Thus, the particle velocities are independent and the “quantum-mechanical” potential reduces to a sum of terms, one involving only x and the other involving only y. This means that the particles move independently. Moreover, the probability density, P = R02(x) × M02(y, t ), is a product of a function of x and a function of y, indicating that the distribution in x is statistically independent of that in y. We conclude, then, that whenever the wave function can be expressed as a product of two factors, each involving only the coordinates of a single system, then the two systems are completely independent of each other.
As soon as the wave packet in y space reaches the neighborhood of the atom, the two systems begin to interact. If we solve Schroedinger’s equation for the combined system, we obtain a wave function that can be expressed in terms of the following series:
(25)
where the f n (y, t ) are the expansion coefficients of the complete set of functions, ψn (x). The asymptotic form of the wave function isiy
(26)
where
(27)
The additional terms in the above equation represent outgoing wave packets, in which the particle speed, hkn/m, is correlated with the wave function, ψn (x), representing the state in which the hydrogen atom is left. The center of the nth packet occurs at
(28)
It is clear that because the speed depends on the hydrogen atom quantum number, n, every one of these packets will eventually be separated by distances which are so large that this separation is classically describable.
When the wave function takes the form (25), the two particles system must be described as a single six-dimensional system and not as a sum of two independent three-dimensional subsystems, for at this time, if we try to express the wave function as ψ (x, y) = R (x, y) × exp[iS(x, y)/ h], we find that the resulting expressions for R and S depend on x and y in a very complicated way. The particle momenta, p1 = ∇2 S (x, y) and p2 = ∇yS(x, y), therefore become inextricably interdependent. The “quantum-mechanical” potential,
ceases to be expressible as the sum of a term involving x and a term involving y. The probability density, R 2(x, y) can no longer be written as a product of a function of x and a function of y, from which we conclude that the probability distributions of the two particles are no longer statistically independent. Moreover, the motion of the particle is exceedingly complicated, because the expressions for R and S are somewhat analogous to those obtained in the simpler problem of a nonstationary state of a single particle [see Eqs. (19) and (20)]. In the region where the scattered waves ψn (x) f n (y, t) have an amplitude comparable with that of the incident wave, ψ0(x) f 0 (y, t ), the functions R and S, and therefore the “quantum-mechanical” potential and the particle momenta, undergo rapid and violent fluctuations, both as functions of position and of time. Because the quantum-mechanical potential has R (x, y, t ) in the denominator, these fluctuations may become very large in this region where R is small. If the particles happen to enter such a region, they may exchange very large quantities of energy and momentum in a very short time, even if the classical potential, V (x, y) is very small. A small value of V (x, y) implies, however, a correspondingly small value of the scattered wave amplitudes, fn(y, t ). Since the fluctuations become large only in the region where the scattered wave amplitude is comparable with the incident wave amplitude and since the probability that the particles shall enter a given region of x, y space is proportional to R 2(x, y), it is clear that a large transfer of energy is improbable (although still always possible) when V (x, y) is small.
While interaction between the two particles takes place then, their orbits are subject to wild fluctuations. Eventually, however, the behavior of the system quiets down and becomes simple again. For after the wave function takes its asymptotic form (26), and the packets corresponding to different values of n have obtained classically describable separations, we can deduce that because the probability density is |ψ |2, the outgoing particle must enter one of these packets and stay with that packet thereafter (since it does not enter the space between packets in which the probability density is negligibly different from zero). In the calculation of the particle velocities, V1 = ∇xS/m, V2 = ∇ y S / m , and of the quantum-mechanical potential, U = (−h 2 /2m R )(∇x 2 R + ∇ y 2 R ), we can therefore ignore all parts of the wave function other than the one actually containing the outgoing particle. It follows that the system acts as if it had the wave function
(29)
where n denotes the packet actually containing the outgoing particle. This means that for all practical purposes the complete wave function (26) of the system may be replaced by Eq. (29), which corresponds to an atomic electron in its nth quantum state, and to an outgoing particle with a correlated energy, En ’= h 2kn 2/2m. Because the wave function is a product of a function of x and a function of y, each system once again acts independently of the other. The wave function can now be renormalized because the multiplication of Ψ n by a constant changes no physically significant quantity, such as the particle velocity or the “quantum-mechanical” potential. As shown in Sec. 5, when the electronic wave function is ψn (x)exp(−i E n t / h ), its energy must be En. Thus, we have obtained a description of how it comes about that the energy is always transferred in quanta of size En − E 0.
