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the eix·A all commute, and consequently the Aj commute. QED.

Quantum mechanics forced a major change in the notion of reality.
The position and momentum of a particle could no longer be thought
of as properties of the particle. They had no real existence before mea-
surement of the one with a given accuracy precluded the measurement of
the other with too great an accuracy, in accordance with the uncertainty
principle. This point of view was elaborated by Bohr under the slogan
of “complementarity”, and Heisenberg wrote a book [35] explaining the
physical basis of the new theory.
At the Solvay Congress in 1927, Einstein was very quiet, but when
pressed objected that ψ could not be the complete description of the state.

For example, the wave function in Fig. 4 would have axial symmetry, but
the place of arrival of an individual particle on the hemispherical screen
does not have this symmetry. The answer of quantum mechanics is that
the symmetrical wave function ψ describes the state of the system before
a measurement is made, but the act of measurement changes ψ.

Figure 4

To understand the rˆle of probability in quantum mechanics it is nec-
essary to discuss measurement. The quantum theory of measurement was
created by von Neumann [33]. A very readable summary is the book by
London and Bauer [36]. See also two papers of Wigner [37] [38], which
we follow now.
Consider a physical system with wave function ψ in a state of super-
position of the two orthogonal states ψ1 and ψ2 , so that ψ = ±1 ψ1 + ±2 ψ2 .
We want to perform an experiment to determine whether it is in the state
ψ1 or the state ψ2 . (We know that the probabilities are respectively |±1 |2
and |±2 |2 , but we want to know which it is in.)
If A is any observable, the expected value of A is

±1 ψ1 + ±2 ψ2 , A(±1 ψ1 + ±2 ψ2 ) =
|±1 |2 (ψ1 , Aψ1 ) + |±2 |2 (ψ2 , Aψ2 ) + ±1 ±2 (ψ1 , Aψ2 ) + ±2 ±1 (ψ2 , Aψ1 ).
¯ ¯

Suppose now we couple the system to an apparatus designed to measure
whether the system is in the state ψ1 or ψ2 , and that after the interaction

the system plus apparatus is in the state
• = ±1 ψ1 — χ1 + ±2 ψ2 — χ2 ,
where χ1 and χ2 are orthogonal states of the apparatus. If A is any
observable pertaining to the system alone, then
(•, A — 1•) = |±1 |2 (ψ1 , Aψ1 ) + |±2 |2 (ψ2 , Aψ2 ).
Thus, after the interaction with the apparatus, the system behaves like a
mixture of ψ1 and ψ2 rather than a superposition. It is a measurement of
whether the system is in the state ψ1 or ψ2 .
However, observe that knowing the state of the system plus apparatus
after the interaction tells us nothing about which state the system is in!
The state • is a complete description of the system plus apparatus. It is
causally determined by ψ and the initial state of the apparatus, according
to the Schr¨dinger equation governing the interaction. Thus letting the
system interact with an apparatus can never give us more information.
If we knew that after the interaction the apparatus is in the state χ1
we would know that the system is in the state ψ1 . But how do we tell
whether the apparatus is in state χ1 or χ2 ? We might couple it to another
apparatus, but this threatens an in¬nite regress.
In practice, however, the apparatus is macroscopic, like a spot on a
photographic plate or the pointer on a meter, and I merely look to see
which state it is in, χ1 or χ2 . After I have become aware of the state χ1 or
χ2 , the act of measurement is complete. If I see that the apparatus is in
state χ1 , the system plus apparatus is in state ψ1 —χ1 and the system is in
state ψ1 . (As we already knew, this will happen with probability |±1 |2 .)
Similarly for χ2 . After the interaction but before awareness has dawned
in me, the state of the system plus apparatus is ±1 ψ1 — χ1 + ±2 ψ2 — χ2 ;
the instant I become aware of the state of the apparatus, the state of
the system plus apparatus is either ψ1 — χ1 or ψ2 — χ2 . Thus the state
can change in two ways: continuously, linearly, and causally by means of
the Schr¨dinger equation or abruptly, nonlinearly, and probabilistically
by means of my consciousness. The latter change is called “reduction of
the wave packet”. Concerning the reduction of the wave packet, Wigner
[39] writes:
“This takes place whenever the result of an observation enters the
consciousness of the observer”or, to be even more painfully precise, my
own consciousness, since I am the only observer, all other people being
only subjects of my observations.”
100 CHAPTER 14

