Download Stochastic Models in Classical and Quantum Mechanics∗

Stochastic Models in Classical and
Quantum Mechanics∗
V.Yu. Terebizh†
Moscow State University, Russia
Crimean Astrophysical Observatory, Ukraine
27 August 2013
PACS 05.10.Gg, 03.65.-w, 42.50.Xa, 03.65.Ud.
Keywords: Statistical physics, Quantum mechanics.
Abstract
Characteristic features of the stochastic models used in classical
mechanics, statistical physics, and quantum theory are discussed. The
viewpoint according to which there is no room for probabilities in Nature, as such, is consistently substantiated; the researcher is forced
to introduce probabilistic concepts and the corresponding models in
conditions that provide only partial predictability of the phenomena
being studied. This approach allows one to achieve a consistent interpretation of some important physical phenomena, in particular, the
relationship between instability of processes and their irreversibility
in time, the stochastic evolution of systems in the theory of deterministic chaos, Boltzmann’s H-theorem, and paradoxes of quantum
mechanics.
∗
“Advances in Quantum Systems Research”, Chapter 8. Nova Publishers, Zoheir
Ezziane (Ed.), ISBN: 978-1-62948-645-1.
†
98409 Nauchny, Crimea, Ukraine; E-mail: [email protected]
1
A year or so ago, while Philip Candelas (of the physics
department at Texas) and I were waiting for an elevator,
our conversation turned to a young theorist who had been
quite promising as a graduate student and who had then
dropped out of sight. I asked Phil what had interfered
with the ex-student’s research. Phil shook his head sadly
and said, “He tried to understand quantum mechanics.”
Steven Weinberg, “Dreams of a Final Theory”, 1992
1. Introduction
Maximalism characteristic of youth so pronounced in the epigraph would unlikely
remain after several years of specific research. In the same book, Weinberg notes,
“Most physicists use quantum mechanics every day in their working lives without
needing to worry about the fundamental problem of its interpretation... But I
admit to some discomfort in working all my life in a theoretical framework that
no one fully understands.”
Another outstanding physicist, Richard Feynman [1985], conveyed his attitude
toward the orthodox quantum theory even more emotionally, “I can’t explain why
Nature behaves in this peculiar way. . . The theory of quantum electrodynamics
describes Nature as absurd from the point of view of common sense. And it agrees
fully with experiment. So I hope you can accept Nature as She is – absurd.” The
negative attitude of Albert Einstein and Erwin Schrödinger in this respect is well
known.
What did not suit the scientists to whom physics owes the introduction of the
concept of photons, the fundamental equation for the wave function, the overcoming of serious difficulties of quantum electrodynamics, and the creation of a
unified theory of electroweak interactions? (Nobel Prizes were awarded for all four
contributions.) It is often said that Einstein did not accept the probabilistic interpretation of the reality that quantum mechanics brought in. However, it should be
remembered that even in his first works, Einstein gave a theory of Brownian motion, independently built a basic framework of statistical mechanics, anticipated
an important probabilistic expansion that many years later was associated with
names of Karhunen and Loéve, introduced the probabilities of transitions in atoms,
and performed a number of other studies clearly showing that he subtly understood the probabilistic problems and had a good command of the corresponding
technique. Of course, that’s not the point; what matters is the understanding of
the researcher–Nature interrelationship expressed in Einstein’s famous aphorism,
“I cannot believe that God plays dice.”
2
It is natural to suppose that the task of science is to construct the simplest
model for the range of phenomena being studied that is consistent with experimental data and that has a forecasting power. Previously, the term theory was
commonly used instead of the term model; the latter term reflects better the incomplete, transient character of our understanding of Nature. Why on earth were
the probabilistic models of classical physics accepted by the physical community1 ,
while there is no universally accepted interpretation of quantum mechanics as yet?
And this is despite the fact that extensive literature, including an in-depth analysis of real and thought experiments, is devoted to the analysis of probabilistic
foundations of physics and the corresponding interpretation of quantum theory
(see, in particular, Bohr 1935; Mandelstam 1950; Fok 1957; Wigner 1960; Born
1963; Chirikov 1978; Kravtsov 1989; Alimov and Kravtsov 1992; Mermin 1998).
The approach suggested by quantum mechanics will become clearer if we initially trace the emergence of probabilistic models in classical physics. Even an
examination of a simple experiment with die gives clear evidence of this kind. The
objective of this chapter is, as far as possible, to facilitate familiarization with
the probabilistic problems of physics to avoid situations like that described in the
epigraph2 . We consistently substantiate the viewpoint according to which there is
no room for probabilities in Nature, as such; the researcher is forced to introduce
probabilistic concepts and the corresponding models in conditions that provide
only partial predictability of the phenomena being studied in both classical and
quantum mechanics. Light was diffracted by holes, atoms retained stability, and
dice and systems consisting of a large number of molecules demonstrated certain
regularities long before Christiaan Huygens, Blaise Pascal, Pierre Fermat, and
Jakob Bernoulli laid the foundations of the probability theory.
For coherence, we had to cursorily touch on some of the facts that are covered well in textbooks. From the probability theory, to understand the subsequent
material, it will suffice to know that a discrete random variable ξ is specified by
a set of values x1 , x2 , . . . , xN that it can take in an experiment and by the corresponding probabilities of these values p1 , p2 , . . . , pN . The number of values N
can be infinitely large; the probabilities must be nonnegative and add up to one.
It is very useful not to confuse the notations of the random variable itself and its
possible values. In applied research, such confusion often leads to misunderstandings; experts on the probability theory occasionally agree to that when there is a
shortage of symbols.
1
Not always unconditionally, though; it will suffice to recall the long-term debate concerning Boltzmann’s H-theorem.
2
See also the preliminary paper by Terebizh [2010].
3
2. Throws of a die and other unstable processes
Consider once again the old problem of the throws of a die by focusing attention
on the physical aspect of the situation. Suppose that such throws are made in
the following conditions: (1) the die has the shape of a cube whose edges and
angles are identical with a microscopic accuracy; (2) the inscriptions 1, 2, . . . , 6
that distinguish the faces from one another are made in such a way that their
mass is smaller than that of the die by several orders of magnitude; (3) the die and
the table on which it falls are made of an elastic material; (4) the initial orientation
of the die is always the same, say, face ‘6’ is directed upward and face ‘4’ is oriented
northward3 ; (5) the die is released from fixation in some delicate way that is now
not concretized; (6) the number written on the upper face of the stopped die is
taken as the result of a single throw. The problem consists in predicting the result
of a single throw of the die. We do not go into further details of the experiment –
the aforesaid is enough to understand the essence of the problem.
First, let the initial height h of the die above the table not exceed the cube edge
length a. Clearly, at such a small height, ‘6’ will prevail in a series of throws; the
faces that have been initially sideways will occur rarely, while many throws would
have to be made until the occurrence of ‘1’. In these conditions, the theory (model)
that allows the results of the experiment to be predicted successfully does not need
to invoke any concepts of the branch of mathematics called the probability theory.
We can just reckon ‘6’ to be the only possible result, consider the occurrence of
one of the side faces as a consequence of inexact adherence to the experimental
conditions, and consider the occurrence of ‘1’ as an exceptional event that requires
an additional study.
At h/a ≃ 1 − 2, the results of experiments will become more varied. For
their interpretation, we can invoke sophisticated means of recording the initial
position of the die and then calculate its motion on a supercomputer using the
corresponding equations of aerodynamics and elasticity theory. Obviously, in this
way, we will be able to achieve such a good accuracy of predictions that the
dynamical model of the process will be considered acceptable.
However, a much simpler, probabilistic, model turns out to be also useful even
here. It does not require a high accuracy of the information about the initial
position of the die and laborious calculations. This model postulates that the
number written on the upper face is a random variable ξ that takes one of the
values 1, 2, . . . , 6 with probabilities p1 , p2 , . . . , p6 . Analysis of the results of a series
3
The die is assumed to be numbered in a standard way, so that the sums of the numbers
on opposite faces are equal to 7.
4
of throws of the die from a fixed initial height shows that for the subsequent
results to be predicted successfully, we must assign the highest probability, p6 ,
to the upper face, slightly lower probabilities, p2 = p3 = p4 = p5 , to the four
side faces, and the minimum probability, p1 , to the lower face. Of course, all
of the introduced probabilities depend on the initial height of the die. As the
ratio h/a increases, the dynamical model leads to errors increasingly often, while
the probabilistic model retains its efficiency, when the probability distribution
{pk (h)}6k=1 is properly redefined. In particular, if the observed frequencies of
occurrence of the faces at h/a ≫ 1 do not differ from 1/6 within the boundaries
prescribed by mathematical statistics4 , we have no reason to suspect asymmetry
of the die. Otherwise, both models require a scrupulous study of the degree of
homogeneity of the die material and the experimental conditions.
Why are we forced to abandon the deterministic model at a significant initial
height of the die? The reason is that as h/a increases, the factors that we disregarded play an increasingly important role: a small spread in initial positions
of the die, nonidentical operations of the die release mechanism, the influence of
the environment, etc. In those cases where the result of the experiment changes
significantly with small variations in initial conditions and behavior of the process,
the latter is called unstable. In classical mechanics, instability had not been considered as a fundamental difficulty for a long time: the problem seemed to consist
only in a large volume of required information on the initial state of the system.
Present-day studies have shown that the development of instability with time is
often exponential, so that extremely small deviations rapidly reach a significant
value (see, e.g., Lichtenberg and Lieberman 1983; Zaslavsky 1984, 1999; Schuster
1984). Since some uncertainty in the initial and boundary conditions is unavoidable in view of the atomic structure of the material, many processes of the world
that surrounds us are fundamentally unstable (these issues are discussed in more
detail in the next section).
The applicability of probabilistic models to the description of processes similar
to the throwing of a die caused no particular disagreement. The stochasticity of
the quantum behavior, about which we will talk below, apart, it could be assumed
that as all details of the process are refined (at least mentally), its result becomes
increasingly definite. The Brownian motion of particles ∼ 1 µm in size in a liquid
can serve as an example. By invoking the simplest model of a random walk to
describe the motion of an individual particle, we draw useful probabilistic conclusions about the behavior of an ensemble of many such particles. At the same
time, analysis of the long-term observations of one specific particle using a refined
theory of the motion of small particles in a liquid allows us to take into account
4
This discipline ascertains the results of imaginary and, hence, completely ‘pure’ experiments for samples of an arbitrarily large size.
5
the particle inertia and the viscosity effects and, hence, to predict its behavior
with a greater certainty than that of the prediction provided by the model of a
purely random walk (Chandrasekhar 1943). Similarly, it is impossible to create a
final, complete theory of turbulence; various probabilistic models of this complex
phenomenon reflect to some degree the properties of viscous fluid flow.
The traditional description of traffic in big cities may be considered as a good
example of natural transition to a probabilistic model. Any driver will undoubtedly
reject the assumption that his trips on Friday are random: he initially drove to
work, subsequently visited several predetermined shops, and, finally, taking his
family, drove to the country cottage. In principle, municipal authorities could
collect information about the plans of each car owner beforehand, but such an
extensive picture on the scales of the city is not required at all – it will suffice
to introduce a stochastic model of traffic along major highways, correcting the
parameters of the corresponding probability distribution depending on the time of
the day.
3. Statistical mechanics
3.1. Instability of motion and irreversibility
The irreversibility of the evolution of large ensembles of particles is usually illustrated by an example of a set of molecules of two types (by convention, ‘white’
and ‘blue’) that fill a closed vessel. Initially, the white and blue molecules are
separated by a baffle. Experiment shows that after the removal of the baffle, both
types mix between themselves, so that the contents of the vessel appear ‘cyan’ after
the relaxation time. Why has the inverse process, when the initial cyan mixture
is separated with time and the white molecules are in one part of the vessel, while
the blue molecules are in its other part, been never observed?
To get a convincing answer to this question, let us first turn to a very simple
thought experiment. Suppose that small identical balls move in a closed box in
the shape of a rectangular parallelepiped. At the initial time, all balls touch one
of the walls, their velocities are directed exactly perpendicular to this wall, and
the minimum separation between the centers of the balls exceeds their diameter.
