We have
(A13)
(A14)
Obtaining f(x) from Eq. (A2), h(x) from Eq. (A3), from Eq. (A10), Ψ0 (R) from Eq. (A9), we readily show that
(A15)
This means that the wave packet implies an excess over zero-point energy that is localized within a region in which the packet function, g(x) is appreciable, where
(A16)
We are now ready to treat the photoelectric and Compton effects. The entire treatment is to similar to that of the Franck-Hertz experiment (Paper I, Sec. 7) that we need merely sketch it here. We begin by adding to the radiation Hamiltonian, H ( R ), the particle Hamiltonian,
(A17)
(We restrict ourselves here to nonrelativistic treatment.) The photoelectric effect corresponds to the transition of a radiation oscillator from an excited state to the ground state, while the atomic electron is ejected, with an energy E = hv − I , where I is the ionization potential of the atom. The initial super wave field, corresponding to an incident packet containing only one quantum, plus an atom in the ground state is (see Eq. (A11))
(A18)
By solving Schroedinger’s equation for the combined system, we obtain an asymptotic wave field analogous to Paper I, Eq. (26), containing terms corresponding to the photoelectric effect. These terms, which must be added to Ψi, to yield the complete superfield, are (asymptotically)
(A19)
where the energy of the outgoing electron is E = h2k′2 /2m = hkc+ E0. The function g µ(θ , φ , k ”) is the amplitude associated with the ψ -field of the outgoing electron. This quantity can be calculated from the matrix element of the interaction term, −(e/c )p · A(x), by methods that are easily deducible from the usual perturbation theory.6
The outgoing electron packet has its center at r = (hk′/m)t. Eventually, this packet will become completely separated from the initial electron wave function, ψ0(x). If the electron happens to enter the outgoing packet, the initial wave function can subsequently be ignored. The system then acts for all practical purposes as if its wave field were given by Eq. (A9), from which we conclude that the radiation field is in the ground state, while the electron has been liberated. It is readily shown that, as in the usual interpretation, the probability that the electron appears in the direction θ , φ can be calculated from |gµ(θ, φ, k′)|2 (see Paper I, Sec. 7).
To describe the Compton effect, we need only add to the super wave field the term corresponding to the appearance of an outgoing electromagnetic wave, as well as an outgoing electron. This part is asymptotically
(A20)
where
The quantity, c k”,µ”k,µ is proportioned to the matrix element for a transition in which the k, µ-radiation oscillator falls from the first excited state, to the ground state, while the k”, µ”-oscillator rises from the ground state to the first excited state. This matrix element is determined mainly by the term (e2 /8mc2) A2 (x) in the hamiltonian.
It is easily seen that the outgoing electron packet eventually becomes completely separated both from the initial wave field, Ψi (x, . . . qk,µ . . .), and from the packet for the photoelectric effect, δΨa [defined in Eq. (A19)]. If the electron should happen to enter this packet, then the others can be ignored, and the system acts for all practical purposes like an outgoing electron, plus an independent outgoing light quantum. The reader will readily verify that the probability that the light quantum kʹ, µʹ appears along with an electron with angles θ , φ is precisely the same as in the usual interpretation.
APPENDIX B. A DISCUSSION OF INTERPRETATIONS OF THE QUANTUM THEORY PROPOSED BY DE BROGLIE AND ROSEN
After this article had been prepared, the author’s attention was called to two papers in which an interpretation of the quantum theory similar to that suggested here was proposed, first by L. de Broglie,jm and later by N. Rosen.jn In both of these papers, it was suggested that if one writes ψ = R exp(is / h ), then one can regard R2 as a probability density of particles having a velocity, v = ∇s / m . De Broglie regarded the ψ-field as an agent “guiding” the particle, and therefore referred to ψ as a “pilot wave.” Both of these authors came to the conclusion that this interpretation could not consistently be carried through in those cases in which the field contained a linear combination of stationary state wave functions. As we shall see in this appendix, however, the difficulties encountered by the above authors could have been overcome by them, if only they had carried their ideas to a logical conclusion.
De Broglie’s suggestions met strong objections on the part of Pauli,jo in connection with the problem of inelastic scattering of a particle by a rigid rotator. Since this problem is conceptually equivalent to that of inelastic scattering of a particle by a hydrogen atom, which we have already treated in Paper I, Sec. 7, we shall discuss the objections raised by Pauli in terms of the latter example.