This theory is known as the orthodox theory of measurement in quan-
tum mechanics. The word “orthodox” is well chosen: one suspects that
many practicing physicists are not entirely orthodox in their beliefs.
Of those founders of quantum mechanics whom we labeled “reac-
tionaries”, none permanently accepted the orthodox interpretation of
quantum mechanics. Schr¨dinger writes [40, p. 16]:
“For it must have given to de Broglie the same shock and disappoint-
ment as it gave to me, when we learnt that a sort of transcendental,
almost psychical interpretation of the wave phenomenon had been put
forward, which was very soon hailed by the majority of leading theo-
rists as the only one reconcilable with experiments, and which has now
become the orthodox creed, accepted by almost everybody, with a few
notable exceptions.”
The literature on the interpretation of quantum mechanics contains
much of interest, but I shall discuss only three memorable paradoxes:
the paradox of Schr¨dinger™s cat [41, p. 812], the paradox of the nervous
student [42] [43] [44], and the paradox of Wigner™s friend [38]. The original
accounts of these paradoxes make very lively reading.
One is inclined to accept rather abstruse descriptions of electrons and
atoms, which one has never seen. Consider, however, a cat that is en-
closed in a vault with the following infernal machine, located out of reach
of the cat. A small amount of a radioactive substance is present, with a
half-life such that the probability of a single decay in an hour is about
one half. If a radioactive decay occurs, a counter activates a device that
breaks a phial of prussic acid, killing the cat. The only point of this para-
dox is to consider what, according to quantum mechanics, is a complete
description of the state of a¬airs in the vault at the end of an hour. One
cannot say that the cat is alive or dead, but that the state of it and the
infernal machine is described by a superposition of various states contain-
ing dead and alive cats, in which the cat variables are mixed up with the
machine variables. This precise state of a¬airs is the ineluctable outcome
of the initial conditions. Unlike most thought experiments, this one could
actually be performed, were it not inhumane.
The ¬rst explanations of the uncertainty principle (see [35]) made it
seem the natural result of the fact that, for example, observing the posi-
tion of a particle involves giving the particle a kick and thereby changing
its momentum. Einstein, Podolsky, and Rosen [42] showed that the situa-
tion is not that simple. Consider two particles with one degree of freedom.
Let x1 and p1 denote the position and momentum operators of the ¬rst,

x2 and p2 those of the second. Now x = x1 ’ x2 and p = p1 + p2 are
commuting operators and so can simultaneously have sharp values, say
x and p respectively. Suppose that x is very large, so that the two par-
ticles are very far apart. Then we can measure x2 , obtaining the value
x2 , say, without in any way a¬ecting the ¬rst particle. A measurement
of x1 then must give x + x2 . Since the measurement of x2 cannot have
a¬ected the ¬rst particle (which is very far away), there must have been
something about the condition of the value x1 = x +x2 which meant that
x1 if measured would give the value x1 = x + x2 . Similarly for position
measurements. To quote Schr¨dinger [44]:
“Yet since I can predict either x1 or p1 without interfering with system
No. 1 and since system No. 1, like a scholar in an examination, cannot
possibly know which of the two questions I am going to ask it ¬rst: it so
seems that our scholar is prepared to give the right answer to the ¬rst
question he is asked, anyhow. Therefore he must know both answers;
which is an amazing knowledge, quite irrespective of the fact that after
having given his ¬rst answer our scholar is invariably so disconcerted or
tired out, that all the following answers ˜wrong™.”
The paradox of Wigner™s friend must be told in the ¬rst person. There
is a physical system in the state ψ = ±1 ψ1 +±2 ψ2 which, if in the state ψ1 ,
produces a ¬‚ash, and if in the state ψ2 , does not. In the description of
the measurement process given earlier, for the apparatus I substitute my
friend. After the interaction of system and friend they are in the state
• = ±1 ψ1 —χ1 +±2 ψ2 —χ2 . I ask my friend if he saw a ¬‚ash. If I receive the
answer “yes” the state changes abruptly (reduction of the wave packet)
to ψ1 — χ1 ; if I receive the answer “no” the state changes to ψ2 — χ2 .
Now suppose I ask my friend, “What did you feel about the ¬‚ash before
I asked you?” He will answer that he already knew that there was (or
was not) a ¬‚ash before I asked him. If I accept this, I must accept that
the state was ψ1 — χ1 (or ψ2 — χ2 ), rather than •, in violation of the
laws of quantum mechanics. One possibility is to deny the existence of
consciousness in my friend (solipsism). Wigner prefers to believe that the
laws of quantum mechanics are violated by his friend™s consciousness.
[29]. R. P. Feynman and A. R. Hibbs, “Quantum Mechanics and Path
Integrals”, McGraw-Hill, New York, 1965.
[30]. Erwin Schr¨dinger, “Collected Papers on Wave Mechanics”, trans-
lated by J. F. Shearer and W. M. Deans, Blackie & Son limited, London,
102 CHAPTER 14