The initial velocities need not be identical; it is interesting to choose them to be
different in accordance with some law. We assume the wall surfaces, along with
the ball surfaces, to be perfectly smooth and the collisions to be absolutely elastic;
there is no mutual attraction. Classical mechanics easily predicts the behavior of
this system for an arbitrarily long time interval: each of the balls independently
of others vibrates along a straight line segment perpendicular to the two walls. If
the initial velocities were not chosen in a special way (all such ways can be easily
6
specified in advance), then in the course of time the set of balls will scatter more or
less uniformly between the two walls of the box. At the same time, if all velocities
are reversed at an arbitrary instant of time, then the set of balls will return exactly
to its initial state. In a similar way, having mixed after many laps on the stadium,
the long-distance runners will simultaneously return to the starting line if they
turn back by the signal of the referee and each will retain his speed.
Let us complicate the experimental conditions only slightly: suppose that there
is a small asymmetric convexity on the box wall at the point of collision of one of
the balls. At the very first impact on the wall, the ball under consideration will
be reflected slightly sideways and, some finite time later, will hit one of the side
walls. Generally, the subsequent trajectory of the ball is very complex; specially
chosen initial conditions apart, it can be asserted that, after a lapse of time, the
balls will begin to collide not only with the box walls but also between themselves.
Nevertheless, classical mechanics insists on the possibility of predicting the state
of the system after an arbitrarily long time interval as accurately as is wished;
however, this requires knowing the initial positions and velocities of the balls with
an infinitely high accuracy (Krylov 1950; Sinai 1970). The fully developed state
of such models is called deterministic chaos (Schuster 1984). Determinacy is attributable to the complete absence of random factors (noise), so that the evolution
is described by a system of differential or difference equations, while the allusion
to chaos is attributable to extreme entanglement of the trajectories even in the
case of systems with only a few degrees of freedom. We emphasize: there is no
chaos here in the true sense of this word, which implies dominant stochasticity of
the behavior.
It should be kept in mind that states appropriately called illusory chaos can be
realized in some macroscopic systems. Such systems have a long-term ‘memory’
of their past states. We will give an example of a real experiment from Brewer
and Hahn [1984]: “A viscous fluid is placed in the ring-shaped space between two
concentric plastic cylinders. Whereas the outer cylinder is stationary, the inner
one is free to rotate about its axis. A streak of colored dye, representing an initial
alignment of particles, is injected into the fluid. When the inner cylinder is turned,
the dye disperses throughout the liquid. If one were to show the volume between
the cylinders to a thermodynamicist, he or she would say that the dye is completely
disordered... Reversal of the rotation of the inner cylinder reverses the mixing
process; after an equal number of reverse rotations the dye streak reappears.” The
cited paper includes impressive photographs. Obviously, a situation similar to the
above example with runners is realized here.
The phenomenon of nuclear spin echo discovered by E.L. Hahn in 1950 also
demonstrates the possibility of a long-lived memory in systems with stable motion
but now on the atomic scale: “A sample of glycerin was placed in a magnetic
7
field and exposed to two short bursts of electromagnetic radio-frequency radiation,
separated by an interval t of a few hundredths of a second. The sample retained a
memory of the pulse sequence, and at time 2t seconds after the first radio-frequency
pulse the sample itself emitted a third pulse, an echo” (Brewer and Hahn 1984).
The experiment can be explained as follows. Glycerin is initially prepared by
orienting the proton spins parallel to the external magnetic field. The first pulse
triggers a complex precession of the proton spins and the second pulse turns the
spins through 180◦ , so that all spins are again oriented identically after time t. At
this moment, the atoms emit the echo pulse of radiation.
The evolution of the system in the last two experiments may be called regular,
stable, because small changes in initial or external conditions do not lead to a
significant change in its state after a long time interval. The very possibility of
a regular evolution of macroscopic systems is of fundamental importance for the
understanding of statistical mechanics: Generally speaking, we do not always have
to refer to probabilistic models.
On the other hand, even the early works by Henri Poincare (1892) and Jacques
Hadamard (1898) and, presently, many of the studies that followed the paper by
E.N. Lorenz (1963) showed that “...stable regular classical motion is the exception,
contrary to what is implied in many texts” (Schuster 1984). As a rule, even strictly
classical systems with a small number of degrees of freedom in the absence of noise
exhibit instability of motion (generally, behavior): negligibly small variations in
initial conditions lead to a radical difference of the final pictures. In contrast
to regular evolution, where the divergence of the phase elements with time is no
higher than the linear one, the divergence in unstable systems is very fast – it
is exponential. Any finite accuracy of specifying the initial conditions guarantees
the possibility of keeping track of the evolution of an unstable system only for a
short time; its subsequent behavior is indistinguishable from the evolution of a
system with a different initial state. The questions of what the relaxation time
is, whether a stationary, on average, density distribution will be established, and
whether this distribution will be uniform, or the regions of avoidance will remain,
as well as many other problems concerning deterministic chaos, have been solved
only partially.
We are forced to conclude that there are no tools at the disposal of classical
mechanics that would allow one to keep track of the evolution of an even ideal
system with a small number of degrees of freedom for a long time if it is unstable.
This is all the more true with regard to real systems. For example, in the experiment we consider, not only the very complex pattern of roughness of the walls
and ball surfaces but also – since their constituent atoms move – the variability
of these characteristics with time, the inelasticity of the impacts, and many other
phenomena should have been taken into account. Only recently the role of yet
8
another factor that effectively influences the behavior of a classical gas in a closed
vessel has been assessed, the interaction of molecules with the radiation field. It
can still be imagined how to isolate the gas from thermal radiation from the vessel walls, but the collisions between molecules even at moderate temperatures are
inevitably accompanied by low-frequency electromagnetic radiation. Gertsenstein
and Kravtsov [2000] showed that this mechanism leads to a significant deviation
of the trajectory of a molecule from the results of purely Newtonian calculations
in an astonishingly short time, which only a few times longer the mean free-flight
time of the molecules. So, with the aid of classical mechanics, only some general
features of the evolution of a many-particle system can be established and other
models should be invoked to create a real picture.
Two factors necessarily considered jointly – the character of the initial state and
the instability of evolution – allow us to answer the question of why we see so many
irreversible processes in the world that surrounds us, say, the sea waves breaking
against the shore do not recover their shape and do not go back. Many systems
were initially prepared – ordered – by Nature or man and then the instability of
evolution intervened. The same roughly ordered sequence of sea waves is generated
in a natural way – by a strong wind, whereas the recovery of a regular system of
waves requires enormous purposeful work. As Richard Feynman [1965a] believed,
the hypothesis that the Universe was more ordered in the past should be added to
the known physical laws.
Now, we are ready to return to the experiment with two types of molecules
in a closed vessel. Clearly, in conditions that approach the real ones at least in
part, the motion of an arbitrarily chosen molecule will be unstable. For example,
if we launch several realizations of the mixing process from the same – within
the theoretical possibilities of the experiment – initial state, with the external
conditions being retained as scrupulously as possible, then this molecule will be in
completely different regions of the vessel after a finite time in different realizations
of the process.
So far, when discussing the thought experiments concerning the evolution of
many-particle systems, we have said no word about probabilities. Their introduction is inevitable, because the evolution of typical systems is unstable; for this
reason, real processes evolve in a way unpredictable for the researcher. (Einstein
said, “God does not care about our mathematical difficulties. He integrates empirically.”) In these conditions, which are much more complicated than those in the
experiment with the throw of a die, the researcher is forced to invoke a particular
probabilistic model by specifying an appropriate stochastic apparatus. Only afterward and only within the framework of the adopted stochastic model do we have
the right to say that the initially separated state will eventually become uniformly
colored with a particular probability and the latter can return to the original state
9
with a low probability. Even in the model of an ideal gas, reversing the velocities of
all molecules will not return the system to the separated state because arbitrarily
weak stochasticity admitted by the researcher will not allow this to happen. This
is all the more true for the models that represent real systems with a distinct
instability of motion.
As an example of a successful probabilistic model, we will mention the ‘dogand-flea model’ suggested by Boltzmann and considered in detail by Paul and
Tatyana Ehrenfest in 1907 (see Kac 1957). The model illustrates the transition
to statistical equilibrium of the gas that nonuniformly fills a closed volume. Initially, 2N numbered balls are distributed among two boxes in some specified way.
Subsequently, a random number generator creates an integer that is uniformly
distributed on the set 1, 2, . . . , 2N ; the ball with this number is moved from the
box where it lies to the other box. The procedure is repeated many times. This
simple model admits an exhaustive analytical study; it is now easy to perform a
number of corresponding computer realizations of the process as well. In particular, it is curious to trace its evolution from the state when all balls were in the
same box. Above, we deliberately emphasized that the distribution of the random
ball number is uniform: in general, some different discrete probability distribution
can also be specified. In this case, a new probabilistic model of the process will
be introduced which may describe better the actual behavior of the specific gas
sample. Obviously, at N ≫ 1, the system that reached statistical equilibrium in
the adopted probabilistic model will return to the original state only with a very
low probability.
It is easy to continue examples similar to those considered above, by successively passing from simple situations to more complicated ones, from Boltzmann’s
ideas to Gibbs’ ensembles (see, in particular, the lectures by Uhlenbeck and Ford
1963). Analysis convincingly indicates that the deterministic model proposed by
classical mechanics is unproductive when systems with an unstable behavior are
studied. The main objective of statistical physics is to construct adequate probabilistic models of such phenomena. This is also suggested by the name of this field
of physics itself5 .
5
However, the word ‘statistical’ is illegitimately used in the literature instead of ‘probabilistic’ or ‘stochastic’. The latter terms emphasize the presence of random factors in the
model and must be contrasted with the term ‘deterministic’. In contrast, mathematical
statistics deals with the problem that is inverse to the main range of problems of the
probability theory, namely, the reconstruction of information about the probabilistic laws
from a specified random realization. Therefore, one cannot say that the behavior of the
system has a ‘statistical’ character.
10
3.2. Boltzmann’s H-theorem
The best known case of prolonged debates in classical physics spawned by the
probabilistic treatment of the phenomenon is related to the H-theorem. In 1872,
Ludwig Boltzmann concluded that some function of time H(t) that characterizes
the state of a rarefied gas either decreases with time or retains its value6 . In this
form, the H-theorem is in conflict with the symmetry of the laws of mechanics
relative to time reversal and with the theorem proven by Henri Poincare according to which a closed mechanical system returns to an arbitrarily small vicinity
of almost any initial state after a fairly long time interval. These contradictions
called the Loschmidt and Zermelo paradoxes forced Boltzmann to turn to the
probabilistic treatment of the process of approaching equilibrium: the change in
H(t) describes only the most probable evolution of the system. The present-day
formulation (Huang 1963) of the H-theorem also includes an important refinement
of the initial state: “If at a given instant t the state of the gas satisfies the assumption of molecular chaos, then with an overwhelming probability at the instant
t + ǫ (ǫ → 0) dH/dt ≤ 0.”
The aforesaid clearly shows that the H-theorem says not about the behavior of
an ensemble of classical particles but only about some probabilistic model intended
to sufficiently describe the approach of the system being studied to equilibrium.
The physical mechanism that destroys the order is the instability of motion, while
the law of entropy increase is only our statement of this objective phenomenon.
Entropy is a useful model concept but by no means an objective property of Nature.
This is evidenced at least by the fact that the value of entropy depends on the
adopted discretization of the phase µ-space into cells: as the cell sizes decrease,
the entropy decreases (see, e.g., Pauli 1954).
As long as one talks about the behavior of a deterministic classical gas, the
arguments of Loschmidt and Zermelo remain valid. In contrast, the ‘gas’ introduced in the appropriate stochastic model need not obey all the laws of classical
mechanics. A good model developed for a specific situation has the right to disregard in full extent the classical reversibility in time, thus, not to obey Poincare’s
recurrent theorem (especially since the corresponding cycle is excessively long!).
For example, the H-theorem invokes a specific model of a rarefied gas that sufficiently describes its evolution from the hypothetical state of molecular chaos
being realized in practice only with a limited accuracy. Precisely “This statistical assumption [about molecular chaos] introduces irreversibility in time” (Uhlenbeck and Ford 1963). Present-day models of statistical physics, in particular, the
6
For a rarefied gas, H(t) coincides with entropy taken with the opposite sign, so the
H-theorem may be considered as a special case of the law of entropy increase in a closed
system.
11
well-known Bogolyubov-Born-Green-Kirkwood-Yvon approach, introduce similar
probabilistic assumptions but on a deeper level than that of Boltzmann’s theory.
Clearly, being forced to choose some probabilistic model, we can no longer purport to describe in detail the process, which we hoped to do within the framework
of classical dynamics.
