zotero-db/storage/N8A7HUFQ/.zotero-ft-cache

.DESC RI PT ION OF P H YSI CAL REALITY

of lanthanum is 7/2, hence the nuclear magnetic moment as determined by this analysis is 2.5 nuclear magnetons. This is in fair agreement with the value 2.8 nuclear magnetons deter-
mined, from La III hyperfine structures by the writer and N. S. Grace. 9
' M. F. Crawford and N. S. Grace, Phys. Rev. 4'7, 536
(1935).

This investigation was carried out under the
supervision of Professor G. Breit, and, I wish to
thank him for the invaluable advice and assis-
tance so freely given. I also take this opportunity to acknowledge the award of a Fellowship by the Royal Society of Canada, and to thank the
University of Wisconsin and the Department of
Physics for the privilege of working here.

MAY 15, 1935

PH YSI CAL REVI EW

VOLUM E 4 7

Can Quantum-Mechanical Description of Physical Reality Be Considered Complete' ?
A. EINsTEIN, B. PQDoLsKY AND N. RosEN, Institute for Advanced Study, Princeton, New Jersey
(Received March 25, 1935)

In a complete theory there is an element corresponding to each element of reality. A sufFicient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system. In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in

quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality. Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false. One is thus led to conclude that the description of reality as given by a wave function is not complete.

A NY serious consideration of a physical theory must take into account the dis-
tinction between the objective reality, which is
independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves.
In attempting to judge the success of a physical theory, we may ask ourselves two ques-
tions: (1) "Is the theory correct?" and (2) "Is
the description given by the theory complete?"
It is only in the case in which positive answers
may be given to both of these questions, that the concepts of the theory may be said to be satisfactory. The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience. This experience, which alone enables us to make inferences about reality, in physics takes the
form of experiment and measurement. It is the
second question that we wish to consider here, as
applied to quantum mechanics.

Whatever the meaning assigned to the term conzp/eEe, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counter part in the physical theory We shall ca. 11 this the condition of completeness. The second question is thus easily answered, as soon as we are able to
decide what are the elements of the physical
reality. The elements of the physical reality cannot
be determined by a priori philosophical considerations, but must be found by an appeal to results of experiments and measurements. A
comprehensive definition of reality is, however, unnecessary for our purpose. We shall be satisfied with the following criterion, which we regard as
reasonable. If, without in any way disturbing a
system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical
reality corresponding lo this physical quantity. It
seems to us that this criterion, while far from exhausting all possible ways of recognizing a physical reality, at least provides us with one

778

E I NSTE I N, PODOLSKY AN D ROSE N

such way, whenever the conditions set down in it occur. Regarded not as a necessary, but merely as a sufficient, condition of reality, this criterion is in agreement with classical as well as
quantum-mechanical ideas of reality.
To illustrate the ideas involved let us consider
the quantum-mechanical description of the behavior of a particle having a single degree of freedom. The fundamental concept of the theory is the concept of state, which is supposed to be completely characterized by the wave function P, which is a function of the variables chosen to describe the particle's behavior. Corresponding to each physically observable quantity A there is an operator, which may be designated by the same letter.
If P is an eigenfunction of the operator A, that is, if
A/=a—g,

where a is a number, then the physical quantity A has with certainty the value a whenever the particle is in the state given by P. In accordance with our criterion of reality, for a particle in the state given by P for which Eq. (1) holds, there is an element of physical reality corresponding to the physical quantity A. Let, for example,

— 'p e (pre/ p) ppg

(2)

where h is Planck's constant, po is some constant number, and x the independent variable. Since the operator corresponding to the momentum of the particle is

we obtain

p = (h/2rri) 8/Bx,

p' = pp = (h/2iri) 8$/Bx = p pp

(4)

Thus, in the state given by Eq. (2), the momen-
tum has certainly the value pp. It thus has
meaning to say that the momentum of .the particle in the state given by Eq. (2) is real.
On the other hand if Eq. (1) does not hold,
we can no longer speak of the physical quantity A having a particular value. This is the case, for
example, with the coordinate of the particle. The
operator corresponding to it, say g, is the operator of multiylication by the independent variable.
Thus,

In accordance with quantum mechanics we can only say that the relative probability that a measurement of the coordinate will give a result lying between a and b is
P(a, b) = PPdx= I dx=b a. —(6)
Since this probability is independent of a, but
depends only upon the difference b —a, we see
that all values of the coordinate are equally probable.
A definite value of the coordinate, for a particle in the state given by Eq. (2), is thus not predictable, but may be obtained only by a direct measurement. Such a measurement however disturbs the particle and thus alters its state. After the coordinate is determined, the particle will no longer be in the state given by Eq. (2). The usual conclusion from this in quantum mechanics is that when the momentnm of a particle is known, its coordhnate has no physical
reali ty.
More generally, it is shown in quantum mechanics that, if the operators corresponding to
two physical quantities, say A and B, do not commute, that is, if AB/BA, then the precise
knowledge of one of them precludes such a knowledge of the other. Furthermore, any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the first.
From this follows that either (1) t'he guanturnmechanical description of rea1ity given by the wave function is not cornplele or (2) when the operators
corresponding .to two physical qlantities do not commute the two quantifies cannot have simul-
— — taneous reality. For if both of them had simul-
taneous reality and thus definite values these values would enter into the complete description,
according to the condition of completeness. If then the wave function provided such a complete description of reality, it would contain these values; these would then be predictable. This
not being the case, we are left with the alternatives stated.
In quantum mechanics it is usually assumed that the wave function does contain a complete description of the physical reality of the system in the state to which it corresponds. At first

DES CRI PT ION OF P H YS I CAL REALITY

779

sight this assumption is entirely reasonable, for the information obtainable from a wave function seems to correspond exactly to what can be measured without altering the state of the system. We shall show, however, that this assumption, together with the criterion of reality given above, leads to a contradiction.

