’s complementarity interpretation and gives it a unique place in the history of human thought is the fact that Bohr’s
ideas were not merely abstract speculations but firmly founded on empirical findings. They
were conceived precisely in order to cope with paradoxes of experimental observation.
4.3.1
The Como Lecture
It was before an illustrious audience on September 16 1927 assembled in the auditorium of
the Istituto Carducci that Bohr, in a lecture entitled The Quantum Postulate and the Recent
Development of Atomic Theory [41] presented for the first time in public his ideas on complementarity.
The first item discussed the general conditions for observation and description in atomic physics
as compared to that in classical physics. Two guiding statements opened the discussion: “The
quantum theory is characterized by the acknowledgement of a fundamental limitation in the
classical physical ideas when applied to atomic phenomena. The situation thus created is of
a peculiar nature, since out interpretation of the experimental material rests exclusively upon
classical ideas.” Despite this, an atomic theory exists whose foundation can be expressed by the
quantum postulate “to any atomic process open to direct observation attributes an essential
discontinuity or rather individuality, completely foreign to classical theories.” 13 This quantum
postulate also implied “that no observation of atomic phenomena is possible without an essential
disturbance”. Thus, “the idea of means of observation independent of phenomena (and vice
versa) cannot be maintained. On the other hand, if in order to make observation possible
we permit certain interactions with suitable agencies of measurement, not belonging to the
system, an unambiguous definition of the state of the system is no longer possible”.14 As
consequence of the current state of a↵airs, Bohr continued, one must “regard the space-time
coordination and the claim of causality, the union of which characterises the classical theories,
as complementary features of the description of experience, symbolizing the idealization of
observation and definition respectively”. This statement, in which Bohr introduced the term
“Complementarity” for the first time, postulated the “space-time coordination” and the “Claim
of causality” complementary to each other, expressed the crux of the earliest version of what
later became known as the “Complementary Interpretation” or “Copenhagen interpretation”
of quantum mechanics.
Bohr proposed a dualistic description of the quantum phenomena- i.e., particle and wave
pictures-united by the idea of complementarity. “The ascribing of unqualified classical attributes to micro-objects”, Bohr continued in his Como lecture, “involves a certain ambiguity,
as illustrated, for example, by the well-known dilemma concerning the undulatory and corpuscular properties of light or electrons.” The two views of the nature of light Bohr said “are to
be considered as di↵erent attempts at an interpretation of experimental evidence in which the
limitation of the classical concepts is expressed in complementary ways.” Whereas the electromagnetic theory provides a satisfactory description for the propagation of light in space and
13
Heisenberg, to aid the idea of the “discontinuous change of the wave picture in observation”, maintained that,
“one must therefore take seriously the concept of probability waves. These waves do not have the immediate
reality which we earlier associated with the waves of Maxwell’s theory. One must interpret them as probability
waves, and therefore expect sudden changes with each new observation.” [42, p. 594]
14
Heisenberg later made a similar statement “In quantum mechanics, as professor Bohr has displayed, observation plays quite a peculiar role. One may treat the whole world as one mechanical system, but then only a
mathematical problem remains while the access to observation is closed o↵. To get an observation, one must
therefore cut out a partial system somewhere from the world, and one must make “statements” or “observations”
just about this particular system. Therefore one destroys the fine connections of phenomena, and the very point
where we place the cut between the observed system on the one hand, and observer and his apparatus, on the
other hand, we have to expect difficulties for our conception (anschauung). [42, p. 593]
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time, the causal aspect of optical phenomena, in Bohr’s opinion, “finds it’s adequate expression
just in the light quantum idea put forward by Einstein.” 15
Referring to plane waves Bohr stated the uncertainty relation between conjugate variables
t·
E=
x·
px =
y·
py =
z·
pz = h
expressing a general reciprocal relation between the maximum sharpness in the definition of
space-time variables (t, x, y, z) and energy-momentum variables (E, px , py , pz ) associated with
the micro-object. “This circumstance” continued Bohr, “may be regarded as a simple symbolical expression for the complementary nature of the space-time description and the claims of
causality.”
Bohr maintained that a connection with the classical description could be established with the
aid of the superposition principle. It enables us to identify wave packets with particles, related
to de Broglie’s results in which the group velocity of the wave is equal to the translational
velocity of the particle. The coalition of a particle and wave-packet, Bohr claimed, illustrated the
complementary character of the description: “The use of wave groups is necessarily accompanied
by a lack of sharpness in the definition of period and wavelength, and hence also in the definition
of the corresponding energy and momentum”.
In Bohr’s final remarks was the statement that the “essence [of the quantum theory] may
be expressed in the so-called quantum postulate, which attributes to any atomic process an
essential discontinuity, or rather individuality, completely foreign to the classical theories and
symbolized by Planck’s quantum of action.” Since according to this quantum postulate, energy
exchanges proceed only in discrete steps of finite size, the postulate of the indivisibility of the
quantum of action demands “not only a finite interaction between the object and the measuring
instrument but even a definite latitude in our account of this mutual action”. For, since the
interaction between the object and the instrument, contrary to classical physics, cannot be
neglected, “an independent reality in the ordinary physical sense can neither be ascribed to the
phenomena nor to the agencies of observation.” This leads to the conclusion that the spacetime coordination and the claim of causality are complementary features in the description of
physical observation.
4.3.2
Analysis and Initial Responses
Bohr’s version of complementarity presented at the Como lecture was underwhelming to the
physicists present. Bohr remembers Eugene Wigner summarizing the prevailing feeling: “This
lecture will not induce any of us to change his own opinion about quantum mechanics.” [44]. It
took some time for the real significance of Bohr’s conception to sink in. It wasn’t until the fifth
Solvay congress from October 24th-29th 1927 that Complementarity became more concrete as
a guiding principle in the interpretation of quantum mechanics, eventually even being hailed
as “opening a new chapter in our understanding of the universe we live in” and as “the most
revolutionary philosophical conception of our day”. [45]
To review the situation critically we must accept that Bohr never gave a clear definition of the
term “Complementarity”. The closest he came to defining the term was in a statement in 1929,
when he declared that the quantum postulate “forces us to adopt a new mode of description designated as complementary in the sense that any given application of classical concepts precludes
15
From his views on the renunciation of a completely classical description he rejected Louis de Broglie proposal
to join “the two apparently contradictory sides of the phenomena by regarding the individual particles of lightquanta as singularities in the wave field” as “resting upon the concepts of classical physics,” thus “not suited to
help us over the fundamental difficulties” [43]
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the simultaneous use of other classical concepts which in a di↵erent connection are equally necessary for the elucidation of phenomena.” Referring to this statement it is the concepts or modes
of description that can be described as complementary. 16 In the Como lecture Bohr used the
term “Complementarity” 15 times in almost all cases the complementarity modes of description
are explicitly mentioned as “the space-time coordination” and demand of “causality”, the latter
being stated as the conservation theorems of energy and momentum. He also made a prominent
emphasis on wave-particle duality in the complementarity context.
Although it was difficult to deduce what Bohr meant by “Complementarity” in micro-physics, it
gained clarity in the following years. A given theory T admits a complementarity interpretation
[39, p. 104] if the following conditions are satisfied:
1. T contains (at least) two descriptions D1 and D2 of its substance-matter.
2. D1 and D2 refer to the same universe of discourse U (in Bohr’s case, micro-physics).
3. Neither D1 nor D2 , if taken alone, accounts exhaustively for all phenomena of U .
4. D1 and D2 are mutually exclusive in the sense that their combination into a single description would lead to a logical contradiction.
More succinctly a theory T admits a complementarity interpretation if it contains two mutually
exclusive and exhaustive descriptions D1 and D2 of its universe of discourse U .
In this view we can claim that Bohr made reference to two main complementary pairs of
description in quantum phenomena.
1. Waves and particles: two models that are mutually exclusive and exhaustive in the description of atomic phenomena.
2. The “Space-time coordination” and the demand of “causality”: two physical parameters
that are mutually exclusive but simultaneously necessary for the description of atomic
phenomena.
To understand the two pairs of complementary features it helps to consider that whenever a
fundamental principle is proposed, it can generally be established on the basis of a standard
gedanken-experiment. For Bohr’s principle of complementarity in relation to (1) we can use the
experimental set-up of a micro-object (photon, electron) passing through a slit in a diaphragm.
For complementarity in the relation to (2) we can use the double slit experiment.
1. Suppose first that the diaphragm (Fig 4.2) is rigidly connected to the frame of the local
coordinate system, then the position of the micro-object (up to the, in principle, arbitrarily
small width of the slit) is obtainable. However any information concerning the exact
energy and momentum exchange between the micro-object and the diaphragm is lost
due to the rigid connection. On the other hand suppose the diaphragm with its slit is
suspended by springs (Fig 4.1), then the momentum transfer (manifested by the motion
of the diaphragm with known mass) is obtainable. However any information concerning
the exact position of the passing micro-object has disappeared owing to indeterminate
location of the diaphragm.
Generalizing this result it was understood that descriptions in terms of space-time coordinates and descriptions in terms of energy-momentum transfers cannot be physically sig16
From this point of view Complementarity, as a general concept, has the potential applicability in all areas of
systemic research. In the 1948 issue of Dilectica, which was devoted entirely to discussions about complementarity,
the mathematician and philosopher Ferdinand Gonseth published an article [46, p. 88] in which he claimed
that since knowledge progresses though an unveiling of successive horizons of reality and since the concept of
complementarity refers to the relationship between any two horizons in this dialectic process, complementary has
the potential applicability in all areas of systematic research.
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4.3
Complementarity
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Figure 4.2: Ridgid slit
Figure 4.1: Diaphragm and slit
nificant at the same time, as they require mutually exclusive experimental arrangements.
In terms of Bohr’s standard thought experiment, that the experimental arrangement were
mutually exclusive was warranted by the fact that the existence or absence of a rigid
connection between the diaphragm and the coordinate frame are logically contradictory
to each other. Such experimental arrangements may be called “complementary”, for although mutually exclusive, they are both necessary for an exhaustive description of the
physical situation.
Figure 4.3: Double slit exhibiting wave-particle complementarity
2. A micro-object, for example an electron, is transmitted through the double slit and it
interacts with the film on the other end ionising a single site as if it existed only at
that site upon impact with the film. The sites ionized however build up an interference
pattern that would only be producible by waves. Thus the micro-object exhibits wave
and corpuscular behaviour.
The conflicting natures of the micro-object and the mutual requirement for both in the
description of the experimental result exhibit the essence of wave-particle complementarity. It is however worth noting that a decade later Bohr realised that, as compared to the
previous example, both complementary features can be realised in the same experimental
set up, thus marking a distinction between the two.
The application of complementarity is often misunderstood. Bohr claimed that only when
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4.4
The Copenhagen Interpretation of Quantum Mechanics
The University of Edinburgh
the variables are connected to corresponding complementary experimental arrangements can
they be called complementary. For the principle was formulated precisely to cope with the
contrasting descriptions of the same phenomena found in di↵erent experimental arrangements.
Bohr concluded from his ideas of complementarity that:
“We meet here in a new light the old truth that in our description of nature the purpose is not
to disclose the real essence of the phenomena but only to track down, so far as it is possible,
relations between the manifold aspects of our experience.” [47]
Complementarity was not without its problems. Bohr did not discuss the logical contradictions within his insistence upon the use of classical physics in the measurement of quantum
phenomena, yet the renunciation of the classical mode of description. The situation can be
summarized in a comment by Con Weizacker: “Classical physics has been superseded by quantum theory; quantum theory is verified by experiments; experiments must be described in terms
of classical physics.” [48] The only way to avoid such contradictions would be to impose certain
limitations on the use of classical physics. The contradiction can be avoided if and only if the
use of classical terms is in such a way that it never encompasses a complete classical mode of
description. This condition is clearly satisfied if the experimental experience obtained under
di↵erent arrangements can be described exhaustively only in terms of mutually exclusive sets of
classical conceptions. Moreover, the indeterminacies, expressed by Heisenberg’s uncertainty relations, are precisely the price we have to pay if we nevertheless attempt to apply such mutually
exclusive sets of classical conceptions simultaneously.17
4.4
The Copenhagen Interpretation of Quantum Mechanics
By late 1928, all of the most important ideas that underpinned the modern interpretational
scheme known as the Copenhagen interpretation were in place. It is an uneasy synthesis of the
ideas of Heisenberg, Pauli, Bohr, Dirac, and Born, among others. At the time the interpretation
was not seen as the concrete scheme that exists today and it was not until the 1950s that
Heisenberg assigned “Copenhagen Interpretation” to the collection of ideas that had been slowly
crystalising over the first half of the 20th century [49].