3.3. Arrow of time
The widely discussed question of the difference between the past and the future
is usually assigned to the same range of problems of statistical physics. In view
of aforesaid it seems obvious that it is illegitimate to condition the directionality
of time (an ‘Arrow of time’, according to Arthur Eddington) by the behavior of
large ensembles of particles described by statistical mechanics. Let us add that
otherwise we would have to observe not only fluctuations of the characteristics of
many-particle systems, but time also. On the contrary, just the flow of time allows
us to observe successive states of an unstable system and to describe its evolution
using probabilistic models. As was noted by St. Augustine [398], “Let no man
then tell me that the motions of the heavenly bodies are times, because, when at
the prayer of one the sun stood still in order that he might achieve his victorious
battle, the sun stood still, but time went on.”
Clarifying the nature of time and, in particular, its directionality is a much
deeper problem of physics.
4. Characteristic features of classical stochastic models
Let us summarize the conclusions that follow from the analysis of probabilistic
models of classical physics in order to subsequently trace the corresponding changes
in quantum theory7 .
It is well known that any model of the phenomenon under study is neither
unique nor exhaustive. Even a very successful model has boundaries within which
it is more preferable than other models, but a more perfect model usually replaces
it when the range of studies is extended. For example, Eintein’s general theory of
relativity replaced Newton’s theory of gravitation; the latter retained its efficiency
in the case of low velocities and weak gravitational fields. The choice of one of
the many possible models is determined by the principle formulated by William
7
The author hopes that he will not be rebuked for “playing with words specially made
up for this”.
12
Occam in the 14th century: The researcher must prefer the simplest model from a
number of alternatives that give an explanation for the experimental results.
Obviously, the simplicity of a model should be understood in a comparative
context. Quite often, one has to use sophisticated constructions even within the
framework of the simplest possible models. Say, when the distribution of people in
height is studied, the histogram is approximated by a Gaussian function that contains the seemingly irrelevant constant π (see Wigner’s 1960 well-known essay).
Present-day models for the structure of matter invoke incomparably more complicated tools; as a rule, abstract mathematical constructions are used in them.
Technically, the general theory of relativity is much more complex than Newtonian theory, but the former invokes fewer a priory assumptions, which justifies its
comparative simplicity.
If the model is simple and productive and especially when it works for decades
or even centuries (as is now commonly said, became a paradigm), people begin
to assign the concepts with which their theories operate to Nature itself. For
example, in the past, a wide debate was caused by the gravitational paradox – the
divergence of the Newtonian potential in an unbounded homogeneous Universe.
The divergence of the potential was believed to reject the infinite model. As
usual, the paradox only emphasized the conditionality of the tool we choose; if we
talk about the gravitational force at an arbitrary point of an unbounded and, on
average, homogeneous Universe, then it is finite and obeys Holtzmark’s probability
distribution. To be more precise, if the spatial distribution of stars is described by
a Poisson model, then the probability that the value of the force is larger than a
given value F rapidly approaches zero as F increases (see Chandrasekhar 1943).
Other concepts that became habitual in the working models of phenomena,
for example, entropy, are also conditional. The subjectivity of any model is worth
noting once again, because it plays an important role in the picture of the world
to which quantum mechanics leads (see Section 9).
Strange as it may seem, many debates were caused by the confusion of the
concepts of ‘randomness’ and ‘unpredictability’. Unless you know the schedule
of a bus, its arrival is unpredictable for you, but you are free either to choose
some probabilistic waiting model or to call the dispatcher. The fact that we are
unable to predict the position of a Brownian particle cannot serve as a reason for
considering its motion to be objectively random; only the economical model we
chose is such.
Similarly, the fact that we can not predict the sequence of digits in the infinite
decimal expansion of the number π = 3.14 . . . does not mean that they appear randomly. They are completely determined by Nature, but for a non-specialist in number theory (such as the author) their seemingly random and plausible equiprobable
appearance is a fairly good probabilistic model.
13
The above discussion of the procedure for throwing die, the models for the
Brownian motion of small particles, the traffic in a big city, and typical models of
statistical mechanics clearly illustrate the consistent viewpoint of classical physics
on the fundamental question of how probabilities appear in the theory:
(A) There is no room for probabilities in Nature itself. The researcher
is forced to invoke probabilistic models in describing unstable processes, more specifically, when the initial data and external conditions
for the experiment are known only approximately, while its results depend on these circumstances so significantly that they become partly
unpredictable.
Einstein may have had something of this kind in mind in his aphorism about God
playing dice: It seemed that, in contrast to classical physics, quantum mechanics
prescribes a probabilistic behavior to the objects of investigation themselves. The
incentives for such a radical assertion can be understood by considering an experiment with the passage of light through two slits, but we will defer this discussion
until Section 6. In the next section, we consider an intermediate situation where
the quantum behavior is consistent with the conclusion A in an obvious way.
5. Passage of polarized light through a crystal
of tourmaline
At the turn of the 19th and 20th centuries, a fundamental property of the recording
of light, its discreteness, was found. If the brightness of a light source is gradually
reduced, then the picture being recorded becomes increasingly ‘grainy’. It is easy
to make the light source so weak that individual flashes, photoevents, separated
by long time intervals, say, more than an hour, are recorded. Significantly, when
monochromatic light is used, the same portion of energy is recorded each time.
This phenomenon can be explained by invoking the hypothesis put forward by
Einstein in 1905: light consists of spatially localized quanta, photons, that have
energy and momentum. Einstein obtained further evidence for the discrete model
of light 4 years later, when he analytically found an expression describing the
fluctuations of radiation energy in a closed volume.
The above explanations were needed in connection with the description of an
experiment on the passage of polarized light through a crystal of tourmaline given
by Paul Dirac [1958]:
“It is known experimentally that when plane-polarized light is used for ejecting
photo-electrons, there is a preferential direction for the electron emission. Thus the
14
polarization properties of light are closely connected with its corpuscular properties
and one must ascribe a polarization to the photons.
Suppose we have a beam of light passing through a crystal of tourmaline,
which has the property of letting through only light plane-polarized perpendicular
to its optic axis. Classical electrodynamics tells us what will happen for any given
polarization of the incident beam. If this beam is polarized perpendicular to the
optical axis, it will all go through the crystal; if parallel to the axis, none of it will
go through; while if polarized at an angle α to the axis, a fraction sin2 α will go
through. How are we to understand these results on a photon basis?
A beam that is plane-polarized in a certain direction is to be pictured as made
up of photons each plane-polarized in that direction... A difficulty arises, however,
in the case of the obliquely polarized incident beam. Each of the incident photons
is then obliquely polarized and it is not clear what will happen to such a photon
when it reaches the tourmaline.
A question about what will happen to a particular photon under certain conditions is not really very precise. To make it precise one must imagine some
experiment performed having a bearing on the question and inquire what will be
the result of the experiment. Only questions about the results of experiments
have a real significance and it is only such questions that theoretical physics has
to consider.
In our present example the obvious experiment is to use an incident beam
consisting of only a single photon and to observe what appears on the back side
of the crystal. According to quantum mechanics the result of this experiment will
be that sometimes one will find a whole photon, of energy equal to the energy
of the incident photon, on the back side and other times one will find nothing.
When one finds a whole photon, it will be polarized perpendicular to the optic
axis. One will never find only a part of a photon on the back side. If one repeats
the experiment a large number of times, one will find the photon on the back side
in a fraction sin2 α of the total number of times. Thus we may say that the photon
has a probability sin2 α of passing through the tourmaline and appearing on the
back side polarized perpendicular to the axis and a probability cos2 α of being
absorbed. These values for the probabilities lead to the correct classical results for
an incident beam containing a large number of photons.
In this way we preserve the individuality of the photon in all cases. We are
able to do this, however, only because we abandon the determinacy of the classical
theory. The result of an experiment is not determined, as it would be according to
classical ideas, by the conditions under the control of the experimenter. The most
that can be predicted is a set of possible results, with a probability of occurrence
for each.”
The process of creating a probabilistic model for the phenomenon is described
15
very clearly in these words. As in the case of macroscopic objects, the necessity of
a probabilistic model in the experiment considered stems from the fact that “the
result of an experiment is not determined... by the conditions under the control
of the experimenter.” However, there is also a fundamental difference between
the situations. Whereas in classical physics we can still hope for a refinement
of the conditions in which the experiment is carried out, in the microworld the
experimenter influences the process under study so significantly that turning to a
probabilistic model becomes inevitable. Having stepped on an anthill, one should
not be surprised by the fussiness of its inhabitants.
Nevertheless, if this was the only peculiarity of quantum phenomena, then the
difficulties with the perception of quantum mechanics that were mentioned in the
Introduction would remain incomprehensible. The problem consists not in the rejection of the probabilistic description of the results of measurements; the point is
that the probabilistic model suggested by quantum mechanics is considered not as
an approximate description of some deeper picture of microworld phenomena but
is given as a fundamental property of Nature. The situation is often characterized
by asserting that there are no ‘hidden parameters’ in quantum theory that could
give a more complete description (see, e.g., Faddeev and Yakubovsky 1980, pp.
37-38). Such a strange model emerged in the course of painful attempts to explain
the peculiar features of the quantum behavior and, first of all, the seeming nonlocality of quantum interaction. These features have been repeatedly illustrated
using thought and real experiments on the interference of light, to the description
of which we will now turn. For the subsequent comparison with the quantum procedure, we will briefly repeat an elementary derivation of the expression for the
intensity of light in the interference pattern.
6. Interference of light according to the classical wave theory
Let us first assume (Fig. 1) that there is only one narrow slit in an opaque screen
on which the light from a bright source O is incident; the radiation passed through
the slit is recorded by the detector located on the other side of the screen. If the
light is bright and the detector resolution is low, then a continuous flux distribution
with one maximum at point C lying on the axial line is recorded.
The experiment with one narrow slit imposes no serious constraints on the
model invoked for its interpretation. In contrast, Young’s experiment with two
narrow slits performed in 1801 (Fig. 2) required significant concretization of the
model, namely, the wave theory of light that includes the concept of wave interference. It was necessary to explain the peculiar distribution of the light flux on
16
Figure 1: Light recording in the case of one slit in the screen.
the detector characterized by several maxima gradually decreasing with increasing distance from the axial line. The classical wave theory of light proposed by
Thomas Young, Christiaan Huygens, and Augustin Fresnel excellently coped with
this.
According to the wave model in the form that it attained by the late 19th
century, light is a set of harmonic ether oscillations with various temporal periods T
and spatial wavelengths λ. Outside an ordinary material, the oscillation frequency
ν ≡ 1/T is related to the wavelength by the relation νλ = c, where c is the speed
of light. In the experiment considered in Fig. 2, the primary wave from source O is
incident on the screen and generates electron oscillations in it; this is equivalent to
the fact that each of the slits serves as a source of coherent secondary waves. The
oscillation amplitude at some point Q of the detector is determined by the phase
difference between the waves arriving at this point from both slits. The reaction
of the detector at this point depends on the local light flux, which is proportional
to the wave amplitude squared.
Consider, for simplicity, a monochromatic light source located at equal distances from slits A and B. Near the slits, the source generates harmonic oscillations a sin ωt with amplitude a and angular frequency ω = 2πν = 2πc/λ. Let
D be the separation between the slits and z be the distance of the detector from
the screen; we are interested in the oscillation amplitude at point Q located at
distance x from the axis. If we neglect the difference in the degree of attenuation
of the secondary waves on their way from the screen to the detector due to the
distances ℓ1 (x) and ℓ2 (x) of the point of observation from the slits being unequal,
then the waves of equal amplitudes but with different time delays ℓ1 /c and ℓ2 /c
can be assumed to arrive at point Q. The combined oscillation is proportional to
sin[ω(t − ℓ1 /c)] + sin[ω(t − ℓ2 /c)] =
2 cos[π(ℓ2 − ℓ1 )/λ] sin[ωt − π(ℓ1 + ℓ2 )/λ].
(1)
This is a harmonic oscillation with amplitude 2 cos[π(ℓ2 − ℓ1 )/λ]. As was said, the
17
Figure 2: Light interference in the case of two slits in the screen.
light flux F at distance x from the axis is proportional to the amplitude squared:
F (x) = 4 cos2 (δφ/2) = 2[1 + cos(δφ)],
(2)
δφ(x) = 2π(ℓ2 − ℓ1 )/λ
(3)
where
is the phase difference between the secondary waves. At x ≪ z, we can assume
that ℓ2 − ℓ1 ≃ Dx/z, so that
Dx
F (x) ≃ 2 1 + cos 2π
λz
.