It should be noted that while the wave packets are still separating, the electron energy is not quantized, but has a continuous range of values, which fluctuate rapidly. It is only the final value of the energy, appearing after the interaction is over that must be quantized. A similar result is obtained in the usual interpretation if one notes that because of the uncertainty principle, the energy of either system can become definite only after enough time has elapsed to complete the scattering process.iz
In principle, the actual packet entered by the outgoing particle could be predicted if we knew the initial position of both particles and, of course, the initial form of the wave function of the combined system.ja In practice, however, the particle orbits are very complicated and very sensitively dependent on the precise values of these initial positions. Since we do not at present know how to measure these initial positions precisely, we cannot actually predict the outcome of such an interaction process. The best that we can do is to predict the probability that an outgoing particle enters the nth packet within a given range of solid angle, dΩ, leaving the hydrogen atom in its nth quantum state. In doing this, we use the fact that the probability density in x, y space is |ψ (x, y)|2 and that as long as we are restricted to the nth packet, we can replace the complete wave function (26) by the wave function (29), corresponding to the packet that actually contains the particle. Now, by definition, we have f |ψn (x)|2d x = 1. The remaining integration of
over the region of space corresponding to the nth outgoing packet leads, however, to precisely the same probability of scattering as would have been obtained by applying the usual interpretation. We conclude, then, that if ψ satisfies Schroedinger’s equation, that if v = ∇ S /m, and that if the probability density of particles is P(x, y) = R 2(x, y), we obtain in every respect exactly the same physical predictions for this problem as are obtained when we use the usual interpretation.
There remains only one more problem; namely, to show that if the outgoing packets are subsequently brought together by some arrangement of matter that does not act on the atomic electron, the atomic electron and the scattered particle will continue to act independently. jb To show that these two particles will continue to act independently, we note that in all practical applications, the outgoing particle soon interacts with some classically describable system. Such a system might consist, for example, of the host of atoms of the gas with which it collides or of the walls of a container. In any case, if the scattering process is ever to be observed, the outgoing particle must interact with a classically describable measuring apparatus. Now all classically describable systems have the property that they contain an enormous number of internal “thermo-dynamic” degrees of freedom that ar
e inevitably excited when the outgoing particle interacts with the system. The wave function of the outgoing particle is then coupled to that of these internal thermodynamic degrees of freedom, which we represent as y1, y2, . . . ys . To denote this coupling, we write the wave function for the entire system as
(30)
Now, when the wave function takes this form, the overlapping of different packets in y space is not enough to produce interference between the different ψn (x). To obtain such interference, it is necessary that the packets fn(y,y1,y2, ...ys)overlapineveryoneoftheS +3 dimensions,y,y 1,y2 ...ys. The reader will readily convince himself, by considering a typical case such as a collision of the outgoing particle with a metal wall, that it is overwhelmingly improbable that two of the packets fn(y1, y1, y2 . . . ys ) will overlap with regard to every one of the internal thermodynamic coordinates, y1, y2, . . . ys , even if they are successfully made to overlap in y space. This is because each packet corresponds to a different particle velocity and to a different time of collision with the metal wall. Because the myriads of internal thermodynamic degrees of freedom are so chaotically complicated, it is very likely that as each of the n packets interacts with them, it will encounter different conditions, which will make the combined wave packet fn (y, y 1, ... ys) enter very different regions of y1,y2 ... ys space. Thus, for all practical purposes, we can ignore the possibility that if two of the packets are made to cross in y space, the motion either of the atomic electron or of the outgoing particle will be affected. jc
8. PENETRATION OF A BARRIER
According to classical physics, a particle can never penetrate a potential barrier having a height greater than the particle kinetic energy. In the usual interpretation of the quantum theory, it is said to be able, with a small probability, to “leak” through the barrier. In our interpretation of the quantum theory, however, the potential provided by the Schroedinger ψ -field enables it to “ride” over the barrier, but only a few particles are likely to have trajectories that carry them all the way across without being turned around.
We shall merely sketch in general terms how the above results can be obtained. Since the motion of the particle is strongly affected by its ψ -field, we must first solve for this field with the aid of “Schroedinger’s equation.” Initially, we have a wave packet incident on the potential barrier; and because the probability density is equal to |ψ (x)|2, the particle is certain to be somewhere within this wave packet. When the wave packet strikes the repulsive barrier, the ψ -field undergoes rapid changes which can be calculatedjd if desired, but whose precise form does not interest us here. At this time, the “quantum-mechanical” potential, U = (−h2/2m)∇2 R/R , undergoes rapid and violent fluctuations, analogous to those described in Sec. 7 in connection with Eqs. (19), (20), and (25). The particle orbit then becomes very complicated and, because the potential is time dependent, very sensitive to the precise initial relationship between the particle position and the center of the wave packet. Ultimately, however, the incident wave packet disappears and is replaced by two packets, one of them a reflected packet and the other a transmitted packet having a much smaller intensity. Because the probability density is |ψ |2, the particle must end up in one of these packets. The other packet can, as shown in Sec. 7, subsequently be ignored. Since the reflected packet is usually so much stronger than the transmitted packet, we conclude that during the time when the packet is inside the barrier, most of the particle orbits must be turned around, as a result of the violent fluctuations in the “quantum-mechanical” potential.