The principles of the modern understanding of the Copenhagen interpretation can be summarised as follows:
1. The wave function
completely describes the state of the system. The wave function
evolves in time according to the relevant equations (Schrödinger’s equation in the nonrelativistic scheme and the Dirac or Klein-Gordon equations in the relativistic scheme).
This is true except for at the instant of measurement, at which point the wave function
collapses into an eigenstate of the observable being measured.
2. The description given by the wave function is purely probabilistic and the outcome of a
measurement is given by the squared amplitude of the wave function associated with that
outcome. This is known as the Born rule.
3. It is not possible to know to an arbitrary degree of accuracy all the physical properties of
a system simultaneously. This is known as Heisenberg’s uncertainty principle.
4. Both matter and radiation exhibit a wave-particle duality. In an individual experiment
17
An analogy may be drawn between the Copenhagen approach and the ancient conceptions behind the nature
of measurement. If measurement destroys some of nature’s (otherwise realisable) potentials then every measurement precludes the possibility of obtaining additional (complementary) information. As old as physical thought,
this conception was what deterred the ancient Greeks from developing a systematic experimental method.
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The Copenhagen Interpretation of Quantum Mechanics
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both matter and radiation may exhibit properties of either a particle-like nature, a wavelike nature or both. This is known as Bohr’s complimentarity principle.
5. Measuring devices can only measure classical observables like position or momentum.
6. The quantum mechanical description of a system in the limit of large systems should
closely approximate the classical description. This is known as the correspondence principle.
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5
The Bohr-Einstein Debates
The years following the 1927 Como conference didn’t bring many new applicational aspects to
quantum theory. Instead there was a heightened focus on interpretation, primarily revolving
around two central figures: Einstein and Bohr. This led to what is now known as the BohrEinstein debates as well as a few other interpretational aspects such as the idea of locality and
the Schrödinger’s cat experiment.
5.1
Round 1: Solvay 1927
The Bohr-Einstein debates were a series of heated discussions between Bohr and Einstein taking
place roughly between the years 1927 and 1935. Einstein had not been present at Bohr’s Como
lecture where he introduced complimentarity to the world. A month later, however, Bohr and
Einstein crossed swords at the fifth Solvay Conference in Brussels.
5.1.1
Setting the Scene
At Solvay Bohr gave the exact same speech as he had given in Como a month earlier, which lead
to a general discussion of causality, determinism and probability [9, p. 203]. Einstein, eager to
pinpoint why the Copenhagen interpretation was flawed, presented a thought experiment. A
beam of electrons is fired towards a narrow slit leading to strong di↵raction of the electrons.
Detection of the electrons happen at a photographic film in the shape of a hemisphere as seen
in figure 5.1. In this situation quantum theory dictates that after passing the slit the electrons
must propagate outwards as spherical waves.
Figure 5.1: Einstein’s simple thought-experiment. A source fires a beam of electrons towards a
narrow slit. Upon reaching the slit the electrons are di↵racted in all possible directions before
reaching the film.
Einstein argued that Bohr’s approach to the issue was problematic. Bohr’s theory (not incor-
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porating hidden variables18 ) was complete in the sense that each electron had to be considered
individually. Therefore, for a certain point on the film the wave function for a single electron
dictated the probability of that electron reaching the point. Einstein was critical towards this
view, as in his opinion, this meant that the electron was potentially present over the entire
wavefront as it approached the film. However, once the wavefront reaches the film the electron
is shown to be at a single spot. The idea that the electron could be at multiple places at the
same time was to him an indication of “a contradiction with the relativity postulate” [9, p.
204].
Einstein instead opted for a more statistical approach. He imagined an electron cloud propagating as a spherical wave rather than a single electron being omnipresent. Using this picture
the square of the wave function gives the probability for any one of the electrons in the cloud,
or the ensemble as he referred to it, to reach a certain point on the film. This view completely
eliminates the need to specify what happens as the electron reaches the film.
In the above approach Einstein toyed with the idea of including hidden variables in the picture.
He considered every one of the electrons to have a well-defined, although unknown, position
and momentum. One might draw parallels between Einstein’s ensemble and the classical Gibbs
ensemble in statistical physics in which all variables have precise values for all times [9, pp.
204].
5.1.2
Einstein and the Uncertainty Principle
One of the reasons that the Bohr-Einstein debates have achieved such fame in the history of
physics is perhaps due to the fact that they are very well documented. Bohr contributed an
entire chapter to Paul Schilpp’s biography of Einstein [50, pp. 199-241], in which he detailed
his conversations with Einstein at the 1927 and 1930 Solvay conferences. Although this chapter
highlighted only one side of the story there is a general consensus [9, pp. 205] that the main
points of the account is in agreement with what took place.
In his account Bohr firstly discussed two extensions of Einstein’s setup in figure 5.1.
Figure 5.2: Increased slit size leads to
di↵raction through a smaller angle ✓.
Figure 5.3: The added shutter limits the
extend of the wave train.
In the first extension (fig 5.2), Bohr increased the size of the slit resulting in a decreased
di↵raction up to an angle ✓. In the second extension (fig 5.3), a shutter has been placed in front
of the slit. The shutter dictates when the beam can enter the slit and thus the wave train is of
limited size.
18
Hidden variable theories were a popular alternative to the Copenhagen view. Believers in hidden variables
thought the probabilistic picture to hold on the surface, but that in order to explain various quantum phenomena
the values of position and momentum were at all times exact, although “hidden” to the observer.
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Round 1: Solvay 1927
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Bohr applied this setup to illuminate how the uncertainty principle was present in the entire
apparatus and therefore is not only a property of the electrons. For the case of figure 5.2 the
electrons will acquire a momentum in the vertical direction corresponding to the size of the
slit. The narrower the slit, the larger the momentum. This is in agreement with the positionmomentum uncertainty principle19 . In the case of figure 5.3 the action of the shutter induces
a spread in frequency and therefore a spread in electron energy. Thus, if the slit was open
for a large time T the energy spread would be minimal. This too was in agreement with the
time-energy uncertainty principle.
However, Einstein wasn’t satisfied with leaving the discussion there. He wished to go further
and ultimately disprove the uncertainty principle. Einstein therefore suggested that one could
possibly infer the momentum transfer between particle and screen and energy transfer between
particle and shutter by simply studying the system. Together with the partial knowledge of
position, deduced from the width of the di↵raction pattern, as well as the partial knowledge of
shutter time, one could theoretically disprove the uncertainty relations.
Bohr responded by noting that in the above scenario the screen is stationary and there is no
uncertainty in the speed of the shutter. This can only be the case if screen and shutter have
infinite mass, so that collisions with particles doesn’t a↵ect the system. However, in such
a situation it is impossible to infer anything about the energy or momentum transfer in a
collision. Bohr added that if one were to introduce finite masses of the screen and shutter one
would also inherit uncertainties in these elements. Thus, overall the uncertainty relations would
be obeyed.
5.1.3
Einstein Strikes Again
Einstein, reluctant to give up, came up with another thought-experiment. His system consisted
of three screens (fig 5.4). At screen A the beam is di↵racted as in the original experiment.
Screen B has two slits, B1 and B2 , and is furthermore connected to a weak spring. The last
screen, C, is where the di↵raction pattern is observed. As is common, Einstein considered a
suitably weak beam such that the di↵raction pattern is built up slowly and each electron might
be studied on its own.
Figure 5.4: Einstein’s thought-experiment. An additional screen is inserted in between the
original setup. The screen contains two slits and is attached to a weak spring.
Einstein then proposed that upon collision with the particle the screen B would recoil in the
19
See [50] for derivation
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5.2
Round 2: Solvay 1930
The University of Edinburgh
vertical direction ever so slightly. Supposing the particle went through slit B1 and reached the
point y on screen C, then we would expect screen B to recoil upwards. Similarly, if the particle
went through slit B2 and reached the same point, we would expect B to recoil downwards.
Using this analysis, Einstein hoped to be able to deduce which slit the electron went through
while simultaneously building up the interference pattern.
However, once again Bohr was able to complete the puzzle. He realised that knowledge of the
momentum increase of screen B must imply an uncertainty in the position of the slits, and
that this uncertainty is exactly large enough to “wash out” the double slit interference pattern
leaving e↵ectively two single slit di↵raction patterns. He proclaimed “This point is of great
logical consequence, since it is only the circumstance that we are presented with a choice of
either tracing the path of a particle or observing interference e↵ects, which allows us to escape
from the paradoxical necessity of concluding that the behaviour of an electron or photon should
depend on the presence of a slit in the diaphragm through which it could be proved not to
pass.”
Upon leaving the first round of the debates, Einstein had to admit that Bohr’s logic was superior.
It seemed that there was no straightforward way of disproving the uncertainty principle. Nor
had it been possible to observe the wave and particle aspects of light simultaneously. Bohr
(together with his followers) had been strengthened in his belief in complementarity.
5.1.4
Einstein’s Ensembles
Einstein strongly advocated his theory of ensembles as an opponent to the “standard” Copenhagen interpretation. In Einstein’s ensemble all variables have definite values at all times.
However, not all the variables are necessarily related to the wave function of the ensemble. One
can adopt the image of the electron cloud in relation to Figure 5.1 In this case the wave function
is an eigenfunction of momentum and so all the electrons in the electron cloud, or the ensemble,
have the same definite value of momentum. Similarly, all the electrons in the ensemble have
definite values of position, but these values all di↵er. Within the ensemble theory, rather than
dealing with the ambiguous collapse of the wave function, the measurement is simply a matter
of ”choosing” one of the electrons from the ensemble. Thus superficially the wave function still
appears the same (although it now spans an entire ensemble) and the result of the measurement
is still uncertain to within the requirements of the indeterminacy principle. This, of course, can
be extended to include definite, but unknown, values of any number of variables. Unfortunately
for Einstein the ensemble theory was disproven20 . Einstein, however, simply chose to ignore
these issues [9, pp. 213].
5.2
Round 2: Solvay 1930
The second round of the Bohr-Einstein debates took place three years later at the 1930 Solvay
conference. For this occasion Einstein brought up a thought-experiment that relied on his own
theory of relativity. He imagined, as seen in figure 5.5, a box containing a single hole that
could be covered by a shutter. In theory, if the shutter was opened for a short time interval
it might be timed such that a single photon would enter the box. Furthermore, the box may
be weighted (to an arbitrary uncertainty) before and after this event. Now Einstein applied
two simple equations to get a very powerful result. Firstly, he argued, the photon would have
a non-zero energy of E = h⌫. Secondly, his well-known equation E = mc2 then implies that
the photon should have an e↵ective mass, which would increase the weight of the box after
20
See Appendix F
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Round 2: Solvay 1930
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capture. Finally, Einstein argued that the fact that the weight of the box may be determined
to an uncertainty of zero implies that the uncertainty in the mass of the photon, and hence
the uncertainty in the energy, may be zero. At the same time, however, the uncertainty in the
shutter time is finite, which leads to the astounding result that the time-energy uncertaincy
principle is violated!
Figure 5.5: Einstein’s thought-experiment. A box with a hole that may be covered by a shutter
allows for a single photon to enter [50, pp. 225].
This was an ingenious argument. Whitaker [9, p. 218] quotes Rosenfeld: “It was quite a shock
for Bohr. . . he did not see the solution at once. During the whole evening he was extremely
unhappy, going from one to the other, and trying to persuade them that it couldn’t be true,
that it would be the end of physics if Einstein were right; but he couldn’t produce any refutation.
I shall never forget the vision of the two antagonists leaving. . . Einstein a tall majestic figure,
walking quietly, with a somewhat ironical smile, and Bohr trotting near him, very excited. . . The
next morning came Bohr’s triumph.”
Figure 5.6: Bohr’s more realistic picture of Einstein’s thought-experiment [50, pp. 227].