(4)
Formula (4) describes a periodic flux distribution on the detector. The spatial
period of the pattern that specifies the linear resolution when the structure of the
light source is studied is
∆x = λz/D,
(5)
and the corresponding angular resolution is ∆θ ≡ ∆x/z = λ/D.
In reality, the height of the maxima decreases with increasing distance from
the symmetry axis, which necessitates a more developed theory. In particular, it
should take into account the increase in the distance of the point of observation
from the slits and the finiteness of their width. Allowance for the latter factor
leads to the expression
Dx
F (x) ≃ 2 sinc (bx/λz) 1 + cos 2π
λz
2
,
(6)
where the function sinc(t) ≡ sin(πt)/(πt) and b denotes the width of each slit. The
additional – compared to Eq. (4) – factor sinc2 (bx/λz) predicts the decrease in
brightness with increasing distance from the axis to zero at
x0 = λz/b,
18
(7)
and, since we assume that b ≪ D, we have x0 ≫ ∆x and the resolution of the
pattern is still specified by Eq. (5). A consistent allowance for other features
of the experiment within the framework of a classical model leads to a detailed
interpretation of the observed pattern, with the exception of one significant fact –
discreteness of the detector counts. We will discuss this phenomenon in the next
section.
Let us give an example. Assume that the light wavelength be λ = 0.5 µm,
the slit width be 10 wavelengths, i.e., b = 5 µm, the separation between the slits
be D = 150 µm, and the distance of the detector from the screen be z = 1 m.
According to Eqs. (5) and (7), the period of the interference pattern is ∆x ≃
3.3 mm and the characteristic size of the image modulation due to the finite slit
width is x0 = 100 mm.
Thus, the classical wave theory of light gives a satisfactory description for the
pattern in Young’s experiment averaged over a long time interval. This theory was
also successful in interpreting an enormous number of other experiments.
7. Wave-particle duality
Let us now consider how the interference of light is explained by the models that
take into account its quantum nature.
First, let us turn to the passage of light through a single slit (Fig. 1). As
has already been said, the observed distribution of photo-counts in the shape of
a single-humped curve increasingly exhibits granularity due to the absorption of
individual photons with decreasing brightness of the light source. These photons
are simply like the classical particles (micropellets), because both the discreteness
of counts and the overall shape of their distribution on the light detector can be
explained in this case.
In Young’s experiment with two slits (Fig. 2), the decrease in the brightness
of the light source also reveals the discreteness of photo-counts, but the classical
view of photons as micropellets turns out to be untenable. Indeed, the micropellets must come to the detector either through slit A or through slit B, so that
the expected image is a superposition of two single-humped distributions shifted
relative to the axis – basically, the projections of slits A and B onto the screen
from point O. This pattern clearly differs from the observed distribution, which
is characterized by sharp intensity variations. The real distribution is formed by
identical photo-events, but the arrival of a photon at a given point of the detector
depends significantly on the presence of a second slit. If we close one of the slits,
then we will see a smooth single-humped distribution shifted relative to the axis,
while an interference pattern with a distinct alternation of extrema appears when
19
both slits are open. Try to imagine how opening the second slit in the model
operating with photons-micropellets can reduce the frequency of their falling to
some place of the detector!
Thus, we get the impression that the propagation of light obeys the laws of
the wave theory, while it interacts with the detector as if it consisted of localized particles. It is this situation that is usually characterized by the concept of
wave-particle duality. The brightness of the light source can be made so low that
independent photon recording events will be recorded with a certainty. The distribution of photo-events accumulated over a long time closely coincides with the
interference pattern that is observed in the case of a bright source. To explain
this fact, we have to assume that “each photon then interferes only with itself.
Interference between two different photons never occurs.” (Dirac 1958, p. 9).
This conclusion emphasizes perhaps the most striking peculiarity of the quantum behavior. Indeed, the slits can be separated very far from each other on the
scale of the light wavelength and, nevertheless, both the interference pattern and
the discreteness of counts are retained. In particular, for the example described
at the end of Section 6, the separation between the slits D was 300 wavelengths.
Using lasers allows the ratio D/λ to be increased even more, by many times. The
telescopes constituting stellar interferometers are tens and hundreds of meters
apart.
Albert Einstein was the first to realize the inevitability of wave-particle duality
in describing light in 1908: “I already attempted earlier to show that our current
foundations of the radiation theory have to be abandoned... It is my opinion that
the next phase in the development of theoretical physics will bring us a theory of
light which can be interpreted as a kind of fusion of the wave and the emission
theory.” Recall that the ‘emission theory’ that considered light as a flux of very
unusual particles was suggested by Isaac Newton.
8. Behavior of material particles
However difficult it is to imagine the diffraction of a single photon, but the idea of
electromagnetic waves, which can naturally reach simultaneously two slits spaced
far apart, helps us in the case of light. However, a similar diffraction pattern is
observed if light is replaced by a flux of electrons or other particles with a nonzero
rest mass (for brevity, such particles are often called ‘material’ ones), or even whole
atoms! This is evidenced by various experiments, the first of which was carried
out in the 1920s. In particular, the diffraction of uncharged particles, neutrons,
by the crystal lattice formed by the atomic nuclei of a solid body is observed. A
satisfactory model of this phenomenon suggests the diffraction of neutron waves
20
by a crystal lattice whose spacing exceeds considerably the neutron wavelength.
According to Louis de Broglie (1924), a wave process with the following wavelength is associated with any particle whose rest mass m0 is nonzero:
λ=
h q
1 − v 2 /c2 ,
m0 v
(8)
where h ≃ 6.626 · 10−27 ergs·sec is Planck’s constant and v is the particle velocity.
For example, when an electron (its rest mass is m0 ≃ 9.11 · 10−28 g) is accelerated
in an electric field with a potential difference of 1 kilovolt, it reaches speed v ≃
1.9 · 109 cm/s; in this case, the de Broglie wavelength is λ ≃ 0.4 · 10−8 cm ≃ 0.4 Å.
If such a beam of electrons is directed to a nickel single crystal, for which the
lattice spacing is about 2 Å, then the appearance of interference extrema in the
distribution of scattered electrons should be expected. This pattern was first
observed in 1927 by C. Davisson and L. Germer.
Thus, the experimental data suggest the wave nature of not only photons but
also material particles. Since the electron can be likened neither to a micropellet
nor to a wave, we have to abandon the seemingly only possible alternatives when
considering the diffraction of electrons by two slits: (1) the electron passes either
through one slit or through the other; (2) it passes through both slits simultaneously. The point is that we unjustifiably transfer the concept of body ‘trajectory’
worked out by macroscopic experience to the quantum world. This is discussed in
more detail in the next section.
The wave-particle duality has been discussed for about a century. Excellent explanations in this connection were given by E.V. Shpol’skiy [1974] in his course on
atomic physics: “Since the properties of particles and waves are not only too different but also, in many respects, exclude each other and the electrons undoubtedly
have a single nature, we have to conclude that the electrons are actually neither
the former nor the latter, so that the pictures of waves and particles are suitable
in some cases and unsuitable in other cases. The properties of microparticles are
so peculiar and their behavior is so different from that of the macroscopic bodies
that surround us in everyday life that we have no suitable images for them. However, it is clear that since we are forced to use both wave and particle pictures to
describe the same objects, we can no longer ascribe all properties of particles and
all properties of waves to these objects.”
New names have been repeatedly proposed for microworld objects. For example, Feynman once used the term wavicles, a derivative of the words waves and
particles. Unfortunately, none of the names was so apt to be widely used in the
literature. Habitually, quantum structures are most often called microparticles,
but the aforesaid should always be kept in mind.
21
9. Quantum mechanics
Attempts to more deeply understand the nature of the wave-particle duality led in
the mid-1920th to the creation of quantum mechanics – the theory of phenomena
in describing which Planck’s constant plays a crucial role. Quantum mechanics
and its development including the special theory of relativity – quantum field
theory – give a satisfactory description of the entire set of phenomena in the
world that surrounds us, with the exception of gravity. Some of the aspects of
this description achieved remarkable agreement with experimental data, ∼ 10−10 ,
which is indicative of a high efficiency of the model created by Erwin Schrödinger,
Werner Heisenberg, Max Born, and Paul Dirac. At the same time, the words of
outstanding physicists given at the beginning of this chapter suggest that, while
providing a consistent formal procedure for calculating the results of experiments
in the domain of atomic phenomena, quantum mechanics raises difficult conceptual questions. John Bell [1987], who proposed well-known experiments to test
quantum mechanics, reached a bitter conclusion: “When I look at quantum mechanics I see that it’s a dirty theory: You have a theory which is fundamentally
ambiguous.”
9.1. Peculiarities of the apparatus
Consider the explanation of the experiment on the diffraction of particles, say,
electrons, by two narrow slits proposed by quantum mechanics (this description
can also be equally extended to the experiment with photons discussed in Section
6). Analysis of this experiment allows the formal aspect of the quantum-theory
calculations to be perceived. For simplicity, consider a stationary process where
source O provides, on average, a constant number of electrons in unit time (Fig. 2).
It is required to find the mean particle flux at point Q located at distance x from
the axis. In the case of two open slits, the calculation is as follows.
With the aid of the procedure briefly described below two complex numbers
should be formed: ϕ1 (x) and ϕ2 (x) – the amplitudes of the probability of electron
passage through slits A and B, respectively, followed by their detecting at Q(x). By
definition, the total amplitude of the probability of detecting at Q that we denote
by ϕ(x) is the sum of the amplitudes corresponding to all mutually exclusive paths;
in our case,
ϕ(x) = ϕ1 (x) + ϕ2 (x).
(9)
Knowledge of the amplitude allows us to find the probability f (x)dx that an arbitrary particle emerged from O will fall into an infinitely small interval of width
dx near point Q:
f (x)dx = |ϕ(x)|2 dx.
(10)
22
The electron flux near point Q and the number of photo-events per unit time are
proportional to the probability density f (x). It follows from the two previous
formulas that
f (x) = |ϕ1 (x)|2 + |ϕ2 (x)|2 + 2 ℜ[ϕ∗1 (x)ϕ2 (x)],
(11)
where the symbols ℜ and ∗ correspond to the separation of the real part of the
complex number and complex conjugation, respectively. The classical description
would be restricted to the first two terms in Eq. (11), which define the probability of
passage either through the first slit or through the second one. Quantum mechanics
introduces the third term dependent on the phases; it is the relationship between
the phases ϕ1 (x) and ϕ2 (x) that allows the interference of microparticles to be
properly described. All of this resembles the operations that we performed in
Section 6 when analyzing the diffraction of light by two slits, but the probability
amplitudes cannot be treated as waves in ordinary space.
Obviously, before we turn to Eq. (11), we should specify how the probability
amplitudes are calculated and what the rules for handling these quantities are.
We will only cursorily touch on these questions here within the framework of
nonrelativistic quantum mechanics; for a detailed description, see the lectures by
Feynman [1965b] and textbooks on quantum mechanics. Feynman showed that
the probability amplitude ϕ for some path is defined by the action S, i.e., the
time integral of the difference between the kinetic and potential energies along
this path:
ϕ ∝ eiS/h̄ ,
(12)
where i is the imaginary unit and h̄ is Planck’s constant divided by 2π. The
transition amplitude, considered as a function of the final state, is the famous ‘psifunction’ ψ(x, t), which forms the basis for the adopted description of quantum
phenomena. To find ψ(x, t), one should either solve Schrödinger’s equation or
calculate the action S and use Eq. (12).
Note, incidentally, that representation (12) elucidates the nature of the wellknown principle of least action. Since Planck’s constant is small, paths with greatly
differing S are characterized by an enormous phase difference and, hence, their
contributions cancel each other out; only for the paths near the S extremum is the
phase variation small, so that the amplitudes are added constructively. Therefore,
material particles, like photons, ‘choose’ the paths on which the action is extremal.
The main rule for handling the probability amplitudes says: If a given finite
state is attainable along several independent paths, then the total probability
amplitude of the process is the sum of the amplitudes for all paths considered
separately. We emphasize that the linearity of the system holds for the probability
amplitudes, while the probabilities themselves are related to the amplitudes in a
quadratic way. Strictly speaking, in the problem of particle diffraction by two slits
23
of finite width, we should have added the amplitudes for the set of all paths from
O to Q enclosed by the slits, but, as a first approximation, when the slit width is
small, each of the beams can be replaced by one path. It is this approximation
that is implied in Eq. (9).