9. POSSIBLE MODIFICATIONS IN MATHEMATICAL FORMULATION LEADING TO EXPERIMENTAL PROOF THAT NEW INTERPRETATION IS NEEDED
We have already seen in a number of cases and in Paper II we shall prove in general, that as long as we assume that ψ satisfies Schroedinger’s equation, that v = ∇ S (x)/m, and that we have a statistical ensemble with a probability density equal to |ψ (x)|2, our interpretation of the quantum theory leads to physical results that are identical with those obtained from the usual interpretation. Evidence indicating the need for adopting our interpretation instead of the usual one could therefore come only from experiments, such as those involving phenomena associated with distances of the order of 10−13 cm or less, which are not now adequately understood in terms of the existing theory. In this paper we shall not, however, actually suggest any specific experimental methods of distinguishing between our interpretation and the usual one, but shall confine ourselves to demonstrating that such experiments are conceivable.
Now, there are an infinite number of ways of modifying the mathematical form of the theory that are consistent with our interpretation and not with the usual interpretation. We shall confine ourselves here, however, to suggesting two such modifications, which have already been indicated in Sec. 4, namely, to give up the assumption that v is necessarily equal to ∇ S (x)/m, and to give up the assumption that ψ must necessarily satisfy a homogeneous linear equation of the general type suggested by Schroedinger. As we shall see, giving up either of those first two assumptions will in general also require us to give up the assumption of a statistical ensemble of particles, with a probability density equal to |ψ (x)|2.
We begin by noting that it is consistent with our interpretation to modify the equations of motion of a particle (8a) by adding any conceivable force term to the right-hand side. Let us, for the sake of illustration, consider a force that tends to make the difference, p − ∇ S (x), decay rapidly with time, with a mean decay time of the order of τ = 10−13/c seconds, where c is the velocity of light. To achieve this result, we write
(31)
where f(p − ∇ S (x)) is assumed to be a function which vanishes when p = ∇ S (x) and more generally takes such a form that it implies a force tending to make p − ∇ S (x) decrease rapidly with the passage of time. It is clear, moreover, that f can be so chosen that it is large only in processes involving very short distances (where ∇ S (x) should be large).
If the correct equations of motion resembled Eq. (31), then the usual interpretation would be applicable only over times much longer than τ , for only after such times have elapsed will the relation p = ∇ S (x) be a good approximation. Moreover, it is clear that such modifications of the theory cannot even be described in the usual interpretation, because they involve the precisely definable particle variables which are not postulated in the usual interpretation.
Let us now consider a modification that makes the equation governing ψ inhomogeneous. Such a modification is
(32)
Here, H is the usual Hamiltonian operator, xi, represents the actual location of the particle, and ξ is a function that vanishes when p = ∇ S (xi). Now, if the particle equations are chosen, as in Eq. (31), to make p − ∇ S (xi) decay rapidly with time, it follows that in atomic processes, the inhomogeneous term in Eq. (32) will become negligibly small, so that Schroedinger’s equation is a good approximation. Nevertheless, in processes involving very short distances and very short times, the inhomogeneities would be important, and the ψ-field would, as in the case of the electromagnetic field, depend to some extent on the actual location of the particle.
It is clear that Eq. (32) is inconsistent with the usual interpretation of the theory. Moreover, we can contemplate further generalizations of Eq. (32), in the direction of introducing nonlinear terms that are large only for processes involving small distances. Since the usual interpretation is based on the hypothesis of linear superposition of “state vectors” in a Hilbert space, it follows that the usual interpretation could not be made consistent with such a nonlinear equation for a one-particle theory. In a many-particle theory, operators can be introduced, satisfying a nonlinear generalization of Schroedinger’s equation; but these must ultimately operate on wave functions that satisfy a linear homogeneous Schroedinger equation.
Finally, we repeat a point already made in Sec. 4, namely, that if the theory is generalized in any of the ways indicated here, the probability density of particles will cease to equa
l |ψ (x)|2. Thus, experiments would become conceivable that distinguish between |ψ (x))|2 and this probability; and in this way we could obtain an experimental proof that the usual interpretation, which gives |ψ (x)|2 only a probability interpretation, must be inadequate. Moreover, we shall show in Paper II that modifications like those suggested here would permit the particle position and momentum to be measured simultaneously, so that the uncertainty principle could be violated.
ACKNOWLEDGMENT
The author wishes to thank Dr. Einstein for several interesting and stimulating discussions.
A SUGGESTED INTERPRETATION OF THE QUANTUM THEORY IN TERMS OF “HIDDEN” VARIABLES. II
BY
DAVID BOHM
In this paper, we shall show how the theory of measurements is to be understood from the point of view of a physical interpretation of the quantum theory in terms of “hidden” variables, developed in a previous paper. We find that in principle, these “hidden” variables determine the precise results of each individual measurement process. In practice, however, in measurements that we now know how to carry out, the observing apparatus disturbs the observed system in an unpredictable and uncontrollable, way, so that the uncertainty principle is obtained as a practical limitation on the possible precision of measurements. This limitation is not, however, inherent in the conceptual structure of our interpretation. We shall see, for example, that simultaneous measurements of position and momentum having unlimited precision would in principle be possible if, as suggested in the previous paper, the mathematical formulation of the quantum theory needs to be modified at very short distances in certain ways that are consistent with our interpretation but not with the usual interpretation.
The Dreams That Stuff is Made of Page 49