In order to suit the problem at hand Bohr had produced a setup as seen in 5.6. Einstein’s
box was now hanging from a spring and a pointer was attached in order for the weight to be
measured on a scale fixed to the balance support. Loads attached underneath the box allowed
for a weighing of arbitrary accuracy m. The important bit is then, that any measurement of
the position of the pointer has a quantum mechanical accuracy q. As the pointer is attached to
the box this implies an uncertainty in the momentum of the box following the usual uncertainty
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5.3
Round 3: EPR
The University of Edinburgh
relation. Bohr then argued that this uncertainty must naturally be less than the total impulse
in the time interval (the shutter time) T. Generally, impulse is simply force times time interval,
so in this case, Bohr debated, the total impulse is that of a gravitational field due to a body of
mass m:
h
<T ·g·
q
p⇡
m
(29)
where as usual g is the gravitational constant. Bohr then noted that the immediate implication
is that for a certain accuracy m, a great accuracy of the reading of the pointer implies the
need for a large time interval T .
It was at this point Bohr showed true brilliance. Not only would he succeed in disprove Einstein,
but would do so using Einstein’s own framework of General Relativity [50, pp. 227]: “according
to general relativity theory, a clock, when displaced in the direction of the gravitational force by
an amount of q, will change its rate in such a way its reading in the course of a time interval
T will di↵er by an amount T given by the relation’:
T
1
= 2g q
T
c
Combining equations (29) and (30) by elimination of
h·g·T
<T ·g·
T c2
)
m
(30)
q gives
T >
h
mc2
(31)
Finally, E = mc2 recovers the time-energy uncertainty relation
E·
T >h
(32)
Once again Einstein had to accept that there was no way of disproving Bohr’s complementarity
by appealing to thought-experiments of this kind. At this point it is believed that Einstein
finally ended his crusade against the legitimacy of the uncertainty relations [9, p. 219].
5.3
5.3.1
Round 3: EPR
Einstein Moves to Princeton
Around late 1931 Einstein arrived at Caltech in California for a short sabbatical. He was
approached by the American Abraham Flexner, who was in the process of founding the Institute
for Advanced Study at Princeton, where the brightest scientific minds of the day would be be
able to focus their time on research, free from any teaching. Einstein agreed to spend half his
time in Princeton and the other half at his current position in Berlin. However, upon returning
to Europe in early 1933, he found Germany in the grip of Nazi power. Fuelled by disgust at
Hitler and fears for his own safety, he surrendered his German passport in Brussels and lived
for a short time on the Belgian coast. As Nazism spread further across Europe and fears for
his safety grew, he moved first to England, and then finally to a permanent position at the
Institute for Advanced Study.
Safe from the Nazi threat, Einstein once more concentrated on physics. Quantum theory had
moved on without him as the leading physicists now ignored any question of interpretation,
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5.3
Round 3: EPR
The University of Edinburgh
concentrating on the myriad of applications, most notably the discovery of the neutron by
James Chadwick. One of the few remaining top physicists critical of quantum theory, Einstein
conceded “here at Princeton I am considered an old fool”. [7]
Despite his growing isolation within the physics community, Einstein was joined in Princeton
by the young researchers Nathan Rosen and Boris Podolsky. Together they worked on one last
attack on quantum mechanics, and one that would have the most profound implications.
5.3.2
EPR
Einstein knew he needed a change of tack. At the previous two Solvay conferences he had
attacked the uncertainty principle, determined to show that quantum mechanics was inconsistent. However, finding this was not so, Einstein turned his attention to showing that quantum
mechanics could not possibly be a complete description of reality.
Over a few weeks in early 1935, Einstein, Rosen and Podolsky laid down the foundations of
their idea and Can Quantum-Mechanical Description of Reality Be Considered Complete? was
published in Physical Review on May 5th of that year. Though only four pages long, the
paper was buoyed by Einstein’s fame, the New York Times reporting that Einstein would
“attack science’s important theory of quantum mechanics, a theory of which he was a sort of
grandfather” [7].
The paper begins [51] by specifying two criterion which any successful theory must satisfy in
order to be considered true:
1. Is the theory correct?
2. Is the description given by the theory complete?
It was the first of these that Einstein had attacked at the Solvay conferences. In this paper, he
turned his attention to the second. The paper then gave a necessary requirement for a theory
to be considered complete: every element in the physical reality must have a counterpart in the
physical theory. Laying the final foundation of the paper, a reasonable definition of reality was
given: If, without in any way disturbing a system, we can predict with certainty the value of a
physical quantity, then there exists an element of physical reality corresponding to this physical
quantity [51].
It went on to consider non-commuting observables. The fundamental state of the system is
“supposedly” completely characterised by the wave function , and each physical observable A
has a corresponding operator A. If the wave function was an eigenfunction of A then
0
=A =a
(33)
and the physical quantity A would have the value a with certainty whenever the system was in
the state . However, if eq(33) did not hold then a did not have a definite value. If a second
physical quantity B was introduced that did not commute with A - in other words they had no
common eigenbasis - then measurement of one would disturb the other, and eq (33) would not
hold. Such is the case with position and momentum in one dimension, where measurement of
position will destroy any knowledge of momentum, and vice versa.
From this the paper drew a startling conclusion. Either
1. The quantum mechanical description of reality given by the wave function is incomplete
or
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Round 3: EPR
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2. When the operators corresponding to two physical quantities do not commute the two
quantities cannot have a simultaneous reality
Einstein asserted that either our description of reality is incomplete, or reality itself is incomplete. Having always considered the second of these to be absurd, he now set out to prove
so.
Suppose two systems, I & II, interact between t = 0 and t = T , after which time there is
no interaction. For t > T one could calculate the state of the combined I + II system using
Schrödinger’s equation, designated . After this time no more can be known about the wave
function without further measurement.
Now suppose a1 , a2 , a3 , ... were the eigenvalues of some physical quantity A associated with
system I, with eigenfunctions u1 (x1 ), u2 (x1 ), u3 (x1 ), ... where co-ordinate x1 corresponded to
system I. could then be expanded in the {un } basis21
(x1 , x2 ) = ⌃1
n=1
n (x2 )un (x1 )
with x2 as the variable corresponding to system II. Measurement of A on would then collapse
into a given eigenfunction uk (x1 ), and the system would then, with certainty, be in the state
k (x2 )uk (x1 ).
On the other hand we could choose some other operator B, also corresponding to system I,
with eigenvalues b1 , b2 , b3 , ..., eigenfunctions v1 , v2 , v3 , ... and expand the system in the {vn }
basis:
(x1 , x2 ) = ⌃1
n=1
n (x2 )vn (x1 )
Suppose then that we measured B on . Given a result br we would know with certainty that
system I is in state vr and system II is in state r .
Depending on the choice of measurement on system I, system II would be in a state described
by one of two wave functions, either k or r . Therefore, it is possible to assign two di↵erent
wave functions to the same reality. It may also be the case that k and r are the eigenfunctions
of two non-commuting operators P and Q.22 Then, by measuring either A or B on system I
we could predict with certainty the eigenstate of system II and therefore the value of either P
or Q. As the systems are physically separated, the measurement of I can have no e↵ect on II,
and so a simultaneous reality must exist for both P and Q.
The analysis was began by stating that either (1) the quantum mechanical description of reality
given by the wave function is incomplete or (2) when the operators corresponding to two physical
quantities do not commute the two quantities cannot have a simultaneous reality. Since it was
found that the quantities corresponding to two such operators do have a simultaneous reality, the
only conclusion is exactly what Einstein set out to prove: the quantum mechanical description
of reality given by the wave function is incomplete.
Einstein, Podolsky and Rosen had fought against the quantum mechanical dogma that no
underlying reality exists, in an e↵ort to save a physics they considered reasonable and real.
The paper finished with
21
{ n (x2 )} can simply be regarded as the coefficients in the expansion of in the {un (x1 )} basis.
Despite being only four pages long, the original EPR paper works through an example for non-commuting
P and Q.
22
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Round 3: EPR
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“While we have thus shown that the wave function does not provide a complete
description of the physical reality, we left open the question of whether or not such
a description exists. We believe, however, that such a theory is possible” [51]
a triumphant nod to the future and hopes for a unified field theory that would save realism.
5.3.3
Response to EPR
Upon publication of the paper there was much disquiet among the leading quantum physicists
at the time. Pauli described it as a “disaster” whenever Einstein wrote a paper on quantum
theory. It was Bohr who spent the following four months drafting a reply. In the same journal he
published his response, identically entitled Can Quantum-Mechanical Description of Reality Be
Considered Complete?, though this time Bohr desperately hoped the answer would be yes.
Unable to find fault in Einstein’s reasoning, Bohr resorted to attacking the assumptions in
EPR, in particular that there must be a reality without disturbing the physical system. He
began by criticising EPR for their definition of physical reality, claiming that it should come
from experiment and quantum mechanics itself, as opposed to simply being some dreamt up
philosophical ideas.
However, Bohr’s main objection to EPR was that, while it is perfectly reasonable to make
assumptions about system II from measuring system I, the physical reality only becomes real
when measurements are made on system II itself. He asserted that the idea of position, momentum or any observable is so intrinsically linked to the measuring apparatus that the idea of
them independently is meaningless. Even the experimental set-up is inherently connected with
what we can say about observables, that we tell them apart by “rational discrimination between
essentially di↵erent experimental arrangements and procedures suited either for unambiguous
use of the idea of space location, or for a legitimate application of the conservation theorem of
momentum.” [52]
Bohr went on to consider the possibility that system II could have somehow communicated
with system I. He asserted that there was “no question of mechanical disturbance” [52] - the
exchange of some particle or force between the two systems. However, in what was at first
considered bizarre and vague, he emphasised that there was the “question of an influence on the
very conditions which define the possible types of predictions regarding the future behaviour
of the system.” [52] In rather vague language, Bohr suggests that because the two systems
were once the same they are forever entwined, and each can instantaneously a↵ect the other.
This argument violates the very concept of locality - so central to EPR’s argument - that
no interaction at a distance can take place faster than the speed of light, a rule enforced by
Einstein’s own special relativity.
Upon publication Bohr’s paper was deemed confusing and unclear, he himself saying it contained
“inefficiency of expression” [7]. Einstein criticised Bohr’s response as “spooky” and “telepathic”
[7], while Dirac proclaimed “Now we have to start all over again, because Einstein proved that
it [quantum theory] does not work.” [7] However, as the successful experimental predictions of
quantum theory continued many quantum theorists chose to ignore EPR and despite his poor
response, Bohr was considered victorious.
It was Schrödinger who continued to probe the more subtle implications of EPR in various
correspondence with Bohr23 . He was the first to use verschränkung - translated as entanglement
- to describe the connection the two particles share once they have separated. In his prominent
paper published that year, he wrote:
23
For a discussion of Schrödinger’s cat see Appendix G
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5.4
Further Developments on EPR, Non-Locality and Bell
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“Any entanglement of predictions that takes place can obviously go back to the fact
that the two bodies at some earlier time formed in a true sense one system, that is
were interacting, and have left behind traces on each other.”
Schrödinger had set the stage for the idea of entanglement and its future development and
mysterious consequences.
5.4
Further Developments on EPR, Non-Locality and Bell
In the years following 1935 there were many attempts to reconcile the ideas of EPR. Bohr’s
response had been unimpressive, yet the experimental validations of quantum theory continued.
Theorists worked at and simplified the underlying concepts of EPR, settling on two major points
of contention24 :
1. Dynamics: If the state of any physical system is given at some initial time, it is possible
to calculate its state at a later time assuming the forces and constrains involved are known.
In other words, there exist a “dynamics” of the state vector i.e. there are deterministic
laws regarding how the state vector, subject to certain forces and constraints, evolves in
time. The equation describing this time evolution is known as the Schrödinger equation.
2. Collapse: Immediately after a measurement that gave a certain result i.e. eigenvalue,
the system is in the eigenstate corresponding to that eigenvalue. In other words, the state
is such that if the measurement is repeated, it is guaranteed that the same result will be
obtained. This implies that the e↵ect of measuring and observable is to change the state
vector of the system by collapsing it, from whatever state it may have been in, into the
eigenvector of the measured observable. However, which particular eigenstate it collapses
into is a matter of probability for the outcome of the measurement.
In order to more deeply understand EPR and its consequences, it was formulated in many
di↵erent ways. The most well known of these considers electron spin as the property of the
systems to measure. Suppose one has now measured the x-spin of a particular electron. After the
measurement, one is in a position to predict with certainty what the outcome of a measurement
of x-spin of that electron would yield at a later time. Therefore, EPR reality condition entails
that the value of spin-x must now be an element of reality of the electron which is also implied
by the quantum mechanical formalism. On the other hand, in the case of measuring z-spin of
an electron and then the x-spin of the same electron, EPR reality condition does not entail that
x-spin is an element of reality of this electron since in order to measure x-spin the system has
been disturbed by the measurement. This is also in accordance with quantum mechanics.