Now, according to Pauli’s argument, the initial wave function in the scattering problem should be Ψ = exp(ip0 · y/ h)ψ0(x). This corresponds to a stationary state for the combined system, in which the particle momentum is p0, while the hydrogen atom is in its ground state, with a wave function, ψ0(x). After interaction between the incident particle and the hydrogen atom, the combined wave function can be represented as
(B1)
ψ =Σn fn (y)ψn(x) where ψn (x) is the wave function for the nth excited state of the hydrogen atom, and fn(y) is the associated expansion coefficient. It is easily shownjp that asymptotically, fn(y) takes the form of an outgoing wave, fn(y) ∼ gn(θ, φ)e i knr /r , where (hkn)2 /2m = [(hk0)2 /2m ] + En − E0. Now, if we write ψ = R exp (iS / h ), we find that the particle momenta, px = ∇x S(x, y) and py = ∇y S(x, y), fluctuate violently in a way that depends strongly on the position of each particle. Thus, neither atom nor the outgoing particle ever seem to approach a stationary energy. On the other hand, we know from experiment that both the atom and the outgoing particle do eventually obtain definite (but presumably unpredictable) energy values. Pauli therefore concluded that the interpretation proposed by de Broglie was untenable. De Broglie seems to have agreed with the conclusion, since he subsequently gave up his suggested interpretation.9
Our answer to Pauli’s objection is already contained in Paper I, Sec. 7, as well as in Sec. 2 of this paper. For as is well known, the use of an incident plane wave of infinite extent is an excessive abstraction, not realizable in practice. Actually, both the incident and outgoing parts of the ψ-field will always take the form of bounded packets. Moreover, as shown in Paper I, Sec. 7, all packets corresponding to different values of n will ultimately obtain classically describable separations. The outgoing particle must enter one of these packets, and it will remain with that particular packet thereafter, leaving the hydrogen atom in a definite but correlated stationary state. Thus, Pauli’s objection is seen to be based on the use of the excessively abstract model of an infinite plane wave.
Although the above constitutes a complete answer to Pauli’s specific objections to our suggested interpretation, we wish here to amplify our discussion somewhat, in order to anticipate certain additional objections that might be made along similar lines. For at this point, one might argue that even though the wave packet is bounded, it can nevertheless in principle be made arbitrarily large in extent by means of a suitable adjustment of initial conditions. Our interpretation predicts that in the region in which incident and outgoing ψ-waves overlap, the momentum of each particle will fluctuate violently, as a result of corresponding fluctuations in the “quantum-mechanical” potential produced by the ψ-field. The question arises, however, as to whether such fluctuations can really be in accord with experimental fact, especially since in principle they could occur when the particles were separated by distances much greater than that over which the “classical” interaction potential, V (x, y), was appreciable.
To show that these fluctuations are not in disagreement with any experimental facts now available, we first point out that even in the usual interpretation the energy of each particle can
not correctly be regarded as definite under the conditions which are assumed here, namely, that the incident and outgoing wave packets overlap. For as long as interference between two stationary state wave function is possible, the system acts as if it, in some sense, covered both states simultaneously.jq In such a situation, the usual interpretation implies that a precisely defined value for the energy of either particle is meaningless. From such a wave function, one can predict only the probability that if the energy is measured, a definite value will be obtained. On the other hand, the very experimental conditions needed for measuring the energy play a key role in making a definite value of the energy possible because the effect of the measuring apparatus is to destroy interference between parts of the wave function corresponding to different values of the energy.jr
In our interpretation, the overlap of incident and outgoing wave packets signifies not that the precise value of the energy of either particle can be given no meaning, but rather that this value fluctuates violently in an, in practice, unpredictable and uncontrollable way. When the energy of either particle is measured, however, then our interpretation predicts, in agreement with the usual interpretation, that the energy of each particle will become definite and constant, as a result of the effects of the energy-measuring apparatus on the observed system. To show how this happens, let us suppose that the energy of the hydrogen atom is measured by means of an interaction in which the “classical” potential, V, is a function only of the variables associated with the electron in the hydrogen atom and with the apparatus, but is not a function of variables associated with the outgoing particle. Let z be the coordinate of the measuring apparatus. Then as shown in Sec. 2, interaction with an apparatus that measures the energy of the hydrogen atom will transform the Ψ-function (B1), into
(B2)
Now, we have seen that if the product at is large enough to make a distinct measurement possible, packets corresponding to different values of n will ultimately obtain classically describable separations in z space. The apparatus variable, z, must enter one of these packets; and, thereafter, all other packets can for practical purposes be ignored. The hydrogen atom is then left in a state having a definite and constant energy, while the outgoing particle has a correspondingly definite but correlated constant value for its energy. Thus, we find that as with the usual interpretation, our interpretation predicts that whenever we measure the energy of either particle by methods that are now available, a definite and constant value will always be obtained. Nevertheless, under conditions in which incident and outgoing wave packets overlap, and in which neither particle interacts with an energy-measuring device, our interpretation states unambiguously that real fluctuations in the energy of each particle will occur. These fluctuations are moreover, at least in principle, observable (for example, by methods discussed in Sec. 6). Meanwhile, under conditions in which we are limited by present methods of observation, our interpretation leads to predictions that are precisely the same as those obtained from the usual interpretation, so that no experiments supporting the usual interpretation can possibly contradict our interpretation.
