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CAN QUANTUM-MECHANICAL DESCRIPTION OF PHYSICAL REALITY BE CONSIDERED COMPLETE?
Albert Einstein, Boris Podolsky, and Nathan Rosen
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.
1.
Any serious consideration of a physical
theory must take into account the distinction 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 questions: (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
complete, the following requirement for a complete theory seems to be a necessary one: every
element of the physical reality must have a counterpart in the physical theory. We shall call 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.
In quantum mechanics, expectation values can generally only be specified probabilistically, with confirmation provided statistically.
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 to 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
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
ψ, 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 ψ is an eigenfunction of the operator A, that
is, if
For Einstein, a statistical theory is incomplete
ψ' ≡ Aψ = aψ, (1)
where a is a number, then the physical quantity
A has with certainty the value α whenever the
particle is in the state given by ψ. In accordance
with our criterion of reality, for a particle in the
state given by ψ for which Eq. (1) holds, there
is an element of physical reality corresponding
to the physical quantity A. Let, for example,
ψ = e(2πi/h) p0x, (2)
where h is Planck's constant, p0 is some constant
number, and x the independent variable. Since
the operator corresponding to the momentum of
the particle is
p = ( h/2πi ) δ / δx = p0x, (3)
we obtain
ψ' = pψ = ( h/2πi ) δψ / δx, (4)
Thus, in the state given by Eq. (2), the momentum has certainly the value p0. 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 q, is the operator
of multiplication by the independent variable.
Thus,
qψ = qψ ≠ aψ, (5)
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) = ∫ab ψ*ψdx = 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 momentum
of a particle is known, its coordinate has no physical
reality.
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) 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. For if both of them had simultaneous 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
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
Schrödinger's equation the state of the combined
system I + II at any subsequent time; in particular, for any t > T. Let us designate the corresponding wave function by Ψ. We 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 a1, a2, a3,... be the eigenvalues of some
physical quantity A pertaining to system I and u1 (x1), u2 (x1), u3 (x1), ... the corresponding eigenfunctions, where x1 stands for the variables used to describe the first system. Then Ψ, considered as a function of x1, can be expressed as
Ψ (x1 , x2) = Σ n=1 ∞ ψn (x2 ) un (x1), (7)
where x2 stands for the variables used to describe
the second system. Here ψn (x2) are to be regarded
merely as the coefficients of the expansion of Ψ
into a series of orthogonal functions un (x1).
Suppose now that the quantity A is measured
and it is found that it has the value ak. It is then
concluded that after the measurement the first
system is left in the state given by the
wave function uk (x1), and that the second system is
left in the state given by the wave function
ψk (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
ψk (x2) uk (x1)
The set of functions un (x1) is determined by
the choice of the physical quantity A. If, instead
of this, we had chosen another quantity, say B,
having the eigenvalues b1, b2, b3,...and eigen-
functions v1 (x1), v2 (x1), v3 (x1), ... we should
have obtained, instead of Eq. (7), the expansion
Ψ (x1 , x2) = Σ s=1 ∞ φs (x2 ) vs (x1), (8)
where φs's are the new coefficients. If now the
quantity B is measured and is found to have the
value br, we conclude that after the measurement
the first system is left in the state given by vr (x1)
and the second system is left in the state given
by φs (x2 ).
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.
Here is the error in most discussions of EPR. At the time of the measurement, a coherent two-particle wave function ψ12 describes both particles. Measurement that locates one particle simultaneously determines the properties of the second particle (in the preferred frame in which the particles source is at rest).
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 ψk and
φr to the same reality (the second system after
the interaction with the first).
Now, it may happen that the two wave functions, ψk and
φr, are eigenfunctions of two non-commuting 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
Only then does ψ12 → ψ1 ψ2
Ψ (x1 , x2) = ∫ -∞ +∞ e ( 2πi / h ) ( x1 - x2 + x0 ) p dp, (9)
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
up( x1 ) = e ( 2πi / h ) p x1 , (10)
corresponding to the eigenvalue p. Since we have
here the case of a continuous spectrum, Eq. (7)
will now be written
Ψ (x1 , x2) = ∫ -∞ +∞ ψp (x2 ) up (x1) dp, (11)
where
ψp (x2 ) = e - ( 2πi / h ) ( x2 - x0 ) p ,
(12)
(12)
This ψp however is the eigenfunction of the
operator
P = (h / 2 π i) δ/ δx2,
(13)
corresponding to the eigenvalue — p of the
momentum of the second particle. On the other
hand, if B is the coordinate of the first particle,
it has for eigenfunctions
v ( x1 ) = δ ( x1 - x )
(14)
corresponding to the eigenvalue x, where
δ ( x1 - x ) is the well-known Dirac delta-function.
Eq. (8) in this case becomes
Ψ (x1 , x2) = ∫ -∞ +∞ φ x (x2 ) vx (x1) dx, (15)
where
φx ( x2 ) = ∫ -∞ +∞ e - ( 2πi / h ) ( x2 - x0 ) p d p
= h δ ( x - x2 + x0 ) (16)
This φx, however, is the eigenfunction of the
operator
Q = x2, (17)
corresponding to the eigenvalue x + x0 of the
coordinate of the second particle. Since
P Q - Q P = h / 2πi, (18)
we have shown that it is in general possible for
ψk and φr 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 ψk and φr are indeed eigenfunctions of some noncommuting operators P and Q, corresponding to
the eigenvalues pk and qr, respectively. Thus, by
measuring either A or B 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 pk) or the value of the
quantity Q (that is qr). In accordance with our
criterion of reality, in the first case we must
consider the quantity P 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 ψk and φr 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 noncommuting 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 description 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 sufficiently 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 when they can be
simultaneously measured or predicted. On this
point of view, since either one or the other, but
not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously 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.
For a modern analysis of this paradox, see EPR. Physical Review article Niels Bohr's reply For Teachers
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