Consider a system of two electrons. Let their non-seperable state in the x-spin basis be
1
|Ai = |x" i1 |x# i2
2
1
|x# i1 |x" i2
2
(34)
One can easily check, using equations 69 to 72, that |Ai can also be written as follows in the
z-spin basis
1
1
|Ai = |z" i1 |z# i2
|z# i1 |z" i2
(35)
2
2
By equation (34), if we make a measurement of the x-spin of electron 1, the outcome can either
be x" or x# with equal probabilities. In the event that the measurement yields x" , from quantum
theory as well as experimental verification, the outcome of any subsequent measurement of xspin of the second electron is necessarily x# and vice versa. EPR assumed that the experiment
could, in principle, be set up in such a way to guarantee that measuring x-spin of the first
24
For a full treatment of the quantum mechanical formalism see appendix H.
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5.4
Further Developments on EPR, Non-Locality and Bell
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electron produces no physical disturbance on that of the second electron. This assumption is
known as locality. For example, one could separate two electrons by a large distance or place an
impenetrable walls between them to ensure that no signal gets passed during the experiment.
Because when we measure x-spin of electron 1 the outcome of the measurement of electron 2
is immediately known i.e. the opposite. It follows from the reality condition x-spin must be
an element of reality of electron 2. The same argument holds for z-spin measurement. This
implies that there must be something wrong with the standard way of thinking since there
exists circumstances such that there are simultaneous “matters of fact” about x-spin and z-spin
values even though these observables are incompatible. Furthermore, the formalism must be
incomplete since there are certain elements of reality that do not appear in the formalism.
Putting all this in other words, EPR notices that there was something bizarre about the collapse
postulate of two-particle systems. They realised that it was in fact non-local - if the two
particles were in non-separable states, when a measurement was performed on one it would
bring instantaneous changes to the other no matter how far apart they were or what type
of shield was placed between them. They then suspected that “non locality must merely be
a disposable artefact of this particular mathematical formalism, of this particular procedure
of calculating the statistic of outcomes of experiments; and that there must be other (as yet
undiscovered) such procedures, which give rise to precisely the same statistical predictions but
which are entirely local.”
Thirty years later, Bell demonstrated that this suspicion was wrong. Bell proved that there is in
fact a genuine non-locality in the actual workings of nature, no matter how we try to describe it.
Non-locality is a feature of quantum mechanics itself and in fact, according to Bell’s theorem,
a feature of every possible way of calculation which results in the same statistical predictions
as quantum mechanics. Such predictions are now known to be experimentally correct.
In order to be more precise about what kind of non-locality is exhibited by quantum mechanics,
consider the following experiment. From what we have seen, the outcome of the spin measurements on the second electron depend non-locally on that of the first electron and vice versa.
The important question now is whether to not the probability of the outcome of spin measurements on the second electron depend non-locally on whether a spin measurement is first made
on electron 1. To answer this question, suppose the x-spin of electron 1 is measured after which
a measurement of electron 2 is carried out. The measurement on electron 1 is equally likely
to yield x" or x# . If the former occurs, by the collapse postulate for a two particle system, it
implies that the subsequent measurement of electron 2 will result in x# and vice versa. Hence,
starting with state |Ai, the outcome of measuring x spin of the second electron is equally
likely to be x" or x# regardless of whether a measurement of x-spin of the first electron is made
first.
Now, suppose a measurement of z-spin of electron 1 is performed followed by a measurement
of x-spin of electron 2. By equation (35) the outcome of the former is equally likely to be z"
or z# . If the outcome is, say, z# the collapse postulate implies that the outcome of the second
measurement can be x" or x# with equal probabilities. Same is true when the first measurement
yields z" . In other words, the outcome of measuring x-spin of the second electron is equally
likely to be x" or x# whether a measurement of x-spin, or z-spin of the first electron is performed,
or even if no measurement is performed on electron 1. The argument can be extended to other
types of observables and physical systems as well.
To summarise, given that the prediction of quantum mechanics are right, there are subtle kinds
of non-local influences in nature. “The outcomes of measurement do sometimes depend nonlocally on the outcomes of other, distant, measurements but the outcome of measurements
invariably do not depend non-locally on whether any other, distant, measurements get carried
out.” [53]
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5.5
5.5
The Measurement Problem
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The Measurement Problem
The measurement problem deals with whether the dynamics and the collapse postulate predict
the same thing about what happens to the state vector of a physical system when a measurement
takes place. The dynamics which tells us how the state vector of physical systems evolve,
in general, is fully deterministic. On the other hand, the collapse postulate states how the
state vector evolves when it comes into contact with a measuring device, is not deterministic.
Therefore, the consistency between the two is not clear.
Suppose that the evolution of everything in the world is governed by dynamical equations of
motion. Consider a device for measuring z-spin of an electron. It has a dial with a pointer
that can indicate three di↵erent positions: ‘ready’, ‘z" ’ and ‘z# ’. The pointer initially shows
‘ready’. The electrons are then fed into the device and their z-spins are measured. The outcome
is recorded by the pointer (Fig 5.7).
Figure 5.7: z-spin measurement
For example, if the electron that enters the device has spin z" , we get
|readyim |z" ie ! |z" im |z" ie
(36)
Similarly if the input electron has spin z#
|readyim |z# ie ! |z# im |z# ie
(37)
Now suppose that the device is pointing to “ready” and an electron with x-spin x" is fed in.
Using the linearity of the equations of motion we have the following for the initial state of the
electron and the measuring device
1
1
|readyim |x" ie = p |readyim |z" ie + p |readyim |z# ie
2
2
(38)
When the measuring device gets switched on it evolves into
1
1
p |z" im |z" ie + p |z# im |z# ie
2
2
(39)
which is given by the dynamics.
However, by the collapse postulate we will end up with either
|z" im |z" ie , or |z# im |z# ie
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(40)
5.5
The Measurement Problem
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which is not the same as (39). In fact, the state described in (40) is the right one, which is
how things do actually occur in experiments. However, in (39) there is no ambiguity where
the pointer is pointing which is in contradiction with what the competent observer actually
observes - the observer either sees the pointer pointing to |z" i or |z# i and these two are the only
possibilities.
What we have shown is that carrying out a measurement has a certain fundamental e↵ect which
is the appearance of a definite outcome. This is not predicted by dynamical equations of motion
whereas the collapse postulate seems to be correct. However, the dynamics seems to be right
about describing what happens to the system when we are not making a measurement. This is
known as “the problem of measurement”. [54]
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6
Di↵erent Interpretations
6.1
The Collapse of the Wave Function, Von Neumann, and Wigner Interpretations
In the late 1930s, John von Neumann tried to tackle the measurement problem in his book
called Mathematical Foundations of Quantum Mechanics. According to him, the only way to
resolve the problem of measurement was to admit that the dynamics was simply wrong at the
point where the measurement occurs. Therefore, he produced two fundamental laws regarding
how the states of a quantum mechanical system evolve:
1. When there is no measurement, the states of all physical systems evolve according to
dynamical equations of motion.
2. When there are measurement, the states of the measured system evolve according to the
postulate of collapse and not according to the dynamical equations of motion.
The trouble here is that these laws will depend on the exact meaning of the word measurement.
Depending on whether or not a measurement is made, only one of them will be obeyed and we
need to decide which one. It turns out that this word does not have an absolute meaning in
ordinary language and therefore these laws cannot determine exactly how the world behaves. In
1961, Wigner also proposed two fundamentally di↵erent physical systems in the world in order
to try and resolve the issue:
1. Purely physical systems: systems that do not contain sentient observes. Therefore, as
long as they are isolated from outside influences, evolve according to dynamical equations
of motion.
2. Conscious systems: systems that do contain sentient observers hence evolving according
other rules. That is, at the point of a measurement the brain of a sentient being enters a
certain state in which states connected with di↵erent conscious experiences are superposed
and so the mind connected to the brain causes system as whole (brain, measuring device
and the measured system) to collapse. This leads to a body-mind dualism.
However, this also does not lead to a genuine physical theory either because now the precise
meaning of consciousness needs to be defined and Wigner did not assign a particular meaning
to this word and later on shifted away from this idea.
It is also hard to set the problem empirically because first we need to find out when are where
exactly the collapse actually occurs. In the previous experiment, we could say that the collapse
occurs at the moment that the state of the electron is measured by the device, or we could
say the collapse occurs at a later time, perhaps when the brain of the competent observer
becomes involved. As it happens, these will have di↵erent consequences especially when making
measurements of incompatible observables of the composite electron system and so by means
of such measurements, in principle, it will be possible to determine exactly where and when
the collapse occurs. However, this also happens to be extremely difficult at experimental level,
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6.2
Bohm’s Interpretation and Bell
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since it is hard to completely isolate the system from its environment. Hence it is not possible
at present to precisely say where and when the collapse occurs. [55]
6.2
Bohm’s Interpretation and Bell
In his book Quantum Theory, published in 1951, Bohm introduced his theories on the Copenhagen Interpretation and the EPR paradox. However, after some discussion with Einstein, his
doubts with regards to this interpretation were deepened. In 1952, he published a series of papers with the aim to seek consistent alternative interpretation in which each individual system
can be thought of a precisely definable state. The change of the system with time are governed
by definite laws analogous the classical equations of motion. Quantum mechanical probabilities
in this context are regarded as merely a practical necessity similar to classical statistical mechanics - they are not treated as a manifestation of an inherent lack of complete determination
in the properties of matter at the quantum level. In this interpretation, the uncertainty principle
is treated not as an intrinsic limitation of the precision with which we can measure momentum
and position simultaneously, but as a practical limitation on how precise the quantities can be
measured which arises due to uncontrollable and unpredicted disturbances by the measuring
apparatus when observing the system. Because of this restriction, the precise values of position and momentum are regarded as hidden. This is the simplest example of hidden variables
interpretation of quantum mechanics. [56]
Heisenberg objected this point of view. He argued that the physics of Bohm’s language is
nothing di↵erent from that of the Copenhagen interpretation, but is unsuitable since it destroys
the symmetry between p and q which is implicit in quantum theory. [56]
According Bohmian mechanics, in a system of particles, the wave function which evolves in
accordance to the Schrödinger equation is only a partial description of the system. The description is complete only when one takes into account the actual positions of the particles. The
latter’s evolution is given by the guiding equation expressing the velocities of the particles in
terms of the wave function. In other words, in Bohmian mechanics, particles are primary and
the wave function is secondary. Note that it is the position of the particles in this mechanics
that are its hidden variables which is an unfortunate terminology. In Bell’s words:
“Absurdly, such theories are known as hidden variable theories. Absurdly, for there it
is not in the wave function that one finds an image of the visible world, and the results
of experiments, but in the complementary hidden(!) variables. Of course the extra
variables are not confined to the visible macroscopic scale. For no sharp definition of
such a scale could be made. The microscopic aspect of the complementary variables
is indeed hidden from us. But to admit things not visible to the gross creatures
that we are is, in my opinion, to show a decent humility, and not just a lamentable
addiction to metaphysics. In any case, the most hidden of all variables, in the pilot
wave picture, is the wave function, which manifests itself to us only by its influence
on the complementary variables.”
Bohmian mechanics is a step from the Schrödinger equation (non-relativistic system of particles)
to a theory describing actual motion of the particles. The state of a system of N particles is
described by
• Its complex wave function
space of the system.
= (q1 , ..., qN ) = (q) where q belongs to the configurational
• The actual configuration of the system Q which is defined by the actual positions Q1 , ..., QN
of the particles.
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6.2
Bohm’s Interpretation and Bell
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Therefore, the theory required two evolution equations:
• The Schrödinger equation: i~ @@t = H
⇤
@k
~
@
@
@
k
• The Guiding equation: dQ
dt = mk Im[ ⇤ psi ](Q1 , ..., QN ) where @k = ( @xk , @yk , @zk ) and
qk = (xk , yk , zk ) is the coordinates of the k th particle.
Thus for a system of N particles the the equations above completely specify the Bohmian
mechanics which accounts for all non-relativistic phenomena of quantum mechanics such as
interference and spectral lines. This mechanics also takes spin into account. In this model,
all the devices used to measure or perform the experiment and the measured system itself are
both taken into account. The hidden variables model is then obtained by taking the initial
configuration of the big system as random, whose distribution is given by | |2 . The initial
configuration is then transformed into the final one by the guiding equation for the big system
implying that the final configuration will also be distributed in a quantum mechanical sense.