In his book,9 de Broglie raises objections to his own suggested interpretation of the quantum theory, which are very similar to those raised by Pauli. It is therefore not necessary to answer de Broglie’s objections in detail here, since the answer is essentially the same as that which has been given to Pauli. We wish, however, to add one point. De Broglie assumes that not only electrons, but also light quanta, are associated with particles. A consistent application of the interpretation suggested here requires, however, as shown in Appendix A, that light quanta be described as electromagnetic wave packets. The only precisely definable quantities in such a packet are the Fourier components, qk, µ, of the vector potential and the corresponding canonically conjugate momenta, Πk,µ. Such packets have many particle-like properties, including the ability to transfer rapidly a full quantum of energy at great distances. Nevertheless, it would not be consistent to assume the existence of a “photon” particle, associated with each light quantum.
We shall now discuss Rosen’s paper briefly.10 Rosen gave up his suggested interpretation of the quantum theory, because of difficulties arising in connection with the interpretation of standing waves. In the case of the stationary states of a free particle in a box, which we have already discussed in Sec. 8, our interpretation leads to the conclusion that the particle is standing still. Rosen did not wish to accept this conclusion, because it seemed to disagree with the statement of the usual interpretation that in such a state the electron is moving with equal probability that the motion is in either direction. To answer Rosen’s objections, we need merely point out again that the usual interpretation can give no meaning to the motion of particles in a stationary state; at best, it can only predict the probability that a given result will be obtained, if the velocity is measured. As we saw in Sec. 8, however, our interpretation leads to precisely the same predictions as are obtained from the usual interpretation, for any process which could actually provide us with a measurement of the velocity of the electron. One must remember, however, that the value of the momentum “observable” as it is now “measured” is not necessarily equal to the particle momentum existing before interaction with the measuring apparatus took place.
We conclude that the objections raised by Pauli, de Broglie, and Rosen, to interpretations of the quantum theory similar to that suggested here, can all be answered by carrying every aspect of our suggested interpretation to its logical conclusion.
ON THE EINSTEIN PODOLSKY ROSEN PARADOX
BY
JOHN S. BELL
Originally published in Physics, 1, 195–200 (1964).
I. Introduction
The paradox of Einstein, Podolsky and Rosen [1] was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality [2]. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts [3] to show that even without such a separability or locality requirement no “hidden variable” interpretation of quantum mechanics is possible. These attempts have been examined elsewhere [4] and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory [5] has been explicitly constructed. That particular interpretation has indeed a grossly nonlocal structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions.
II. Formulation
With the example advocated by Bohm and Aharonov [6], the EPR argument is the following. Consider a pair of spin one-half particles* Work supported in part by the U.S. Atomic Energy Commission
†On leave of absence from SLAC and CERN
formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins 1 and 2. If measurement of the component , where is some unit vector, yields the value +1 then, according to quantum mechanics, measurement of · must yield the value—1 and vice versa. Now we make the hypothesis [2], and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of , by previously measuring the same component of , it follows that the result of any such measurement must actually be predetermined. Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.
Let this more complete specification be effected by means of parameters λ. It is a matter of indifference in the following whether λ denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous. However, we write as if λ were a single continuous parameter. The result A of measuring is then determined by and λ, and the result B of measuring in the same instance is determined by and λ, and
(1)
The vital assumption [2] is that the result B for particle 2 does not depend on the setting , of the magnet for particle 1, nor A on .
If ρ(λ) is the probability distribution of λ then the expectation value of the product of the two components and is
(2)
This should equal the quantum mechanical expectation value, which for the singlet state is
(3)
But it will be shown that this is not possible.
Some might prefer a formulation in which the hidden variables fall into two sets, with A dependent on one and B on the other; this possibility is contained in the above, since λ stands for any number of variables and the dependences thereon of A and B are unrestricted. In a complete physical theory of the type envisaged by Einstein, the hidden variables would have dynamical significance and laws of motion; our λ can then be thought of as initial values of these variables at some suitable instant.
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