When the probability amplitude is calculated for a path of complicated shape,
the following rule turns out to be useful: for any route, the probability amplitude can be represented as the product of the amplitudes corresponding to the
motion along separate parts of this route. For example, in our problem, the amplitude ϕ1 (x) is the product of ϕ(O → A), the transition amplitude from O to slit
A, and ϕ(A → Q), the transition amplitude from slit A to point Q(x). Solving
Schrödinger’s equation or performing calculations according to Eq. (12) for separate paths of two possible transition ways from the source to the detector (point
of interest), and then multiplying the corresponding partial amplitudes, we can
find the distribution of the mean particle flux along the detector with the aid of
Eq. (11).
9.2. Interpretation
The above brief description is intended only to emphasize the most important
features of the quantum mechanical approach:
(B) The basic picture of phenomena is described in the language of
probability amplitudes – new concepts that have no classical analogue.
The complex probability amplitudes are calculated according to the
specified set of rules. It is possible then to find the probabilities of
various events that are defined as the squares of the absolute values
of the corresponding amplitudes.
Thus, according to Max Born (1926), quantum mechanics initially dealt with the
probabilistic picture of physical phenomena. This inference, per se, is consistent
with conclusion A in Section 4: If probabilistic models are needed even for the
description of classical unstable models, they are all the more inevitable in describing microworld phenomena, where the predictability of measurements is restricted
by several fundamental circumstances, in particular, by Heisenberg’s uncertainty
relations. For example, when the experiment in which the position x of a particle
and the conjugate momentum p are measured simultaneously is carried out, the
standard deviations σx and σp of the measured quantities obey the inequality
σx σp ≥ h̄/2.
(13)
Restrictions of this type are formally included in the apparatus of quantum mechanics; their necessity becomes clear from the analysis of simple thought exper-
24
iments (see Bohr 1935; Born 1963, Ch. 4, Sect. 7; Shpol’skiy 1974, Sect. 148,
149).
The actually significant difference between quantum mechanics and classical
models is attributable not to the probabilistic way of reasoning but primarily to
the peculiarity of the quantum behavior itself (above all, nonlocal – in the classical sense – interaction), the abandonment of searches for a deeper deterministic
underlying picture, and the clarification of the key role of experimental conditions for the possibilities of describing the phenomenon under study. Collectively,
these features required introducing new concepts, including the concept of wave
function.
Classical physics invoked a probabilistic model for some phenomenon in a situation where the impossibility of describing it exhaustively was obvious either in
view of the instability of its behavior, or due to the extreme complexity of the
accompanying processes. Such is the origin of the models related to the throws
of die, the theory of Brownian motion, and the ensembles of statistical physics.
The reality of the deterministic, in principle, behavior of the system under study
has always been implied, even if it was not possible to give its detailed description8 . Quantum mechanics considers a probabilistic model not as an approximate
description of some deeper picture of microworld phenomena but as a fundamental property of Nature. This statement has repeatedly appeared in various forms
throughout the history of development of quantum theory; it will suffice to cite
the opinion of Wolfgang Pauli [1954]: “It was wave or quantum mechanics that
was first able to assert the existence of primary probabilities in the laws of nature,
which accordingly do not admit of reduction to deterministic natural laws by auxiliary hypotheses, as do for example the thermodynamic probabilities of classical
physics. This revolutionary consequence is regarded as irrevocable by the great
majority of modern theoretical physicists – primarily by M. Born, W. Heisenberg
and N. Bohr, with whom I also associate myself”.
In the early 1950s, when this authoritative statement was made, the crucial
role of the instability of motion in classical physics was not yet so clear; now,
the allusion to the possibility of reducing the probabilistic models of statistical
physics and thermodynamics to deterministic laws seems untenable. As for the
“primary probabilities” of quantum mechanics, Pauli most likely had in mind
only the inevitability of the probabilistic description of the microworld but not
God playing dice. The latter would become inevitable if we were dealing only
with Nature, as such. In reality, however, we are always forced to give a joint
description of the phenomenon and the experimental setup. In this connection,
8
Einstein also had this in mind in 1916 in his theory of interaction of the radiation field
with atoms. However, the Einsteinian transition probabilities were the proclaimers of a
new quantum theory...
25
Max Born [1963] noted that before the creation of the relativity theory, the concept
of ‘simultaneity’ of two spatially separated events was also considered self-evident,
and only the analysis of the experimental foundations of this concept performed by
Einstein showed its dependence on the frame of reference. V.A. Fok characterized
the situation in the microworld as “relativity to the means of observations”.
The following two opinions separated by seven decades clarify well the essence
of the matter.
In his introductory article to the debate between Einstein and Bohr on completeness of the quantum mechanical description of reality, Fok [1936] gave the
following clarifications9 : “Quantum mechanics actually studies the objective properties of Nature in the sense that its laws are dictated by Nature itself, not by
human imagination. However, the concept of state in the quantum sense is not
among the objective concepts. In quantum mechanics, the concept of state merges
with the concept of ‘information about the state obtained through a certain maximally accurate experiment’. The wave function in it describes not the state in
an ordinary sense but this ‘information about the state’... By the maximally accurate experiment we mean such an experiment that allows all of the quantities
that can be known simultaneously to be found. This definition is applicable to
both classical and quantum mechanics. However, in classical mechanics, there was
basically one maximally accurate experiment, namely, the experiment that gave
the values of all mechanical quantities, in particular, the positions and momentum
components. It is because any two maximally accurate experiments in classical
mechanics give the same information about the system that one could talk about
the state of the system there as about something objective, without specifying
through which experiment the information was obtained”.
The second extract allows the viewpoint of a modern researcher, David Mermin, to be judged: “A wave function is a human construction. Its purpose is to
enable us to make sense of our macroscopic observations. My point of view is
exactly the opposite of the many-worlds interpretation. Quantum mechanics is
a device for enabling us to make our observations coherent, and to say that we
are inside of quantum mechanics and that quantum mechanics must apply to our
perceptions is inconsistent.” (a quotation from the paper by Byrne 2007).
The many-worlds interpretation mentioned by Mermin concerns the interpretation of the measurement process proposed by H. Everett in the mid-1950s (for the
history and references, see Byrne 2007). Everett’s dissertation was discussed before its publication in Copenhagen by leading physicists; the reaction was negative.
Fok’s clarifications and Mermin’s remark clearly reveal the reason why Everett’s
idea cannot be considered acceptable (see also comments by Feynman regarding
9
Now, one says ‘complete experiment’ instead of ‘maximally accurate experiment’.
26
the role of an observer cited at the end of this section).
The term ‘objectivity’ concerning the concept of a quantum system’s state is
occasionally perceived not quite unambiguously; therefore, the following example
may prove to be useful. Suppose that we have a set of dice at our disposal, each of
which is made asymmetric in some known way. Say, one of the dice is made so that
number ‘6’ occurs very rarely. Obviously, the result of a throw is determined both
by the (subjective) choice of a die from the set and by the (objective) structure
of this die. Similarly, the result of a quantum mechanical experiment depends
both on the ‘objective state’ of the system unknown to us and on the character of
the question asked by the experimenter by choosing the experimental conditions.
As John Wheeler noted, “Schrödinger’s wave function bears to (the unknowable)
physical reality the same relationship that a weather forecast bears to the weather.”
The above interpretation of the wave function, basically kept within the framework of the orthodox interpretation of quantum mechanics, allows some of the
known paradoxes to be avoided10 . For example, putting the wave function in the
list of objects of physical reality led to a discussion of the problem of its ‘collapse’
with a superluminal speed (!) as the result of carrying out an experiment. However, we should then also say that the probability distribution {pk } of the possible
results of throwing a die collapses similarly when the die stops. This did not happen; paradoxes appear when the concepts introduced by us are ascribed to Nature
itself.
A reasonable interpretation of the experiment on the diffraction of electrons by
two slits should be sought in the same direction. The experimental data strongly
suggest that separate electrons are diffracted as if each of them came to the detector through both slits at the same time. Emphasizing the inevitability of a
joint description of the physical process under study and the recording, necessarily macroscopic, instrument, quantum mechanics requires particular thoroughness
in choosing the words and concepts used. In the case under consideration, the
posed question is formulated as follows: Can the interference pattern and the
passage of electrons through a particular slit be observed simultaneously in some
experiment? A comprehensive analysis shows that using any means that allows the
electron trajectory to be established immediately destroys the interference (see, in
particular, the lectures on quantum mechanics by Feynman 1965b). Therefore,
the question of whether an electron passes through both slits in the experiment on
interference is empty to the same extent as the question about the number of devils
fit at the needle tip widely debated in the Middle Ages. No physical experiment
can answer questions of this kind, hence, physics is forced to abandon the classical
10
We everywhere understand a ‘paradox’ as an ‘imaginary contradiction’. Feynman
emphasized: “In physics there are never any real paradoxes because there is only one
correct answer... So in physics a paradox is only a confusion in our own understanding.”
27
concept of ‘trajectory’ in the cases where the result can be achieved by various
paths to which markedly differing probability amplitudes correspond.
Reality can be successfully described using the rule of addition of the probability amplitudes and the fact that the laws of quantum mechanics seem absurd
to us only says about the degree of discrepancy between our everyday experience
and the microworld laws.
It should be said that, apart from the underestimation of the macroscopic
experimental conditions specified by an observer, the viewpoint that clearly overestimates the role of an observer in studying quantum phenomena is also fairly
popular. More specifically, it is believed that Nature is real only to the extent
to which it appears before the observer. The feelings of an acting physicist that
continuously ponders new experimental data was vividly expressed by Richard
Feynman [1995]: “This is a horrible viewpoint. Do you seriously entertain the
thought that without the observer there is no reality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Was there no reality in the
universe before 109 B.C. when life began? Or are you the observer? Then there is
no reality to the world after you are dead?”
9.3. Some probabilistic aspects of the uncertainty relation
Let us note, referring to the physical meaning of the inequality (13), that quite often met ambiguity in interpretation of this relation caused largely by use of vague
concept of ‘measurements error’. Meanwhile, the inequality concerns only the
relationship between the characteristic widths σX and σP of the probability distributions of X and its conjugate momentum P , which are interpreted as random
variables.
For definiteness, we will continue discussion in a frame of experiment on diffraction of weak flux of electrons on a narrow slit. Obviously, at registration of any
electron, its position is measured with accuracy of an order of width of the slit,
whereas its transverse momentum is defined by accuracy of an estimation the deviation of impact point from an axial line. We can make width of the very massive
slit arbitrarily small, and the measurements with the detector arbitrarily detailed,
so the accuracy of measurement of realizations (x1 , p1 ), (x2 , p2 ),. . ., (xN , pN ) in N
consecutive passages of electrons through the slit is defined only by devices which
are used in experiment. The actually important feature of quantum phenomena
is that narrowing a slit leads to increase of average width of a whole set of impact
points on the detector. Thus, the true uncertainty of experiment is limited not
by accuracy of individual measurements, but the wave nature of microparticles
generating the diffraction phenomenon.
28
From the formal point of view, the inequality (13) is a consequence of the
fact that the coordinate density distribution f (x) and the conjugate momentum
density distribution g(p) are interdependent. Really, for any state |ψi the probability amplitude in coordinate representation, hx|ψi ≡ ψ(x), and the probability
amplitude in momentum representation, hp|ψi ≡ η(p), are connected by Fourier
transform:
η(p) = √
1
2πh̄
Z
e−ipx/h̄ ψ(x) dx,
1
ψ(x) = √
2πh̄
Z
eipx/h̄ η(p) dp.
(14)
The mentioned interrelation of probability densities is caused by their quadratic
definition through the probability amplitudes:
f (x) = |ψ(x)|2 ,
g(p) = |η(p)|2 .
(15)
Hermann Weyl has shown that Heisenberg’s inequality (13) is a direct consequence
of equations (14) and (15); corresponding derivation can be found in the book by
Fok [2007], P. II, Ch. I, §7.
Said above suggests that conjugate random variables X and P possess not only
the particular probability densities, but also a joint probability density f (x, p).
Nevertheless, searches for a physically sensible joint density, the beginning to
which has put Wigner [1932], were unsuccessful. This circumstance deservs to
be included in the list of many ‘oddities’ of quantum mechanics. Ballentine [1998]
has devoted to the quantum mechanics in phase space a special chapter of his
recent monograph.
9.4. Quantum chaos
In classical mechanics, the concept of ‘chaos’ is associated with a distinct instability of particle trajectories under small variations in initial data and ambient
conditions. As we saw, it is the instability that impels us to introduce probabilistic models of classical phenomena. Therefore, the following question is quite
legitimate: Can the probabilistic nature of the quantum theory be attributable to
a similar instability?