Hence the Bohmian model results in the usual quantum mechanical predictions for the results
of the experiment.
In 1964, John S. Bell published a paper in that under certain conditions, no physical theory
which is realistic and also local in a specified sense can agree with all of the statistical implications of quantum mechanics. This is known as the Bell’s theorem. One of the issues this theory
explores is the conjecture that the quantum mechanical state of the system needs to contain a
further reality or hidden variable or complete sets in order to provide a complete description.
The incompleteness in this context would then be the explanation for the statistical nature of
quantum mechanical predictions about the system. By Bell’s theorem, these hidden variable
theories must necessarily be non-local. There are two di↵erent types of hidden variable theories,
the di↵erence between which was pointed about by Bell in 1966 for the first time:
1. Non-contextual: in this theory, the complete state of the system determines the value of
the quantity being measured irrespective of what other quantities are being measured simultaneously or the complete state of the measuring apparatus. (Hidden variable theories
of Kochen and Specker 1967).
2. Contextual: the value of the quantity obtained depends on that quantities are being
measured simultaneously and the complete state of the measuring apparatus.
• Algebraic context: specifies the quantities which are measured jointly with the quantity of primary interest.
• Environment context: a specification of the physical characteristics of the measuring
apparatus whereby it simultaneously measures several distinct co-measurable quantities.
Bohm’s hidden variable theory (1952) is an example of an environmental context. However, in
Bell’s (1966) the context is algebraic.
It is important to mention that Bell’s theorem proves that there cannot be any local hidden
variables theories through the use of an inequality known as the Bell’s inequality. In the 1980s,
Alain Aspect and his collaborators carried out a series experiments to test the Bell’s inequality. They used an optical analogue of the thought experiment involving spin-1/2 particles. A
measurement of linear polarisation of the photons in di↵erent orientations at two points were
performed. Furthermore, di↵erent versions of the same experiment were also carried out to
test the theory in more depth. All the experimental results violated the Bell’s inequality in
excellent agreement with quantum mechanics, proving that local hidden variable theories could
not possibly be valid. [57]
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6.3
Many-Worlds Interpretations
6.3
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Many-Worlds Interpretations
6.3.1
Introduction
The Many-Worlds Interpretation, which stems from the doctoral thesis of Hugh Everett in
1957, attempts to explain Quantum Mechanics without the standard collapse postulate. The
Many-Worlds interpretation is in fact only one of many di↵erent theories that try to explain
Everett’s thesis. We will begin by exploring Everett’s work and then continue with the theories
of other authors which attempt to amend his Relative State Formalism into a form which is
more complete than what was presented initially.
6.3.2
Relative State Formulation
In 1957 while a graduate student at Princeton, Everett wrote his doctoral thesis about a new
formulation of quantum mechanics, named Relative State Formulation. In the thesis he stated
that the leading problem with the standard collapse formulation of quantum mechanics was that
all observers would have to be considered external to the system. To Everett this represented
an enormous problem, since with only external observers how could one describe the entire
universe as a wave function?
His solution to the quantum measurement problem was to disregard collapse of the wave function
entirely from the quantum formulation and take the remaining as a full description of all physical
systems. To explain the standard statistical predictions found with the wave function collapse
he stated that observers, now considered part of the physical system, only have subjective
experiences.
To illustrate Everett’s thoughts, consider a physical system S with spin properties, which can
either be spin up or spin down. The observer J does not disturb the system for simplicity
and is a “good” observer, defined by Everett as an observer which obeys the condition that if
the system is in a determinate spin state then the observer will determinately record the spin
state:
|“ready”iJ |upiS
> |“up”iJ |upiS
|“ready”iJ |downiS
> |“down”iJ |downiS
(41)
This is simple in both the standard formulation and Everett’s formulation. However when the
system is initially in a superposition state such as:
|“ready”iJ (a|upiS + b|downiS )
(42)
then the actual measurement of the observer will be:
|“up”iJ |upiS + b|“down”iJ |downiS
(43)
With the standard formulation the wave function would collapse to a single state either “up”
or “down”, however with the relative state formulation no collapse occurs. This final state is
the center of all the many worlds theories because Everett himself failed to entirely explain the
interpretation of this state and therefore this lead to many reinterpretations of his thesis.
Everett himself admits confusion from this final state in which the observer seems to not be
able to be defined independently. With this he introduces a principle, which he named the fundamental relativity of quantum mechanics, which states that observer J recorded “up” relative
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6.3
Many-Worlds Interpretations
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to the system being in state “up”. Since this principle does not allow Everett to reconcile the
standard collapse final states his intention is unclear. Everett initially attempted to explain in
his non-collapse formulation an observer that would be able to give the same predictions as the
original collapse quantum mechanics, however he ends up with an observer that seems to be
unable to state any certain record.
The disparity in Everett’s thesis has caused many to amend or suggest other theories to explain
what he had meant to say. Such reconstructions need to explain three main problems which
Everett did not manage to solve. Firstly, a reconstruction would somehow explain why the
observer would have a determinate record or atleast appear to do so without the final state
being an eigenstate of a single determinate record. Secondly, the theory would have to account
for the standard quantum mechanical statistical predictions which are given by the collapse
postulate. Finally the theory would o↵er some empirical justification for the Relative State
Formalism to accept it as true. Many theories which attempt to fulfil atleast some of these
conditions have been published and will be examined in the following sections. [58] [59]
6.3.3
Many-Worlds Theory
The Many-Worlds theory is the most popular interpretation of Everett’s original idea. It was
published by DeWitt in 1970 in a paper named Quantum Mechanics and Reality [60]. Commonly
the name Many-Worlds is applied for the entire set of theories which derive from the Relative
State Formalism, but only this theory actually states splitting of the physical world into multiple
future worlds. The Many-Worlds theory states that for each term in the final state of the
measurement described by Everett there is a di↵erent physical world which would allow for
determinate measurement records from every observer. Mathematically this requires the state
to be written in the preferred basis such that each term obeys this condition.
Several problems with Many-Worlds theory have been proposed. A simple objection which
many researchers use to reject the theory is that it violates Occam’s Razor [61]. It seems very
counter-intuitive to create a large number of nearly identical worlds. However, this reduction
of physical laws and rejection of the collapse postulate is also in the spirit of Occam’s Razor. A
less known reason, but more severe problem is the need for a preferred basis in Hilbert space.
A standard quantum mechanical state can be written in any basis in which one pleases, but
the Many-Worlds theory forces a basis for which all terms represent determinate measurements.
The second condition outlined to amend the relative state formalism is to recreate the standard
quantum mechanical statistics. In the example above the state “up” would be reached with
a certain probability, but Many-Worlds tells us that each state is in a physical world, so each
state is in-fact reached with a probability of one. The final problem with Many-Worlds theory
which has been explored is that it is incompatible with special relativity due to the formation
of an entirely new space-time at each split.
6.3.4
Bare Theory
David Albert and Barry Loewer published a paper in 1988 named “Interpreting the Many
Worlds Interpretation” [62] which presented a theory which kept intact Everett’s disregard for
the collapse postulate, though did intend to keep the eigenvalue-eigenstate link. The theory
does not attempt to explain why the observer would end with a determinate experience, but
instead explains why the observer would believe that it obtained a determinate result.
By restating the question of what the result was from the observer to whether the observer got
a specific determinate result, Albert and Loewer were able to justify the observer’s statements,
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6.3
Many-Worlds Interpretations
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since if we consider a result with either “up” or “down” the observer will say it determinately
has this result. Albert and Loewer attempted to explain the observer’s determinate experiences
with an illusion.
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7
Recent Developments and Applications
7.1
Quantum Computing
Quantum computation is the study of possible applications and exploitation of quantum mechanical e↵ects with regards to computation, complexity, cryptography and communication.
The subject is considered to be a hybrid of theoretical computer science and quantum physics.
The field emerged in mid-1980s and currently is subject to wide international research. The are
three main reasons why this area of study is significant:
1. Fundamental issues: There is a deep fundamental concoction between physics and
computation. Computation is “processing” of “information”. Information is always represented in physical degrees of freedom of a physical system so a computer is always a
physical device. The states of the physical system are distinguished by the two bits, 0 and
1. “Processing” is the physical evolution of the system. Therefore, possibilities, limitations of information storage and efficiency of computation all depend on the law of physics
since it characterises the kinds of evolutions that are allowed. In computer science, a theoretical model is first set up to study computations. These standard models are built based
on classical physics. However, taking quantum physics as the starting point instead will
lead to di↵erent models which in turn provide new modes of computation that are not
available in the formalism of standard classical computing. These new models are actually
“available for implementation” in the real world computer technology and are not just a
formal abstract.
2. Technological issues: One of the most important issues in computational complexity
theory is the existence of polynomial time algorithm which is bridged by quantum computation. An example is polynomial time quantum algorithm for integer factorisation
discovered by Peter Shor in 1994 which has no classical analogue. Since 1965 a steady
rate of miniaturisation of computer components has been observed, in accordance with
Moore’s law, by approximately a factor of 4 every 3.5 years. At this rate, classical physics
fails completely and quantum e↵ects become dominant. Therefore, we could either redesign the components in such a way that they provide the same functionality as before,
or instead embrace new quantum e↵ects with the aim to explain them in new types of
computational methods. The next issue addressed why we need to choose the latter
option.
3. Theoretical issues: A quantum computer cannot compute anything that is classically
uncomputable but one of the main benefits of quantum versus classical computing is
the issues of computational complexity. That is to say, quantum computation can solve
computational task exponentially faster that any known classical algorithm. By this we
mean exploiting completely new (quantum) types of computational steps and not merely
increasing the speed of steps. As well as that, quantum physics as great implications for
issues concerning communication, an example of which is quantum teleportation as well
as important cryptographic issues such as implementing secure communication.
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Extensions and Variations of the Copenhagen InterpretationThe University of Edinburgh
Quantum computation is based on a “real” physical theory and is intended to be a realisable
technology. So far, the individual constituents of quantum algorithms have been demonstrated
successfully by experiments. Having said that, the construction of a “scalable” device to carry
out large computations is beyond current technology and lab experimentation. [63]
To develop a feel of the large scale of information that can be stored in quantum computation
consider the following state:
| iA = a0 | "iA + a1 | #iA
(44)
where | iA is a general state what we call a qubit, A. Note that it is in superposition of the
two states. The two arbitrary complex numbers satisfy
|a0 |2 + |a1 |2 = 1
(45)
We now extend the argument to the two qubit states
|
product i
= | iA | iB
(46)
which is not the most general state of the two quit system while | iA and | iB are the general
states for each individual qubit. The EPR state mentioned in one of the previous chapter is an
example of such a state. i.e.
|
EPR |
1
= p (| "iA | #iB
2
| #iA | "iB )
(47)
where the state of quibit A depends on the outcome of measurement of qubit B. We say that
such a state is entangled. Extending further, one can consider N qubits but first consider the
state of N classical bits characterised by N numbers
x = {s1 , s2 , ..., sN }
(48)
with si = 0, 1 being the i th classical bit. The number of possible states is therefore 2N . The
classical bit can either be |0i or |1i but a quibit can be in a superposition of |0i and |1i states.
So here we have:
X
| iN qubits =
cx |xi
(49)
0X<2N
where we have denoted that state of N classical bits by |xi. Clearly,
P the 2general N quibit state
N
is specified by 2 complex amplitudes cx subject to the constant x |cx | . For say, N=100 this
is already a very large number which arises due to the fact that the general state of N quibits
is highly entangled and this entanglement contains a lot of information! [64]
In the last fifty years the number of distinct interpretations of quantum mechanics has nearly
doubled. For the most part these schemes extend older interpretations, though some do exist
almost entirely independently. Possibly as a result of the sophistication of these approaches,
relative to the conventional Copenhagen interpretation; many of them remain only in active
use within the areas of theoretical research where they are most relevant. In addition to their
conceptual complexity, many of them also include additions to the formalism. While these
additions do not change the applicability of more conventional Quantum Mechanics, they do
extend it somewhat in an attempt to tackle what some see as unanswered questions.
7.2
Extensions and Variations of the Copenhagen Interpretation
The Copenhagen interpretation is one of the most commonly applied interpretations of quantum
mechanics and has been said to have almost as many variants as it has adherents. Three of
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7.3
Alternative Interpretations
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these variants that have gained the status of distinct interpretations will be discussed here, the
first being the consistent histories interpretation.