[31]. Max Born, Zur Quantenmechanik der Stossvorg¨nge, Zeitschrift f¨r
a u
Physik 37 (1926), 863“867, 38, 803“827.
[32]. P. A. M. Dirac, “The Principles of Quantum Mechanics”, Oxford,
[33]. John von Neumann, “Mathematical Foundations of Quantum Me-
chanics”, translated by R. T. Beyer, Princeton University Press, Prince-
ton, 1955.
[34]. Frigyes Riesz and B´la Sz.-Nagy, “Functional Analysis”, 2nd Edi-
tion, translated by L. F. Boron, with Appendix: Extensions of Linear
Transformations in Hilbert Space which Extend Beyond This Space, by
B. Sz.-Nagy, Frederick Ungar Publishing Co., New York, 1960.
[35]. Werner Heisenberg, “The Physical Principles of the Quantum The-
ory”, translated by Carl Eckhart and Frank C. Hoyt, Dover Publications,
New York, 1930.
[36]. F. London and E. Bauer, “La th´orie de l™observation en m´canique
e e
quantique”, Hermann et Cie., Paris, 1939.
[37]. Eugene P. Wigner, The problem of measurement, American Journal
of Physics 31 (1963), 6“15.
[38]. Eugene P. Wigner, Remarks on the mind-body question, pp. 284“302
in “The Scientist Speculates”, edited by I. J. Good, 1962.
[39]. Eugene P. Wigner, Two kinds of reality, The Monist 48 (1964),
[40]. Erwin Schr¨dinger, The meaning of wave mechanics, pp. 16“30 in
“Louis de Broglie physicien et penseur”, edited by Andr´ George, ´ditions
e e
Albin Michel, Paris, 1953.
[41]. Erwin Schr¨dinger, Die gegenw¨rtige Situation in der Quanten-
o a
mechanik, published as a serial in Naturwissenschaften 23 (1935), 807“
812, 823“828, 844“849.
[42]. A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical
description of reality be considered complete?, Physical Review 47 (1935),
[43]. N. Bohr, Can quantum-mechanical description of reality be consid-
ered complete?, Physical Review 48 (1935), 696“702.
[44]. Erwin Schr¨dinger, Discussion of probability relations between sep-
arated systems, Proceedings of the Cambridge Philosophical Society 31
(1935), 555-563.

An argument against hidden variables that is much more incisive than
von Neumann™s is presented in a forthcoming paper:
[45]. Simon Kochen and E. P. Specker, The problem of hidden variables in
quantum mechanics, Journal of Mathematics and Mechanics, 17 (1967),
Chapter 15

Brownian motion in the

Let us try to see whether some of the puzzling physical phenomena
that occur on the atomic scale can be explained by postulating a kind
of Brownian motion that agitates all particles of matter, even particles
far removed from all other matter. It is not necessary to think of a
material model of the aether and to imagine the cause of the motion
to be bombardment by grains of the aether. Let us, for the present,
leave the cause of the motion unanalyzed and return to Robert Brown™s
conception of matter as composed of small particles that exhibit a rapid
irregular motion having its origin in the particles themselves (rather like
Mexican jumping beans).
We cannot suppose that the particles experience friction in moving
through the aether as this would imply that uniform rectilinear motion
could be distinguished from absolute rest. Consequently, we cannot base
our discussion on the Langevin equation.
We shall assume that every particle performs a Markov process of the

dx(t) = b x(t), t dt + dw(t), (15.1)

where w is a Wiener process on ‚3 , with w(t) ’ w(s) independent of
x(r) whenever r ¤ s ¤ t. Macroscopic bodies do not appear to move
like this, so we shall postulate that the di¬usion coe¬cient ν is inversely
proportional to the mass m of the particle. We write it as
ν= .

106 CHAPTER 15

The constant h has the dimensions of action. If h is of the order of
¯ ¯
Planck™s constant h then the e¬ect of the Wiener process would indeed
not be noticeable for macroscopic bodies but would be relevant on the
atomic scale. (Later we will see that h = h/2π.) The kinematical as-
sumption (15.1) is non-relativistic, and the theory we are proposing is
meant only as an approximation valid when relativistic e¬ects can safely
be neglected.


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