Obviously, the classical concept of chaos cannot be extended directly to quantum mechanics, which not only insists on the fundamental impossibility of a simultaneous exact measurement of the conjugate coordinates and momenta of particles
but also dispenses with the word ‘trajectory’. If the complete description of reality
is assumed to be given by a wave function, then the concept of ‘quantum chaos’
could be attempted to associate with the manner of time variation of the wave
function defined by Schrödinger’s equation. However, the latter is a first-order
29
equation that always specifies a stable time evolution: two close, in some appropriate sense, states remain so during the entire subsequent evolution11 . As David
Poulin [2002] points out, this conclusion follows even from the requirement that
the system’s energy be real. In contrast, the description of chaotic systems in classical physics is accompanied by differential equations of the second or higher orders
(of course, invoking such equations is not a sufficient condition for the described
phenomenon being chaotic). Thus, a productive definition of quantum chaos, if
there is a need for this concept, should be searched for in a different direction.
These searches are being conducted. In particular, it appears that a small
difference, but now of the Hamiltonians rather than the initial states, can lead to
an exponential divergence of initially close systems. Another important direction
is the development of decoherence of quantum systems due to the influence of the
environment. In view of the peculiar behavior of chaotic systems, we will also mention that it is desirable to refine Ehrenfest’s theorem concerning the passage to the
classical limit. All these studies are of interest in their own right, but the question
about the origins of the probabilistic interpretation of quantum mechanics should
be associated not with them. We have in mind the “relativity to the means of
observations” and the peculiarity of the concept of ‘state’ in quantum mechanics
that were discussed in the previous section. Schrödinger’s equation describes not
the evolution of the system’s state that exists independently of the experimentation but, speaking somewhat simplified, the time behavior of the potentialities
of a specific experiment with regard to the system under study. Therefore, it is
generally illegitimate to expect that the variation in wave function will resemble
the behavior of classical particles.
On the other hand, the instability of evolution is important not in itself; in
classical physics, it determines the fact that serves as a real basis for turning to
probabilistic models, namely, it produces situations where, as Dirac said, “The
result of an experiment is not determined... by the conditions under the control
of the experimenter.” But the latter, for a number of reasons, is also highly
characteristic of our experience in microphysics. It is not only a matter of the
influence of a classical observer on quantum processes. The inappropriateness of
the concept of particle trajectory to the results of quantum experiments reflects
only one aspect of a more general concept – the microparticle identity principle.
Indistinguishability of two particles of a given class (say, electrons) themselves and
their trajectories serves as a basis for the amplitude addition rule, which determines
the characteristic features of the quantum theory.
So far we have restricted ourselves to the discussion of nonrelativistic quantum
mechanics, because the inevitability of introducing a probabilistic model is clear
11
The same is true with regard to the systems for which only the density matrix exists.
30
even within the framework of this theory. However, no interpretation of a wide
range of experimental data is possible without invoking the special theory of relativity. The picture of the world painted by the corresponding model – quantum
field theory – is much richer in colors. For example, it appears that the smaller
the interaction scale, the more intensely the physical vacuum ‘boils’: new particles incessantly ‘evaporate’ from the vacuum and annihilate almost immediately.
Nevertheless, virtual particles often have time to interact with real particles and
atoms, which determines the observed effects (in particular, Einsteinian spontaneous transition coefficient). This picture, which more deserves the name quantum
chaos, clearly suggests that the probabilistic approach in describing the microworld
is quite natural.
10. Spooky action at a distance
The words in the title of this section were used by Einstein, who was the first to see
striking corollaries of the quantum theory. In classical physics, we got accustomed
to the fact that the interrelation between events is attributable either to the action
of one of them on the other or to their common past history. In the opinion of
Einstein, Podolsky, and Rosen [1935], the orthodox quantum theory introduces a
new type of interaction that can manifest itself in the parts of the system that
have already ceased to influence each other in an ordinary sense. This paper is
so often referred to that the abbreviation EPR became common for it and for the
corresponding effect.
In Schrödinger’s papers that followed the EPR work in the same year, the term
‘entanglement’ of the properties of noninteracting systems was used. Over the last
quarter of the century, this phenomenon was subjected to thorough experimental
testing; we will consider an idealized version of the real experiment that retains
the essence of the original (Bohm 1952, Ch. 22). To abstract from insignificant,
in this context, features of the experiment related to the particle charge, we will
deal with a neutral particle, say, a neutron.
10.1. Thought experiment by Einstein, Podolsky, and
Rosen
First, recall some of the peculiarities of quantum measurements using a specific
example. Let it be required to measure the spin of a neutron – its intrinsic mechanical moment and the related magnetic moment. This can be done using the
Stern-Gerlach setup, in which the microparticles fly between the poles of a magnet
that produces a strong and, what is important, nonuniform magnetic field. Since
31
Figure 3: Scheme of the thought experiment on measuring the spins of two
neutrons emerging from source O. Detectors A and B are separated by an
impermeable baffle (hatched), numbers 1, 2, and 3 mark the directions in
which the spin is measured.
the spin of a free neutron is an arbitrarily directed vector, one could expect that
when measuring its projection in the direction specified by the arrangement of the
magnet poles, we would obtain a certain value from a continuous range of values. However, experiment shows that the neutron spin projection vector is always
equal in magnitude to h̄/2 and is directed to one or the other pole of the magnet.
The appearance of Planck’s constant here suggests that the microparticle spin is a
purely quantum property; the angular momentum of a body about an axis passing
through the center of inertia represents its indirect analogy in classical mechanics.
According to the interpretation adopted in quantum mechanics, the choice of one
of the two possible neutron spin projection directions is random. In the experiment
under consideration, the corresponding probabilities are
p+ = cos2 (θ/2),
p− = sin2 (θ/2),
(16)
where θ ∈ [0, π] is the angle between the spin vector and the direction in which it is
measured. As we see from Eqs. (16), if the spin was initially directed to one of the
poles, then its measured projection would retain its direction. These peculiarities
of the experiment are a special case of a general principle of quantum mechanics:
when an observable quantity is measured in a closed system, one of the eigenvalues
corresponding to this quantity will be obtained (Dirac 1958, Sect. 10). In our case,
the eigenvalues of the spin are ±h̄/2.
Let us now turn to the critical experiment (Fig. 3). In source O, the system
of two neutrons is prepared in such a way that its spin is zero. Subsequently, the
32
system spontaneously breaks apart, so that the neutrons fly in opposite directions
toward observers A and B who have Stern-Gerlach-type detectors at their disposal.
For brevity, the detectors and neutrons are denoted by the letters corresponding to
the observers. The source and the observers are considered in a common inertial
frame of reference; the distances OA and OB are assumed to be equal. (The
conclusions do not change fundamentally if, say, OB is slightly larger than OA,
so that the measurement in B is made slightly later than that in A.) If the
detectors are far apart or are separated by an impermeable baffle, then the neutrons
no longer interact, in the classical understanding of this word, with one another
shortly before their recording. Since the result of measuring the spin projection
for one particle is random, it is clear that the result of each individual experiment
to measure the spins of two neutrons with the corresponding detectors will also be
random. The question is how correlated the counts of detectors A and B are.
First, we will attempt to predict the results of the experiment using a semiclassical model that makes it possible to independently consider the remote neutrons
but takes into account the quantum character of the spin measurement expressed
by Eqs. (16) firmly established in experiments. Subsequently, we will give the
conclusions of an analytical study of the same problem in terms of the quantum
theory and, finally, will present the corresponding experimental data.
10.2. Semiclassical model
Taking into account spin conservation in a closed system, we should consider the
total spin of a system of two neutrons flying apart to be always zero. In the
classical approximation, if this was the case for the spin, it could be assumed that
the neutrons are independent and their spins are directed oppositely along some
straight line whose orientation can change arbitrarily in successive experiments.
We will consider the direction from the south pole of the detector to its north pole
to be positive. The subsequent explanations will be simplified if we equip each
detector with two lamps, green and red, and adopt the following condition: the
green (G) and red (R) lamps turn on if the recorded neutron spin projection is
oriented, respectively, in the positive and negative directions of a given detector.
Thus, one of the four combinations of turned-on lamp colors can be realized in a
separate experiment: GG, GR, RG, or RR. Let us find the probabilities of these
events by assuming, for simplicity, that the detectors are oriented in the same
direction, say, in direction 1 (Fig. 3).
Denote the angle between the positive direction 1 and the direction of the
spin vector for neutron A realized in a given experiment by Θ ∈ [0, π]; the angle
between the spin of neutron B and the same direction 1+ will be π − Θ. As was
said, the spin projection measured in A will be randomly oriented in the positive or
33
1
0.9
0.8
0.7
g(Θ)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
Θ
2
2.5
3
Figure 4: Recording probability of oppositely directed neutron spin projections versus angle between the detector and spin orientations for the semiclassical model.
negative direction of vector 1. Denote the possible results of the measurement for
neutron A by A1+ and A1− , respectively; for neutron B, the possible results will
be B1+ and B1− . Taking into account the independence of the neutron recording
events and Eq. (16), we obtain for the sought probabilities of the events:

+ · B1+ ) = cos2 Θ cos2 π−Θ = cos2 (Θ/2) sin2 (Θ/2),

P
r(GG)
=
P
r(A1



2 2


 P r(GR) = P r(A1+ · B1− ) = cos2 Θ sin2 π−Θ = cos4 (Θ/2),
2
2 π−Θ
2 Θ
4
−
+
2


 P r(RG) = P r(A1 · B1 ) = sin 2 cos 2 = sin (Θ/2),



 P r(RR) = P r(A1− · B1− ) = sin2 Θ sin2 π−Θ = sin2 (Θ/2) cos2 (Θ/2).
2
2
(17)
It is easy to verify that the sum of all probabilities (17) is equal to 1.
We are particularly interested in the probability that oppositely directed spin
projections are recorded, i.e., the lamps of different colors will turn on:
P r(GR) + P r(RG) ≡ g(Θ) = 1 −
1
sin2 Θ.
2
(18)
The function g(Θ) has the meaning of a conditional probability of occurrence an
event given angle Θ. As Fig. 4 shows, only at Θ = 0 and Θ = π, i.e., when the
‘spin-line’ is oriented in the same way as the detectors, are oppositely directed spin
projections recorded with confidence; for noncoincident orientations, the values of
g(Θ) lie between 0.5 and 1.
Obviously, the model with fixed Θ is insufficient to explain real experiments;
it is more appropriate to assume that this angle changes in some way in successive
34
experiments. This means that Θ should be considered as a random variable. In
particular, for an isotropic orientation of the straight line along which the neutron
spins are directed, the probability that this straight line will fall within the solid
angle dω is dω/4π = sin θdθdϕ/4π, where the factor (1/2) sin θ is the distribution
density of the polar angle Θ and 1/2π is the distribution density of the azimuth
angle Φ. Averaging p+ from (16) over all angles yields the probability that detector
A in a given experiment will record the positive spin direction, i.e., the green lamp
will turn on:
1
4π
Z
0
2π
dϕ
Z
π
0
1
p+ (θ) sin θdθ =
4π
Z
2π
dϕ
0
Z
π
cos2 (θ/2) sin θdθ = 1/2.
(19)
0
As would be expected, each of the detectors records one or the other spin projection with a probability of 1/2, so in a long series of experiments, the lamps of
different colors turn on equally frequently. However, we are more interested in
the correlation between the results, which requires calculating the unconditional
probability pd of recording oppositely directed spins. This is achieved by averaging
Eq. (18) for the function g(Θ) over all angles:
pd =
1
4π
Z
0
2π
dϕ
Z
π
0
1−
1
sin2 θ sin θdθ = 2/3.
2
(20)
Thus, the semiclassical model predicts that for an isotropic distribution of the
neutron spin direction, on average, 2/3 and 1/3 of the experiments will lead to the
recording of oppositely and identically directed spin projections, respectively.
The isotropic model we considered cannot be reckoned to be mandatory; it is
only an example of one of the possible spin direction distributions for the particles
flying apart. Only the fact that follows from the form of the function g(Θ) in (18) is
actually important: for any spin direction distribution that admits a deviation from
the orientation of the detectors, the probability of obtaining oppositely directed
spin projections in the case of independent recording events is less than 1.