The consistent histories interpretation, first proposed in 1984 [65] maintains the original postulates of the Copenhagen interpretation as stated in section 4.4 with the addition of ‘histories’.
These are formal constructs that are represented as ‘History Projection Operators’, made up
of the tensor product of a time ordered set of projection operators. Each of the projection
operators is associated with a time and represents a proposition about the state of the system
in question. In e↵ect the history operator represents a series of questions that can be asked
about the system at their respective times with their respective answers. Within this extension
of the formalism is contained a method to assign probabilities to each history and measure the
consistency of the histories of a system [66]. Using this method it is possible to to ascribe a
set of such consistent histories to a system. According to this interpretation it is this set of
consistent histories which determines what questions about the system have simultaneously well
defined answers. It is important to note that these histories are constructed from the standard
formalism of quantum mechanics and therefore this interpretation is in a sense a generalisation
of the Copenhagen interpretation.
The second of the three interpretations that could be seen as an extension of the Copenhagen
interpretation is the class of interpretations known as the ‘Objective Collapse Theories’. Within
this class of models, both the wave function and its collapse are considered to be objectively
real, and the wave function collapse is stimulated by some physical threshold being reached,
without reference to any observer. Inevitably these interpretations come up against the problem
of describing this objective collapse and in an e↵ort to overcome it have, in some cases, proposed
modifications to the formalism. Thus in the strictest sense many interpretations of this form
are technically theories. One particular example of this kind of ‘interpretation’ is the Penrose
interpretation proposed in 1989 by Roger Penrose [67] as an attempt to marry quantum mechanics and general relativity. Within this scheme as with all objective collapse interpretations
the wave function represents a physical wave and the particle that it represents are thought of
as actually occupying several places at once. If the energy di↵erence between the possible states
of the particle is of the order of the Planck mass then, according to Penrose, the superposition
breaks down, the wave function collapses and the particle becomes localised.
The third and most modern interpretation discussed here is the relational interpretation which
is attributed to Carlo Rovelli and was developed in 1996 and is known as relational quantum
mechanics [68]. This interpretation of quantum mechanics borrows from Einstein’s theory of
relativity in that its core idea is that of only defining physical observables with respect to some
frame of reference, or more precisely, some observer. Another important aspect of the interpretation is that all systems are quantum systems, that is to say that observation is downgraded to
interaction, and the observer to another quantum system. This has a remarkable consequence
for the interpretation of the state vector; it is not a representation of the absolute state of a
system, but instead a representation of the correlations of a set of degrees of freedom of a of
a pair of quantum systems. This approach is completely compatible with the consistent histories interpretation and does not edit or add to the formalism of quantum mechanics in any
way.
7.3
Alternative Interpretations
In this section two interpretations will be discussed - the elementary cycles interpretation and
the is the transactional interpretation which attempt to explain wave function collapse as the
superposition of forward and backward in time quantum wave functions.
The elementary cycles interpretation was developed in 2012 by Donatello Dolce. It formulates
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Unscrambling the Foundations of the Theory
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relativistic quantum field theory by starting from the postulate of space-time periodicity. This
postulate originates with de Broglie who, in his PhD thesis assumed the “existence of a certain
periodic phenomenon of a yet to be determined character, which is to be attributed to each and
every isolated energy parcel (elementary particle) [69]. The application of modern mathematical
technology to this idea of intrinsic periodicity can be used to derive, according to Dolce, the
well known results of modern quantum field theories [70].
Possibly the most exotic interpretation discussed in this section is the transactional interpretation. It involves the use of both the Schrödinger equation and its complex conjugate to obtain
solutions that travel both forward and backward in time respectively (in the non-relativistic
formulation). In this interpretation, developed in 1986 by John Cramer, the superposition of
the forward and backward propagating solutions provides a systematic explanation for wave
function collapse [71]. An event such as the exchange of a photon between a emitter and a
detector could be summarised in a slightly simplified form as follows: The emitter produces advanced and retarded waves which are exactly time symmetric about t = 0, as does the detector
at about t = td . Outside the region 0  t  td the waves exactly cancel and within the region
they constructively interfere. This means that for t > td what looks like a collapse of the wave
function can be understood purely as an interference phenomenon.
7.4
Unscrambling the Foundations of the Theory
At this stage it is helpful to reflect on Schrödinger’s failed attempts25 to find, if it existed,
some physical reality in | i. Although a consistent “physical” interpretation for | i has not
been revealed since its inception in 1925, the question of whether it represents anything real
about physical reality can be asked. Inherent in this debate is the ancient ontological problem
between subject and object: can we distinguish between objective reality and our subjective
representation of it? Does | i represent our subjective knowledge about phenomena or does it
represent something objectively real? In a recent paper On the reality of the quantum state [72],
published in Nature on 6 May 2012, the authors consider just this question. The paper considers
two options for the nature of | i:
1. | i corresponds to a physical property of the system.
2. | i is a representation of the observer’s information about some aspects of the physical
properties rather than a physical property itself.
The authors acknowledge that if (1) is true then the wave function is an “odd kind of wave”
being defined in abstract configuration space rather than three- dimensional, the collapse must
also then be some mysterious physical process. However, certain phenomena such as interference - exhibited in the double-slit experiment - are understood precisely though interference of
real physical waves conforming to the interpretation summarised in (1). If (2) is true then the
collapse need not be mysterious but merely the “instantaneous Bayesian updating of a probability distribution upon obtaining new information”. However, this provides no interpretation
of the detailed dynamics of experiments such as the double-slit. The paper quotes Edwin T.
Jaynes:
“But our present[quantum mechanical] formalism is not purely epistemological; it is
a peculiar mixture describing in part realities of nature, in part incomplete human
information about nature - all scrambled up by Heisenberg and Bohr into an omelette
that nobody has seen how to unscramble. Yet we think that the unscrambling is
a prerequisite for any further advance in basic physical theory. For, if we cannot
25
see section 3.2.1
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Unscrambling the Foundations of the Theory
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separate the subjective and objective aspects of the formalism, we cannot know what
we are talking about; it is just that simple.”
The paper presents a “no-go” theorem that claims (2) must be false: “If the quantum state
merely represents information about the real physical state of a system, then experimental
predictions are obtained which contradict those of quantum theory”. They begin by defining,
“physical property: some function of the physical state” and later giving the example of energy
H(x, p). They refer to classical physics commenting that sometimes the exact physical state of
the particle might be uncertain, but there is nonetheless a well-defined probability distribution
µ(x, p). Though its time evolution is dictated by Liouville’s equation, it does not directly
represent reality but rather the experimenter’s “state of knowledge” about the physical state of
the particle.
The main assumptions in the theorem are:
1. For any micro-system there exists a “real physical state” - not necessarily completely
described by quantum theory, but objective and independent of the observer (which only
needs to hold for systems in isolation and not entangled with other systems).
2. Systems prepared independently have independent physical states
Both of these assumptions are in general very acceptable classically. Where the final statement
of the first: “independent of the observer” being more debatable, but still completely possible,
within quantum mechanics. Additionally assume some theory or model, perhaps undiscovered,
associates a physical state to the system. If the quantum state is prepared in a particular way
then quantum theory associates a quantum state . need not be uniquely fixed by preparation
but the preparation produces, as unknown to the observer or apparatus, some physical state
according to some probability distribution µ ( ). Then for the purposes of the theorem
consider the preparation of two distinct quantum states | 0 i and | 1 i with distributions µ 0 ( )
and µ 1 ( ) respectively. To understand the distinction between a quantum state representing
information or reality consider (Fig 7.1).
Figure 7.1: Consider a collection of probability distributions µl ( ) with denoting the systems
physical state. If every pair of distributions are disjoint as in a then the label l is uniquely fixed
by and we call it a physical property. If however there exists a pair of labels l and l0 with
distributions that both assign positive probability to some overlapping region , as in b, a
from
is consistent with either label and neither l or l0 can be a physical property
Classically we could consider position, x and momentum, p such that H(x, p) = E with distribution µE (x, p). Since the energy is a physical property of the system, di↵erent values of energy
E and E 0 correspond to di↵erent state distributions µE (x, p) and µE 0 (x, p) that are disjoint,
i.e. two di↵erent values of E cannot have the same (x, p) phase space coordinates. In comparison is like (x, p) such that is methodologically comparative to the energy - a physical
property.
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The main result of the theorem is to arrive at a contradiction if we begin with the hypothesis:
| i is a state of knowledge - its probability distribution µ ( ) can overlap with another.
Begin by preparing two distinct quantum
states | 0 i and | 1 i with distributions µ 0 ( ) and
p
µ 1 ( ) respectively with h 0 | 1 i = 1/ 2, choosing a basis of the Hilbert space so that
|
0i
= |0i , |
1i
p
= (|0i + |1i)/ 2
p
|±i = (|0i ± |1i)/ 2
(50)
If the distributions µ 0 ( ) and µ 1 ( ) overlap then there exists q > 0 such that the preparation
of either quantum state result in a from the overlap region with probability at least q. With
probability q 2 > 0 it happens that the physical states 0 and 1 (corresponding to quantum
states | 0 i and | 1 i) are both from the overlapping region . This means that the physical
state of the two systems is one of 4 possible quantum states 7.4.
{|0i ⌦ |0i, |0i ⌦ |+i, |+i ⌦ |0i, |+i ⌦ |+i}
(51)
The two systems are then brought together and measured. The measurement is an entangled
measurement, which projects onto the four orthogonal states 52.
1
|⇠1 i = p (|0i ⌦ |1i + |1i ⌦ |0i)
2
1
|⇠2 i = p (|0i ⌦ | i + |1i ⌦ |+i)
2
1
|⇠3 i = p (|+i ⌦ |1i + | i ⌦ |0i)
2
1
|⇠4 i = p (|+i ⌦ | i + | i ⌦ |+i)
2
(52)
The first possible outcome |⇠1 i is orthogonal to |0i ⌦ |0i and hence the predicted outcome has a
probability of zero if the quantum state was prepared as |0i ⌦ |0i. Similarly |⇠2 i has probability
zero if the sate is |0i ⌦ |+i, |⇠3 i if |+i ⌦ |0i and |⇠4 i if |+i ⌦ |+i. Thus at least q 2 of the time, we
could obtain no outcome - probability 0 outcome, and thus the probability distributions µ 0 ( )
and µ 1 ( ) can not overlap - a contradiction with the assumption that | 0 i and | 1 i represent
a states of knowledge and that their probability distributions overlap.
The paper then proves this for general | 0 i and | 1 i and then presents an argument that works
even in the presence of error and noise. If one accepts the assumptions and conclusion of the
theorem then the quantum collapse must correspond to a poorly defined physical process. The
theorem does not prove that | i has a unique physical reality but it provides a mechanism to
disprove the idea that | i is merely a state of knowledge and has no manifest in reality. If
experimental measurements provide evidence for validity of the claim then theories such as de
Broglie-Bohm pilot wave theory that claim | i represents something physically real would gain
increasing favour. The paper itself does seem a little muddled in semantics. The assumptions
could be clearer yet the mathematics is straightforward for the basic theorem. If assuming the
theorem is correct it appears that the conclusion is in support of hidden variables. They define
a physical property: ”some function of the physical state”. If the theorem is correct - | i cannot
represent our state of knowledge and that it is objectively real, we arrive at hidden variables,
and thus to a non local theory. This conclusion of course must be taken with a pinch of salt
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Unscrambling the Foundations of the Theory
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as they do not suggest an explicit mechanism by which the objective | i is a “function” of the
physical state and they do not necessarily prove that | i even is real, yet this does seem to be
the only reasonable choice. The character of the paper is pioneering in its aims to narrow down
the possible interpretations and as a result is useful in illustrating the types of questions that
can be asked, regardless of the validity of the answer.
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8
Conclusion
The question of the nature of the quantum state is one that has preoccupied some of the best
minds of the last hundred years. In the thirty years starting from last decade of the 19th
century, physics transformed from a state of complete self-assuredness to a state of fracture and
self-doubt. The scramble to develop and interpret the new quantum theory from the ashes of
the classical was imperative for physics to be able to continue. For the first time there was a
theory whose elements could not be intuitively assigned physical meaning in the way that had
made the classical theory so appealing. The first and by now most conventional understanding
of this theory was the Copenhagen interpretation; though widely used this interpretation is
not without difficulties. Issues such as the reality of the wave function and the nature of its
collapse have pushed physicists to develop many varied understandings of the theory including
ideas as exotic as parallel universes and atemporal interpretations.