2.

For this purpose let us suppose that we have

two systems, I and II, which we permit to inter-
act from the time t =0 to t = T, after which time

we suppose that there is no longer any interaction

between the two parts. We suppose further that
the states of the two systems before t=0 were

known. We can then calculate with the help of

Schrodinger's equation the state of the combined
t) system I+II at any subsequent time; in par-
ticular, for any T. Let us designate the cor-
responding wave function by +. Ke cannot,

however, calculate the state in which either one

of the two systems is left after the interaction.

This, according to quantum mechanics, can be

done only with the help of further measurements,

by a process known as the reduction of the wave

packet. Let us consider the essentials of this

process.

Let a~, a2, a3,

be the eigenvalues of some

physical quantity A pertaining to system I and

u((x(), u2(x)), us(x(), ~ the corresponding

eigenfunctions, where x& stands for the variables

used to describe the first system. Then +, con-

sidered as a function of x~, can be expressed as

+(x(, xm) = Q ))(.(xm)u. (x(),

where x2 stands for the variables used to describe
the second system. Here P„(x&) are to be regarded
merely as the coefficients of the expansion of +
into a series of orthogonal functions u„(x)). Suppose now that the quantity A is measured and it is found that it has the value af, . It is then concluded that after the measurement the first
system is left in the state given by the wave function uh(x(), and that the second system is
left in the state given by the wave function ph(x2). This is the process of reduction of the
wave packet; the wave packet given by the

infinite series (7) is reduced to a single term

)t h(xg)uh(x().
The set of functions u„(x() is determined by the choice of the physical quantity A. If, instead

of this, we had chosen another quantity, say B,

having the eigenvalues b~, b2, b3,

and eigen-

functions v((x(), v2(x(), v3(x(),

we should

have obtained, instead of Eq. (7), the expansion

4'(x), xm) = Q ((),(x2)v, (x(),
s=l
where y, 's are the new coeAicients. If now the
8 quantity is measured and is found to have the
value b„we conclude that after the measurement the first system is left in the state given by v„(x()
(. and the second system is left in the state given
by (»)
We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is
meant by the absence of an interaction between
the two systems. Thus, it is possible to assign two
different wave functions (in our example )Ih and
e„) to the same reality (the second system after the interaction with the first).
Now, it may happen that the two wave functions, )th and e„, are eigenfunctions of two noncommuting operators corresponding to some
physical quantities P and Q, respectively. That
this may actually be the case can best be shown by an example. Let us suppose that the two systems are two particles, and that

— +(x& x2)

e(2&&ih) (&s—&2+&o) ndp

—co

where x0 is some constant. Let A be the momentum of the first particle; then, as we have seen in Eq. (4), its eigenfunctions will be

(x ) e(2m(/h) yes,

(io)

corresponding to the eigenvalue p. Since we have here the case of a continuous spectrum, Eq. (7)
will now be written

780

EINSTEIN, PODOLSKY AND ROSEN

%(x(, xg) = P„(x,)u, (x))dP,

4

where

— (x ) s—(sw(l h) (zg —zo) P

(12)

This P„however is the eigenfunction of the
operator

P = (8/2')(7/Bx2,

(13)

corresponding to the eigenvalue —p of the
momentum of the second particle. On the other
8 hand, if is the coordinate of the first particle,
it has for eigenfunctions

corresponding to the eigenvalue x, where
()(xq —x) is the well-known Dirac delta-function.
Eq. (8) in this case becomes

where
e(21r i/h) (x—F2+$0) gldp
= h()(x —xp+xo). (16)
This q, however, is the eigenfunction of the
operator (17)
corresponding to the eigenvalue x+xo of the
coordinate of the second particle. Since
PQ QP = 1(/2mi, —
we have shown that it is in general possible for P~ and p„ to be eigenfunctions of two noncommuting operators, corresponding to physical quantities.
Returning now to the general case contemplated in Eqs. (7) and (8),. we assume that P),
and y„are indeed eigenfunctions of some non-
commuting operators P and Q, corresponding to
the eigenvalues pI, and q„, respectively. Thus, by
8 measuring either A or we are in a position to
predict with certainty, and without in any way

disturbing the second system, either the value
of the quantity P (that is p)„) or the ~alue of the
quantity Q (that is q„). In accordance with our criterion of reality, in the first case we must
consider the quantity I' as being an element of
reality, in the second case the quantity Q is an element of reality. But, as we have seen, both wave functions P), and q, belong to the same reality.
Previously we proved that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical
quantities do not commute the two quantities
cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with rioncommuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only
other alternative (2). We are thus forced to conclude that the quantum-mechanical descrip-
tion of physical reality given by wave functions is not complete.
One could object to this conclusion on the grounds that our criterion of reality is not suf-
ficiently restrictive. Indeed, one would not arrive
at our conclusion if one insisted that two or more
physical quantities can be regarded as simultaneous elements of reality only +hen they can be simultaneously measured or predi cted. On this point of' view, since either one or the other, but
not both simultaneously, of the quantities I'
and Q can be predicted, they are not simultane-
ously real. This makes the reality of P and Q
depend upon the process of measurement carried out on the first system, which does, not disturb
the second system in any way. No reasonable definition of reality could be expected to permit
this. While we have thus shown that the wave
function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.