10.3. Results of quantum mechanical calculations
We will now briefly present the conclusions that follow from rigorous quantum
mechanical calculations. David Bohm [1952] performed such calculations for a
situation where the detectors were oriented in the same direction and the recording
events occurred simultaneously12 . His conclusions are:
12
The basically analogous process of positronium annihilation is considered in Feynman’s
lectures on physics [1965b], Sect. 18.3. This solution is briefly reproduced in the Appendix
to this chapter.
35
1. In the initial state of the system, when its total spin is fixed, it is admissible to talk about the spins of individual neutrons only provisionally. The
possible spin states of a pair of neutrons depicted as ↑↓ and ↓↑ interfere
between themselves and just this interference provides both the fixed total
spin during the experiment and the conjugacy of the results of measuring
the spins for remote neutrons.
2. In individual experiments, the possible values of the spin projection for each
of the neutrons, positive and negative, occur randomly with a probability of
1/2; detectors A and B always record the opposite spin directions.
3. The unequivocal connection between the results of measuring the particle
spins does not point to any influence of one of them on the other after the
termination of the interaction understood in the classical sense.
Thus, the predictions of the two theories are significantly different: according
to quantum mechanics, oppositely directed particle spin projections are always
recorded, i.e., pd = 1, while for all nontrivial semiclassical models there is a nonzero
probability of identically directed projections and pd < 1.
Why does not the assumption of the independence of the neutron states hold
while the quantum theory confirms that there is no interaction between the neutrons? The reason is that both neutrons always constitute a single quantum system
that turns to one of its two eigenstates as a result of the measurement. In each
of the system’s eigenstates, the neutron spin projections are opposite. “If there
are two particles in nature which are interacting, there is no way of describing
what happens to one of the particles by trying to write down a wave function for
it alone. The famous paradoxes... where the measurements made on one particle
were claimed to be able to tell what was going to happen to another particle, or
were able to destroy an interference have caused people all sorts of trouble because
they have tried to think of the wave function of one particle alone, rather than the
correct wave function in the coordinates of both particles. The complete description can be given correctly only in terms of functions of the coordinates of both
particles.” (Feynman 1965b, vol. 3, p. 231).
10.4. Verification of Bell’s inequalities
The EPR thought experiment remained as such until the appearance of the papers
by Bell [1964, 1966], who pointed out the possibility of its experimental verification. Imagine that a series of experiments of the type described above is made
in the scheme shown in Fig. 3, but now the detectors are not fixed in the same
direction – the observers randomly choose one of the three possible orientations of
36
their instruments independently of one another. As above, the observer records an
event of one of the two types in an individual experiment: (G) the direction of the
measured spin projection coincides with the arbitrarily chosen positive orientation
of the magnet poles, and (R) the measured spin projection is directed oppositely.
Bell proved that considering the neutrons as independent particles in the classical
sense (in other words, the existence of hidden variables) entailed the fulfillment
of the rigorous inequality for the probabilities of the combinations of events of a
certain type.
Bell’s theorem, especially in the form that was imparted to it by Clauser et
al. [1969], allows the conclusions of the classical and quantum models to be really
compared (one can find the clear formal discussion in a lecture course by Kiselev
2009). The fact that in almost all experiments photons were used instead of
neutrons and the photon polarization rather than the spin projection was measured
does not change the essence of the problem. Pairs of photons with correlated
linear polarizations were produced under two-photon transitions of exited atoms;
the polarization direction of each photon was ascertained using calcite crystals or
other analyzers (Shimony 1988).
Almost all experiments, starting from the first of them performed by Freedman
and Clauser [1972], suggest that Bell’s inequalities break down; in the cases where
the opposite result was obtained subsequent verification revealed shortcomings of
the experiment. As was said above, Bell’s inequalities must hold if the classical understanding of independence is valid, therefore, their breakdown suggests that the
existence of hidden parameters is incompatible with the behavior of the microworld.
The experiments performed by Aspect et al. [1982] became widely known (see also
Aspect 1999; Mermin 1981; Brida et al. 2000). To exclude the possibility of mutual influence of the recording events in a given experiment, the orientation of the
polarization analyzers was chosen during the flight of photons. Experiments on
the correlation of photons in the parts of the setup the separation between which
reached 18 km have been performed recently (Salar et al. 2008); these experiments
have again demonstrated the unavoidability of the quantum behavior of light.
Although the described experiments are spectacular, it should be recognized
that fundamentally they have added little since the debate between Einstein and
Bohr regarding the meaning of the EPR effect and, especially, after the formal
clarification of the problem by Bohm. However, the work of Bell is of great importance in more general relation, namely, substantiation of quantum mechanics. The
point is that in his article of 1966 Bell discovered that the known von Neumann’s
(1932) proof of impossibility to incorporate hidden parameters into quantum mechanics relied on an erroneous assumption (see also an excellent discussion by
Rudolf Peierls 1979). The Bell’s argument has given a new, this time a correct
proof that the theory of classical realism with hidden variables can reproduce the
37
experimentally confirmed predictions of quantum mechanics only by violation an
essential physical requirement, the condition of locality.
10.5. Illusory superluminal speed
The classical interpretation of the above-mentioned experimental data inevitably
leads to the supposition that the signals between remote observers can be transmitted with a superluminal speed, which is in conflict with the experimentally tested
postulate of the special theory of relativity. In numerous publications, not only
popular ones, fairly vague explanations are given on this subject. In particular,
one alludes to the mysterious ‘collapse of the wave function’ (wave packet) that
we have already commented on in Section 9 devoted to the laws of quantum mechanics. Meanwhile, the transmission of a signal (information) between observers
is not required at all within the framework of the latter for both identical and
independent orientations of the detectors. In both cases, the lamps of different
colors on each of the detectors turn on equally frequently and if the observers are
isolated from each other, then the strong correlation between the measurement
results remains unknown to them. The existence of a correlation will be revealed
only after the completion of a fairly long series of experiments, when all data will
be collected in one place. The same is also true for the situation where one of the
observers will change the orientation of his detector, thereby attempting to code
the message to his colleague: the lamps of different colors on each detector turn
on equally frequently, while the correlation between the results remain hidden.
It is also worth noting that knowledge of the quantum laws by the observers
does not promote the transmission of information between them either. Suppose
that the detectors are oriented identically and observer A is slightly closer to the
source than B, so the former will perform a measurement slightly earlier than the
latter. If observer A sees the turn-on of the green lamp, then he immediately
receives one bit of information relative to the result of observer B: the latter will
see the turn-on of the red lamp. However, this information is provided by the set of
all stages of the experiment: preparing the initial two-particle system, providing
the corresponding information to the observers, and, finally, the flight of particles
from the source to the detectors.
Let us explain this by a simple example. Suppose that an inhabitant of Siberia
sends messages to two friends living in Paris and Tokyo. All three agreed in advance that a piece of green paper be randomly (e.g., equiprobably) enclosed in
one of the envelopes and a piece of red paper be enclosed in the other. Obviously, having opened the envelope, the Parisian will immediately learn the color
of the enclosed paper in the envelope of Tokyo’s resident. In this case, no information is transmitted between the remote points – the connectivity of the events
38
is attributable to their common past history.
Basically, we see in the EPR effect the same manifestation of the specific quantum behavior as in the experiment on the diffraction of electrons by two widely
separated slits. This is quite sufficient to once again, as Bohr said, experience a
shock when one familiarizes oneself with quantum mechanics, but it is absolutely
unnecessary to invoke the reasoning about the influence on remote objects with a
superluminal speed. Like most other paradoxes of quantum mechanics, this reasoning is attributable to the improper use of the concept of a state of a quantum
system (see Section 9.2 above). Changing the detector orientation, the researcher
carries out a new experiment whose description enters as a component into the new
state of the entire system. It is quite natural that changing the experimental conditions can affect the result obtained. By choosing the orientation of his detector,
the observer selects the possible alternatives of the experiment, thereby creating
the illusion of influence on a remote object. For the subsequent ‘explanation’ of
the effect, it remains only to repeat what was said in Section 10.3 and to refer to
the analysis given in the Appendix.
Einstein rightly believed the direct action at a distance to be inadmissible
in physical theories, but there is no need to resort to this concept in quantum
mechanics.
11. Schrödinger’s cat could play dice
An impressive illustration of the concept of entanglement of quantum states was
proposed by Erwin Schrödinger in his 1935 papers initiated by the work of Einstein,
Podolsky, and Rosen. The case in point is a thought experiment, in which a cat
is in a superposition of partially alive and partially dead states. Consider the
probabilistic aspect of this construction.
Imagine that the following objects were placed in a closed box: a cat, a grain
of radioactive material, a Geiger counter, an ampoule with a quickly acting poison,
and some actuating device that breaks the ampoule when the counter is triggered.
The decay half-life of the radioactive atoms is 1 hour; poison release leads to immediate death of the cat. The critical question is: How must an external observer
describe the cat’s state after several hours?
One usually reasons as follows. There are two eigenstates of the entire (closed)
system: the first corresponds to the situation where a β-particle has not yet triggered the counter, the ampoule is intact, and the cat is alive; the second state
corresponds to the case where the poison spilled, causing the death of the cat.
The complete description of the system is given by a wave function that is a superposition of eigenstates with weight coefficients determined by the details of the
39
experiment. As has already been noted above, one of the possible states of any
closed system is realized only when the measurement is done. Therefore, the cat
is partially alive and, at the same time, partially dead until the observer opens the
box and, thus, ‘measures’ its state.
The classical experiment tells us that the object is in one of the possible states
irrespective of whether the measurement is made. (Einstein: “The moon exists
even when I don’t look at it.”) Accordingly, the ‘classical cat’ is either alive or
dead but we do not know precisely in which state it is. In contrast, the above
description leads to the conclusion that the cat is in a strange superposition of
eigenstates.
Let us now consider the experiment more carefully. First of all, it should be
emphasized that a subsystem of a closed system, in our case, the cat, cannot
be characterized by a wave function. The state of the subsystem is described
using the density matrix introduced by von Neumann, which corresponds to a
mixture of several pure states taken with fixed weights. In the probability theory,
this corresponds to the so-called randomization of the distribution function over
the possible values of some parameter. Landau and Lifshitz [1963] pointed out,
“For the states that have only the density matrix, there is no complete system
of measurements that would lead to unequivocally predictable results”. Note also
that the evolution of the subsystem depends on all details of its interaction with
the remaining part of the system.
Thus, there is no way of ascertaining the cat’s state until the observer opens
the box, hence, the question about how it feels is vacuous. Physics answers only
reasonable questions13 . The present-day experiments with macroscopic systems
that have two eigenstates are properly described using the apparatus of quantum
mechanics (Blatter 2000).
In Schrödinger’s experiment, the situation is dramatized by the fact that a
habitual breather is involved in it. Nothing will change essentially if we replace
the cat, say, with a pendulum clock and the poison with a stopper. We can go even
farther and place a microlaser or even a vibrating diatomic molecule in a closed
system instead of the clock.
The paradox with Schrödinger’s cat can also be resolved without using such
formal concepts as the density matrix: the terminology itself invoking the concept of a system that is partially in different states is provisional. For example,
analyzing the experiment with the passage of light through a tourmaline crystal
described in Section 5, Dirac [1958] wrote: “Some further description is necessary
in order to correlate the results of this experiment with the results of other experiments that might be performed with photons and to fit them all into a general
13
However, as science in general, physics is able to answer not all reasonable questions.
40
scheme. Such further description should be regarded, not as an attempt to answer
questions outside the domain of science, but as an aid to the formulation of rules
for expressing concisely the results of large numbers of experiments. The further
description provided by quantum mechanics runs as follows. It is supposed that
a photon polarized obliquely to the optic axis may be regarded as being partly
in the state of polarization parallel to the axis and partly in the state of polarization perpendicular to the axis.” Obviously, the cat in two states is no more
surprising than the analogous behavior of a photon. The ‘simultaneous’ passage
of an electron along all possible trajectories in Feynman’s approach should also be
understood in the same sense (see Section 9.1).
Let us also touch on the following curious question: Will the description of
the system change from the standpoint of an external observer if the radioactive
material is replaced with an ordinary die? Say, if the clock hand (or the cat itself)
throws the die from a height very large compared to its sizes after a given time. If
an odd number occurs, the actuating device (or the cat) breaks the ampoule with
poison.