In order to develop a modern understanding of the underlying reality of the quantum state,
if any, a return to the heart of the interpretation is required. As we have seen, there have
been many attempts to address this since the birth of quantum theory. Wrapped up in the
interpretation is the ontological issue of the distinction between subject and object, and as a
prerequisite for progress we must attempt to separate the subjective and objective aspects of
the formalism. The work of John Bell and the recent PBR paper in providing “no-go” theorems
illustrates the possibility of distinguishing true interpretation from theory. The approach taken
in the above examples is instrumental in refining our idea about what we can and cannot
say about the quantum world, and by extension about the wider world. It is through the
insight gained from developments of this type that the application and development of quantum
mechanics gains ever more ground.
Throughout history interpretations have not altered the predictions of quantum mechanics.
Until these interpretations are properly formulated and tested in a rigorous fashion, one should
suspend judgement. As a scientific body we should accept that in our description of nature,
the objective is not necessarily to disclose the “real” essence of phenomena but to formulate
relations between di↵erent aspects of them. Furthermore, we may never know all of what can be
said about the quantum nature of the universe and so the ability to know what cannot be said
must be regarded as a most valuable tool. In conclusion, we should be careful in our approach to
understanding the quantum state, treating it as an entity that leads to predictions, irrespective
of whether or not it carries an underlying “reality”. Any progress made in the interpretation of
the theory must surely come in the form of that which more precisely defines the boundary of
what can be considered an interpretation.
57
Appendices
A
Quantum State Superposition
We start by discussing the Stern-Gerlach experiment with electrons. Consider the two measurable physical properties of electrons - the x and z direction. The spin-x, the electrons can take
only two possible values, “x" ” or “x# ” corresponding to the values 1/2 and 1/2 respectively.
Similarly, electrons can be either “z" ” or “z# ”.
As shown in the figure, “SGX box” measures the the electron spin in the x direction. It has
an input in which the electrons are fed into from the left, and two output apertures. Any
x" electron that enters the box leaves it from the top aperture marked b. Similarly, every x#
electron leaves the box from the top aperture. The spin value of each electron can be later
on inferred from its final position. The situation is similar for the SGZ box. Note that the
measurements are repeatable. Also if, for example, a SGX measurement of an electron yields
the result x" and if the electron is subsequently fed into another SGX box it will certainly
emerge from the x" aperture of the second box as expected. It now becomes interesting to see
whether or not the two properties of the electrons, spin-x and spin-z, might be related in any
way. It turns out that there is no correlation between the two values. Let us examine what will
happen if we set up a sequence of three boxes as follows: SGX-SGZ-SGX.
Figure A.1: Stern-Gerlach Experiment
In standard representation,
|z" i =
✓ ◆
1
0
|z# i =
✓ ◆
0
1
(53)
where |z" i and |z# i are eigenvectors of the spin-z operator
✓
◆
1
1 0
ŝz = ~
0
1
2
with eigenvalues ± 12 . Similarly
1
|x" i = p
2
✓ ◆
1
1
1
|x# i = p
2
58
(54)
✓
1
1
◆
(55)
Quantum State Superposition
with the spin-x operator
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✓
◆
1
0 1
ŝx = ~
1 0
2
(56)
When the experiment is performed half the emergent particles are spin x" and the other half are
spin x# , even though x# ones leave the box from the top aperture and as a result are not input
in SGZ. No matter how the SGZ box is built and no matter what we do to ensure the initial
physical properties of the electrons are the same, the statistics of the result will remain the same
emerging fifty-fifty from the two apertures. Therefore, the measurable physical properties, here
being SGX and SGZ, are said to be “incompatible” with each other since the measurement of
one disrupts that of the other.
We will now change the experimental setup (see figure H.1). The SGZ box is placed on the
left hand corner of the setup. Route d corresponds to z# electrons and route u to z" ones. z#
electrons emerging from that box, travel along route d, whose direction of motion is changed
due to the “mirror” placed in this route. Note that the mirror merely changes the direction of
the electrons and not spin-z or spin-x. Another box, called the m-box is placed on the top right
corner of the setup. This box changes the direction of motion of the electrons without altering
their z-spins. It makes the routes u and d coincide as they pass through it. A z" electron will
emerge along u d as a z" electron. The same goes for a z# electron. Suppose a x# electron
is fed into the SGZ box, in the final stage along u d we can measure its z-spin. Half of such
electrons will end up at u d, having taken route u as z" electrons and the other half having
taken route d as z# ones. This is what we expect since nothing has occurred in between these
stages to alter SGZes. If z" electron are fed into the SGZ box and their x-spins are measured
at u d, after having taken path u, 50% of the time they turn out to be x" and 50% of the
time they will be measure to be x# . Suppose instead that at the start of the experiment a x#
electron is fed into the SGZ box. It turns out the “all” the x# electrons input to the setup at
the start will come out as x# in the end.
Now we alter the experiment a little bit. There is also have a movable wall that can stop
electrons, which can be slid in and out route d. Hence, when the wall is in, it stops electron in
route d and only those it route u make it through to u d. When the experiment is conducted,
the output will be down by half (as expected). However, half will be back and the other half will
be x# . At this point we need to analyse the situation by thinking about the possible routes an
electron can take. Consider an electron passing the apparatus when the wall is out. It cannot
have taken route u since, as we now know, the electrons that take such route have 50-50 SGX
statistics which is in contrast with our results at u d where all electrons are measured to be x# .
An similar situation applies for electrons taking route d. One might suggest that the electron
has taken both routes. However, during the experiment if one stops the experiment to observe
where the electron is, half the time if is found to be on u and half the time on d. Also, we cannot
claim that it has taken neither since if one puts a wall at both routes nothing will appear at
the output. Therefore, the electron is not taking u nor d or both or neither since these are the
only possibilities. The question now is this: what are these electrons doing? We say that the
electron is in a new mode, called “superposition” - the electron is in a superposition of being
on d and being on u.
One of the standard examples explaining superposition is the familiar double slit experiment
where the electron emerging from the source, pass through the slits and hit the fluorescent
screen at the back. If the top slit is closed, they arrive at the screen with intensity shown in
figure A.2.
The case where the bottom slit is closed is similar. However, when both slits are open (fig A.3)
the intensity patterns is characteristic of a wave rather than a particle. Just as we have seen
before it is impossible to tell which slit each individual electron has gone through. In other
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(a) Upper slit blocked
(b) Lower slit blocked
Figure A.2: Double slit experiment with one slit blocked
words, it is in a superposition of both slits.
Figure A.3: Interference pattern for the double slit experiment
One can conclude that we cannot talk about the x-spin and z-spin of the electron at the same
time. In other words, the fact that the electron has a definite z-spin, say, entails that it is
neither x# nor x" , nor both, nor neither. It is what we call “superposition” of being x" and x# .
Yet, we know that any x-spin measurement will either yield z" or z# implying that the outcome
of x-spin measurement must be a matter of “probability”. [73]
B
Einstein’s Derivation of Planck’s Formula from the Bohr
Model
The following provides a superficial but nonetheless insightful derivation of Planck’s formula by
Einstein.
Following the scientific community’s ratification of Bohr’s frequency formula,
h⌫ = En
Em
(57)
Einstein argued that the blackbody radiation spectrum should also abide by this expression.
In particularly he concentrated on jumps between a state 1 and an excited state 2. Firstly he
considered the emission of radiation in time dt. This amounts to N2 ’atoms’ dropping down
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to state 1 with probability P2 / N2 dt. Next, Einstein proposed that N1 atoms could absorb
radation and jump to the excited state 2 with probability P1 / u⌫ N1 dt, where u⌫ is the radiation
density. By including the radiation density Einstein takes into account that only radiation of a
certain frequency may be absorbed according to equation (57).
By adding in constants26 A and B of proportionality and considering a high temperature (classical) regime where N1 ⇡ N2 and u⌫ is large we reason that
Bu⌫ N1 dt
AN2 dt
(58)
emission
(59)
or
absorption
As one would expect the amount of radiation and emission to be equal27 , Einstein introduced
a probability of spontaneous (or induced) emission Pspon = Bu⌫ N2 .
Bu⌫ N1 dt = (A + Bu⌫ )N2 dt
(60)
N2
Bu⌫
=
N1
A + Bu⌫
(61)
i.e.
Combining the Maxwell-Boltzmann distribution and Bohr’s frequency formula gives
N2
=e
N1
(E2 E1 )/kT
)
N2
=e
N1
h⌫/kT
(62)
where as usual E1 and E2 are the energies of the states involved in transition and ⌫ is the
frequency of the radiation. Einstein then went on to equate equations (61) and (62) in order to
obtain
Bu⌫ = (A + Bu⌫ )e
h⌫/kT
(63)
i.e.
u⌫ =
A
1
h⌫/kT
Be
1
(64)
which we recognise as Planck’s formula upon letting
A
8⇡h⌫ 3
=
B
c3
(65)
26
To declare A and B for constants is a bit ambiguous. A and B may in fact be functions of ⌫. However, since
we are working at an (arbitrary) fixed frequency, A and B are constants with respect to u⌫ . In litterature, A and
B are often denoted as Einstein coefficients.
27
In the original 1917 paper [74], Einstein dicusses momentum transfers between molecules (here ’atoms’) and
radiation: In classical theory emission occurs as spherical waves, whereas absorption is directional. Einstein
argues that in quantum theory one ”shall have to assume that all directions of emission are equally probable”.
Hence we may equate radiation and emission.
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Interpretations of the Correspondence Principle
C
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Interpretations of the Correspondence Principle
According to [12], Bohr’s writings gave rise to three di↵erent understandings of the correspondence principle, namely the frequency interpretation, the intensity interpretation, and the
selection rule interpretation. To understand these interpretations, it is worth establishing some
simplifying notation. Suppose an electron jumps from state n0 to state n00 and radiation of
frequency ⌫n0 !n00 is emitted. Then let ⌧ = n0 n00 be the energy di↵erence. Now consider a
classical picture, where an electron is undergoing simple periodic motion with frequency !. One
can then describe this (classical) trajectory as a Fourier series:
x(t) = C1 cos(!t) + C2 cos(2!t) + C3 cos(3!t) + ...
(66)
This is a sum of harmonics, where the ⌧ th harmonic has amplitude C⌧ and frequency !⌧ =
⌧ !.
The frequency definition then states that the quantum frequency equals the classical frequency
in the large n limit:
⌫n0 !n00 = !⌧ = ⌧ !, for large n
(67)
Bohr justified this by referring to the hydrogen atom. Recall from figure 2.1 that in the large n
limit the energy levels gets so close to each other that they resemble a continuous spectrum. In
this limit, the frequencies ⌫n 1 !nn and ⌫n!nn+1 become indistinguishable from each other and
the choice of frequency for ⌫n0 !n00 becomes unimportant and we recover the classical limit [9, pp.
119-122]. For a more detailed description, refer to [12].
The intensity interpretation uses a similar argument of statistical asymptotic agreement, namely
that
Pn0 !n00 / |C⌧ (n)|2 , for large n
(68)
where P is the probability of transition between two stationary states.
Finally and most importantly is the selection rule interpretation. [12] defines it succinctly:
“Bohr’s selection rule states that the transition from a stationary state n0 to another stationary
state n00 is allowed if and only if there exists a ⌧ th harmonic in the classical motion of the electron
in the initial stationary state; if there is no ⌧ th harmonic in the classical motion, then transitions
between stationary states whose separation is ⌧ are not allowed quantum mechanically.” It is
worth noting that the selection rule interpretation makes no assumption about being in the
large n limit.
D
The Formal Equivalence of the Wave and Matrix Formalisms
Von Neumann showed in his 1932 book Mathmematische Grundlagen der Quantenmechanik
that “quantum mechanics can be formalized as a calculus of Hermitian operators in Hilbert
space and that the theories of Heisenberg and Schrödinger are merely particular representations
of this calculus. Heisenberg made use of the sequence space I2 , the set of all infinite sequences of
complex numbers whose squared absolute values yield a finite sum, whereas Schrödinger made
use of the space L2 (1, +1) of all complex-valued square-summable (Lebesque) measurable
functions; but since both spaces, I2 and L2 , are infinite-dimensional realizations of the same
abstract Hilbert space H , and hence isomorphic(and isometric) to each other, there exists a
one-to-one correspondence, or mapping between the wave functions of L2 and the sequences of
complex numbers of I2 , between Hermitian di↵erential operators and Hermitian matrices. Thus
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Bohr-Schrödinger Debate 1926
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solving the eigenvalue problem an operator in L2 is equivalent to diagonalizing the corresponding
matrix in I2 .”
E
Bohr-Schrödinger Debate 1926
Heisenberg recalled the further discussion between Bohr and Schrödinger as follows:
Bohr: “ What you say is absolutely correct. But it does not prove that there are no quantum
jumps. It only proves that we cannot imagine them, that the representational concepts with
which we describe events in daily life and experiments in classical physics are inadequate when
it comes to describing quantum jumps. Nor should we be surprised to find it so, seeing that the
processes involved are not the objects of direct experience.”
Schrödinger: ”I don’t wish to enter into long arguments about formation of concepts; I prefer to
leave that to the philosophers. I wish only to know what happens inside the atom, and if these
are particles- as all of us believe then they must surely move in some way. Right now I am not
concerned with a precise description of this motion, but it ought to be possible to determine
in principle how they behave in the stationary state or during the transition from one state to
the next. But from the mathematical form of the wave or quantum mechanics alone it is clear
that we cannot expect reasonable answers to these questions. The moment, that we change the
picture and say that there are no discrete electrons, only electron waves or waves of matter, then
everything looks quite di↵erent. We no longer wonder about the fine lines. The emission o light
is as easily explained as the transmission of radio waves through the aerial of the transmitter,
and what seems to be an insoluble contradiction has suddenly disappeared.”
Bohr: “I beg to disagree. The contradictions do not disappear; they are simply pushed to
one side. You speak of the emission of light by the atom or more generally of the interactions
between the atom and the surrounding radiation field, and you think that all the problems are
solved once we assume that there are material waves but no quantum jumps. But just take
the case of thermodynamic equilibrium between the atom and the radiation field, remember,
the energy of the atom should assume discrete values and change discontinuously from time to
time, discrete values for the frequencies cannot help us here. You cant seriously be trying to
case doubt on the whole basis of quantum theory!”
Schrödinger: “I don’t for a moment claim that all these relationships have been fully explained.
But then you, too, have so far failed to discover a satisfactory physical interpretation of quantum
mechanics. There is no reason why the application of thermodynamics to the theory of material
waves should not yield a satisfactory explanation of Planck’s formula as well- an explanation
that will admittedly look somewhat di↵erent from all precious one.”
Bohr: “No, there is no hope of that at all. We have known what Planck’s formula means for the
past twenty-five years. And, quite apart from that, we can see the inconsistencies, the sudden
jumps in atomic phenomena quite directly, for instance when we watch sudden flashes of light
on a scintillation screen or the sudden rush of an electron through a cloud chamber. You cannot
simply ignore these observations and behave as if they do not exist at all” [27, p. 824]
F
The Problem with Ensembles
There exist numerous proofs of why Einstein’s ensemble picture can’t be true. Here we simply
consider a few. See [9, pp. 213-217] for more rigorous examples.
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Schrödinger’s Cat
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Our first example is one Schrödinger presented in 1935, which focuses on the (quantum mechanical) simple harmonic oscillator. We start by considering a certain energy level, En . Then if we
denote the potential energy by Ep and the kinetic energy by Ek we must have that En = Ep +Ek
at all times. Following Einstein’s explanation of the ensemble theory we have that both velocity
and position should take definite values for all t and hence should Ek and Ep .
Consider now a special case where Ek is zero and so Ep = En . Since Ek cannot be negative, we
must have that Ep is at a maximum which in turn implies that the value of position must also
be sharply peaked. However, this is in disagreement with experiment and theory which tells us
that there is no threshold value for position [9, pp. 214].
Another relatively simple example is one of a decaying particle. Inside the nucleus and at large
radii the particle has 0 < Ep < Etotal . However, just outside the nucleus there exists a classically
forbidden region of high potential energy. Quantum mechanics tells us that the particle has
a finite probability of tunnelling through this region and reaching the classical region at large
radii. Nonetheless, Einstein’s ensemble dictates that some particles should be present within
the forbidden region giving rise to negative kinetic energy, which, of course, is forbidden.
G
Schrödinger’s Cat
In his sustained correspondence with Einstein, Schrödinger became increasingly disturbed and
critical of the Copenhagen interpretation. While searching for the most absurd consequences
of the theory, he introduced Schrödinger’s Cat, perhaps the most commonly misinterpreted
thought experiment of modern physics:
A cat is penned up in a steel chamber, along with the following device (which must
be secured against direct interference by the cat): in a Geiger counter, there is a tiny
bit of radioactive substance, so small that perhaps in the course of the hour, one
of the atoms decays, but also, with equal probability, perhaps none; if it happens,
the counter tube discharges, and through a relay releases a hammer that shatters
a small flask of hydrocyanic acid. If one has left this entire system to itself for an
hour, one would say that the cat still lives if meanwhile no atom has decayed. The
psi-function of the entire system would express this by having in it the living and
dead cat mixed or smeared out in equal parts.” [add reference]
While all discussion of quantum mechanics had so far concerned unfamiliar microscopic objects
such as electrons, Schrödinger now applied it to macroscopic, everyday objects. By the contemporary thinking there was a wave function that described the cat in a simultaneously dead and
alive state. Furthermore, it is the act of opening the box and observing that kills or saves the
cat, as opposed to the cat being inherently alive or dead. Schrödinger also sought to highlight
the boundary between the observed and observing systems.
Many years later Hungarian physicist Eugene Wigner extended the idea of Schrödinger’s Cat
with Wigner’s Friend - now replacing the cat with a human being. This time the human does
not necessarily have to be poisoned, only observe when the particle decays. The question is
- can the “friend” collapse the wave function of the particle or must he wait for the outside
observer to observe them both and in doing so cause collapse? Wigner believed this implied the
world would be solipsistic. The di↵ering sensations and impressions meant di↵erent underlying
wave functions for the two observers and ultimately implied there was no true reality.
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Quantum Mechanical Formalism
H
The University of Edinburgh
Quantum Mechanical Formalism
The following is an outline of the current requirements of a mathematical formalism of quantum
mechanics, expanded from the end of section five.
We start from the five main principles:
I Physical States: Each physical state is represented by a vector called the state vector. The
states picked by all such vectors correspond to all possible physical situations of the system.
Note that the correspondence is not one to one, due to that fact that a “phase” factor of
unit magnitude does not change the physical state of the system. One can superimpose
states by adding and subtracting two vectors in the space.
II Measurable properties: Observables are represented by Hermitean linear operators on
the vector spaces. This is directly related to the formalism.
III Dynamics: If the state of any physical system is given at some initial time, it is possible
to calculate its state at a later time assuming the forces and constrains involved are known.
In other words, there exist a “dynamics” of the state vector i.e. there are deterministic
laws regarding how the state vector, subject to certain forces and constraints, evolves in
time. The equation describing this time evolution is known as the Schrödinger equation.
IV The Connection with Experiment: From part II, we know that if the state coincides
with one of eigenstate of the system, the result is certain with probability equal to 1, to
yield the eigenvalue corresponding to that eigenvector. Otherwise, we cannot predict with
certainty the outcome of a measurement. When the state is in a superposition of the
eigenstates of say, the SGX operator, the result of the SGZ measurement is a matter of
probability, in this case being equal to 1/2 for getting z" and 1/2 for getting z# .
V Collapse: Immediately after a measurement that gave a certain result i.e. eigenvalue, the
system is in the eigenstate corresponding to that eigenvalue. In other words, the state is
such that if the measurement is repeated, it is guaranteed that the same result will be
obtained. This implies that the e↵ect of measuring and observable is to change the state
vector of the system by collapsing it, from whatever state it may have been in, into the
eigenvector of the measured observable. However, which particular eigenstate in collapse
into is a matter of probability for the outcome of the measurement.
Note that principle V implies that under particular circumstances, for example when a measurement occurs, the state vector evolves in a certain way, namely it collapses onto the eigenvector
of the observable being measured. However, principle III accounts the general evolution of the
state vector under any circumstances. The important point is now about consistency. From
what we now know, it seems as if V is a special case of III deducible from it. But this is nontrivial, because changes in the state vector is probabilistic in the case of V while those in III
are deterministic. We will address this issue more in the section on the measurement problem.
From quantum mechanics we know that a wave function is a complex entity that describes the
quantum state of a system. It encodes all the information about the system. So any measurable property of particles such as momentum and energy it representable as an operator on the
wave function. Going even further, whatever that can be said about about the state vectors of
particles can be translated into wave functions’ language.
We now briefly discuss the situation for multi-particle systems. Consider a two particle system.
Particle A is in state | a i and particle B is in | b i, then the quantum state vector for the pair
is denoted by | a i1 | a i2 = | 1a , 2b i which represents a vector in the state space of the twoparticle system. It is important to understand what the principle of collapse (V) would mean
for such states. It is possible to construct two-particle systems in separable states, in which
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Quantum Mechanical Formalism
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Figure H.1: Two-Path Experiment
case the measurement of an observable of particle A, only a↵ects the state of the measured
particle, without a↵ecting that of particle B. However, this is not the case with non-separable
sates where the measurement made on particle A changes the quantum mechanical description
of the unmeasured particle B.
We now a return to the two path experiment described the the previous section in order to who
how the explanation is also directly encoded in the formalism. The principles of superposition
and incompatibility immediately imply:
1
1
|x" i = p |z" i + p |z# i
2
2
1
|x# i = p |z" i
2
1
|z" i = p |x" i +
2
1
|z# i = p |x" i
2
(69)
1
p |z# i
2
1
p |x# i
2
1
p |x# i
2
(70)
(71)
(72)
One can now follow each step of the experiment (see figure):
• At time t1 , the particles is fed into the apparatus whilst in the state
✓ ◆
1
1
|x# , X = x1 , Y = y1 i = p
|X = x1 , Y = y1 i
1
2
✓ ◆ ✓ ◆
1
1
0
=p
|X = x1 , Y = y1 i
0
1
2
1
1
= p |hi|X = x1 , Y = y1 i p |si|X = x1 , Y = y1 i
|
{z
}
{z
}
2
2|
|ii
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|ji
(73)
Quantum Mechanical Formalism
The University of Edinburgh
which is now represented in the z-spin basis.
• If the state at time t1 only contained the |ii term, then after passing through the SGZ
box, at time t2 , the state would have been
|z" i|X = x2 , Y = y2 i
(74)
Similarly if it only contained the |ji term it would have been
|z# i|X = x3 , Y = y1 i
(75)
However, the state is neither of the above - it is a superposition of the two. Therefore
at time t2 the state will be
1
p |z" i|X = x2 , Y = y2 i
2
1
p |z# i|X = x3 , Y = y1 i
2
(76)
It is important to emphasise that there exist a non-separable correlation between the
spin and coordinate-space properties of the electron. In other words, no coordinate space
properties, such as position, momentum, nor spin, have definite values here. Therefore,
it would for example be meaningless to talk bout whether the electron has taken the z"
path or the z# or both or neither.
• By the same analogy, the state of the electron at time t3 is
1
p |z" i|X = x3 , Y = y3 i
2
1
p |z# i|X = x4 , Y = y2 i
2
(77)
• Similarly at t4 the state will be
1
1
p |z" i|X = x5 , Y = y4 i p |z# i|X = x5 , Y = y4 i
2
2
1
= p (|z" i |z# i) |X = x5 , Y = y4 i
2
=|x# i|X = x5 , Y = y4 i
(78)
Which is what we found before. At this stage the spin and coordinate space states have
become separable again, i.e. the position of the electron has a definite value which results
in a definite x-spin as well.
Suppose now that the experiment is stopped in the middle by making a measurement of position
at t3 . Then, there will be no superposition and a collapse occurs. Hence the state just after the
measurement will be either of the following each with a probability of 12 :
|z" i|X = x3 , Y = y3 i
or
|z# i|X = x4 , Y = y2 i
(79)
or
|z# i|X = x5 , Y = y4 i
(80)
and the state at t4 will be
|z" i|X = x5 , Y = y4 i
respectively. If we put a wall into the z# path at (x3 , y1 ) and measures the position of the
electron at t4 , i.e. if one observes to see whether the electron had emerged from the d-box, the
probability corresponding to finding it at (x5 , y4 ) would be a half. If found at this point, it
would necessarily be a z" electron. A measurement of x-spin would then yield x" and x# with
equal probabilities confirming our previous results. [75]
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