The question touches on the widespread conviction that “...without a β-particle,
nobody could even think about the admission of such a strange superposition”
(Kadomtsev 1999). Only atomic phenomena, in particular, radioactive decay, are
widely believed to provide ‘true’ randomness, unlike the behavior of macroscopic
bodies governed by the laws of classical mechanics. However, in the experiment
with Schrödinger’s cat, if it is considered from the viewpoint of quantum mechanics, it does not matter precisely how the necessity of a probabilistic description
of the subsystem from the standpoint of an external observer is provided. As for
quality of the device that implements stochasticity, analysis of the probabilistic
models of classical physics suggest the following: when the simple conditions that
provide instability of motion are met, the behavior of the die is as unpredictable
as radioactive decay. After all, the die can be reduced to the sizes of a Brownian
particle without violating the fundamental aspect of the experiment.
12. Conclusion
The above examples strongly suggest that God does not play dice; in the quantum
world, as in classical mechanics, the probabilistic concepts are inherent only in
models of real phenomena but not in Nature itself. The word ‘model’ is a key one
here – as soon as the stochastic behavior is ascribed to some real objects, such as
sea waves, billiard balls, molecules, or photons, the appearance of contradictions
becomes inevitable. This assertion seems so obvious that it remains only to wonder how widely the opposite viewpoint is covered in literature, including classical
41
works. The probabilistic concepts, as a branch of mathematics, were created for
several centuries in order to formalize various life experiences and scientific data.
It would be strange to believe that these concepts were ‘built’ in objective reality from the outset; the probability theory is only an efficient tool for modeling
Nature.
Another conclusion is that standard quantum mechanics in terms of which our
discussion was conducted gives a strange but logically consistent explanation for all
of the experiments performed to date. This is enough to leave aside the widespread,
in recent years, attempts to associate the interpretation of experiments with the
consciousness of an observer, the birth of universes at each act of observation,
the transmission of information with a superluminal speed, and other mystical
phenomena. The aforesaid by no means rules out the quest for models different
from the theories of Schrödinger, Heisenberg, and Feynman; they may give a deeper
and clearer picture of quantum processes (in fact, Feynman’s approach serves
as such an example). New experiments may also require a fundamentally new
approach, but it is quite unrealistic to expect the appearance of a deterministic
basic model.
Note also that the probabilistic formulation of models for real processes determines the statistical nature of the inverse problem that consists in ascertaining
the true properties of the processes from their observed manifestations (Terebizh
1995, 2005).
It has been repeatedly pointed out that the difficulties in understanding the
quantum theory primarily stem from the fact that the behavior of quantum systems is unusual: our everyday experience concerns the properties of surrounding
us massive bodies and waves, while the microparticles are neither the former nor
the latter. This is true, but still it is hard to avoid the feeling that the rules of
quantum mechanics are a set of strange procedures justified by nothing but their
unconditional practical efficiency. If there was no statistical physics, similar feelings would also be aroused by the laws of thermodynamics – while being important
in engineering. In a similar context, Richard Feynman described the algorithm for
predicting solar eclipses developed for centuries by South American Indians. From
generation to generation, priests handed ropes with many tied knots to their disciples; it took many years to memorize the rules for handling the knots, but the
meaning of these rules remained completely mysterious for Indians. We will add
that the system worked like the model proposed by Thales of Milet in the 6th
century B.C.; a significant difference between the approaches is that, without explaining the nature of the motion of celestial bodies, Thales’s theory nevertheless
proceeded from a simple model of the Solar system. Critics of quantum mechanics
also would like to have a simpler underlying model of microworld phenomena from
which the set of quantum rules would follow.
42
In response to reasoning of this kind, an advocate of the orthodox interpretation of quantum mechanics can recall the origin of the classical equations of
electrodynamics. James Clerk Maxwell, who wrote a complete system of equations, made much effort to create a mechanistic model of phenomena that led to
the equations of electrodynamics. This proved to be an unsolvable problem, while
Maxwell’s equations per se became habitual in time to an extent that physicists
left aside the question about their origin. A similar situation also arose with regard
to the basic principles of classical mechanics, in particular, the law of inertia.
Eight centuries ago, the King of Castile Alfonso X named ‘The Wise’ had
reasons to note, “Had I been present at the creation of the World, I should have
recommended something simpler.” We can only guess what Alfonso X would say
after familiarizing himself with quantum mechanics. It probably would not be
simpler.
Acknowledgements
I am deeply grateful to V.V. Biryukov (Moscow State University), Yu.A. Kravtsov
(Space Research Institute, Moscow), and M.A. Mnatsakanian (California Institute
of Technology) for a stimulating discussion of the issues under consideration and
constructive suggestions. In particular, Yu.A. Kravtsov pointed to the importance
of allowance for the low-frequency radiation in collisions of molecules and the
necessity of discussing the problem of quantum chaos.
Appendix. Positronium annihilation
In our description of the experiment on the recording of neutrons, we gave only
a reference to the formal solution of the problem in terms of quantum mechanics
performed by Bohm [1952]. Below, we reproduce with minor changes the analytical consideration of a basically similar process – the annihilation of a positronium
atom – contained in Sect. 18.3 of the lectures by Feynman [1965b]. This solution
is all the more instructive, because it is given with clarity characteristic of Richard
Feynman and is performed in the context of a modern approach to quantum mechanics.
A positronium atom is composed of an electron e− and a particle with opposite
charge sign, a positron e+ . The spin of each of these particles is 1/2 (in units of h̄);
initially, we consider an atom at rest with antiparallel spins of its components and
a zero total spin. The characteristic lifetime of this system is 10−10 s, following
which the electron and the positron annihilate with the emission of two γ-ray
quanta. The latter fly apart in opposite directions with equal speeds; the direction
43
of the flight is distributed isotropically. We are interested in the polarization of
the produced photons.
In this case, to describe the photons, it is convenient to choose a state of
circular polarization. Recall that we arbitrarily attribute a right-hand circular
polarization to the monochromatic light wave if the electric field vector rotates
counterclockwise as it travels toward the observer; the photons constituting the
wave, accordingly, are assumed to be right-hand circularly polarized (state |Ri).
In a beam of left-hand circularly polarized photons (state |Li), the electric field
vector rotates clockwise if we look at the approaching wave.
Obviously, two alternate modes of decay that conserve the system’s zero total
spin are admissible (Fig. 5). Two right-hand circularly polarized photons are
produced in the first mode; each of them has an angular momentum of +1 relative
to its momentum direction, while the angular momenta relative to the z axis are
+1 and −1. Denote this state of the system by |R1 R2 i. Two left-hand circularly
polarized photons are produced in the second mode denoted by |L1 L2 i. We will
assume that |R1 R2 i and |L1 L2 i constitute an orthonormal basis.
According to the superposition principle, the final state of the system after
annihilation |F i is a linear combination of the alternate states |R1 R2 i and |L1 L2 i.
To find the corresponding coefficients, we should take into account two conditions:
(1) the initial state – a positronium atom with zero spin – is characterized by odd
parity and the parity of |F i must be the same; (2) the normalization of |F i must
provide a unit sum of the probabilities of all possible realizations. The only linear
combination of the alternative states that satisfies both conditions is
√
(A1)
|F i = (|R1 R2 i − |L1 L2 i) / 2.
Indeed, the operator of spatial inversion P̂ changes both the direction of photon motion and the direction of its polarization; therefore, P̂ |R1 R2 i = |L1 L2 i,
P̂ |L1 L2 i = |R1 R2 i. As a result, P̂ |F i = −|F i, suggesting that parity is conserved
under positronium annihilation.
Having the final state of the system, we can calculate the amplitudes and
probabilities of events of various kinds. In particular, the amplitudes of the two
alternative types of decay that we mentioned above are
√
√
hR1 R2 |F i = 1/ 2,
hL1 L2 |F i = −1/ 2,
(A2)
so the probabilities of both modes are 1/2. Physically, the representation (A1)
means that the detectors placed in the positive and negative directions of the zaxis will always record equiprobably either a pair of right-hand photons or a pair
of left-hand photons.
The scheme of the experiment shown in Fig. 6 is of interest in the context of the
question about the possibility of action at a distance discussed in Section 10. The
44
Figure 5: Alternate states of the pair of photons that resulted from positronium annihilation.
photons that fly apart after positronium annihilation pass through calcite crystals,
as a result they become linearly polarized either in the x- or in the y- direction.
Each of the four possible channels of photon propagation is equipped with a light
detector. It is required to ascertain the way in which this scheme operates when
many annihilation processes are observed successively. More specifically, since the
triggering of one of the four pairs of counters is admissible at each annihilation:
D1x and D2x , D1x and D2y , D1y and D2x or, finally, D1y and D2y , the probabilities
of the corresponding processes should be found.
As an example, let us calculate the amplitude hy1 x2 |F i of the event that consists in the triggering of counters D1y and D2x . Taking into account the representation (A1), we find
√
(A3)
hy1 x2 |F i · 2 = hy1 x2 |R1 R2 i − hy1 x2 |L1 L2 i.
Since the photon recordings by different counters are independent events, we assume that
(A4)
hy1 x2 |R1 R2 i = hy1 |R1 ihx2 |R2 i
and similarly for hy1 x2 |L1 L2 i. As a result, Eq. (A3) takes the form
√
hy1 x2 |F i · 2 = hy1 |R1 ihx2 |R2 i − hy1 |L1 ihx2 |L2 i.
(A5)
Next, we should take into account the fact that the states of right-hand and lefthand circular polarization are related to the states of linear polarization along the
x- and y- directions by the relations
(
√
|Ri = (|xi + i |yi) /√ 2,
(A6)
|Li = (|xi − i |yi) / 2.
45
Figure 6: Scheme for recording the photons produced by positronium annihilation. C1 and C2 are the calcite crystals; x1 , y1 , x2 and y2 are the paths of
the photons linearly polarized in the x and y directions; D1x , D1y , D2x and
D2y are the photon counters.
Taking into account the orthonormality of the system |xi and |yi, we find from
(A6):
(
√
√
hy1 |R1 i = +i/√ 2,
hx2 |R2 i = 1/√ 2,
(A7)
hy1 |L1 i = −i/ 2,
hx2 |L2 i = 1/ 2.
Substituting these expressions into (A5) finally yields
√
hy1 x2 |F i = i/ 2,
(A8)
so that the probability of the corresponding process is 1/2.
The amplitudes of the three remaining processes are calculated similarly. As
a result, we obtain:
(
hx1 x2 |F i = hy1 y2 |F i = 0, √
(A9)
hx1 y2 |F i = hy1 x2 |F i = i/ 2.
These expressions show that one of the pairs of counters in the channels with
mutually orthogonal polarizations of photons is triggered always, and with equal
probabilities: either D1x and D2y or D1y and D2x ; the triggering probabilities of the
pairs with identically directed polarizations of photons are zero. Thus, Feynman’s
analysis of the positronium annihilation process leads to the same conclusions as
those listed in Section 10.3 from Bohm’s calculations concerning the recording of
particles with a spin of 1/2.
Such are the predictions of the quantum theory that are fully consistent with
experimental data. Where is then the paradox in the described situation? Suppose
that observer 2 is slightly farther from the positronium atom than observer 1. As
a consequence, the triggering of a particular counter of observer 1 will allow him to
predict with certainty precisely which counter of observer 2 will be triggered. On
the other hand, the photon flying toward observer 2 is in a superposition of states
|x2 i and |y2 i with linear polarization, therefore, this photon could seemingly reach
one of the detectors of observer 2 with some nonzero probabilities irrespective of the
46
result obtained by observer 1. Why the recording of a photon by a remote observer
completely determines the result of an experiment that has not yet happened?
Does this mean that there is some interaction that propagates with a speed higher
than the speed of light in a vacuum?
In his lectures, Feynman gives a detailed interpretation of the experiment with
which, of course, one should familiarize oneself carefully. In our view, even the
structure of representation (A1) gives an answer to the above questions: The pair of
photons produced by annihilation was initially prepared in such a way that the two
corresponding detectors always recorded a circular polarization of one type. This
‘preparedness’ of the system is retained after the passage of photons through calcite
crystals and determines the fact of triggering the pairs of detectors with mutually
orthogonal polarizations of photons. No influence, not to mention superluminal
one, on the remote process is required. The situation is similar to the example
with the sending of letters to remote points considered in Section 10.5 with the only
significant (but now familiar) difference that the quantum processes are nonlocal.
“Do you still think there is a ‘paradox’ ? Make sure that it is, in fact, a paradox
about the behavior of Nature, by setting up an imaginary experiment for which the
theory of quantum mechanics would predict inconsistent results via two different
arguments. Otherwise the ‘paradox’ is only a conflict between reality and your
feeling of what reality ought to be” (R. Feynman 1965b, p. 261).
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