(2) The process of transfer of a single quantum from observed system to measuring apparatus is inherently unpredictable, uncontrollable, and unanalyzable.
Let us now inquire into the question of whether there are any experiments that could conceivably provide a test for these assumptions. It is often stated in connection with this problem that the mathematical apparatus of the quantum theory and its physical interpretation form a consistent whole and that this combined system of mathematical apparatus and physical interpretation is tested adequately by the extremely wide range of experiments that are in agreement with predictions obtained by using this system. If assumptions (1) and (2) implied a unique mathematical formulation, then such a conclusion would be valid, because experimental predictions could then be found which, if contradicted, would clearly indicate that these assumptions were wrong. Although assumptions (1) and (2) do limit the possible forms of the mathematical theory, they do not limit these forms sufficiently to make possible a unique set of predictions that could in principle permit such an experimental test. Thus, one can contemplate practically arbitrary changes in the Hamiltonian operator, including, for example, the postulation of an unlimited range of new kinds of meson fields each having almost any conceivable rest mass, charge, spin, magnetic moment, etc. And if such postulates should prove to be inadequate, it is conceivable that we may have to introduce nonlocal operators, nonlinear fields, S-matrices, etc. This means that when the theory is found to be inadequate (as now happens, for example, at distances of the order of 10−13 cm), it is always possible, and, in fact, usually quite natural, to assume that the theory can be made to agree with experiment by some as yet unknown change in the mathematical formulation alone, not requiring any fundamental changes in the physical interpretation. This means that as long as we accept the usual physical interpretation of the quantum theory, we cannot be led by any conceivable experiment to give up this interpretation, even if it should happen to be wrong. The usual physical interpretation therefore presents us with a considerable danger of falling into a trap, consisting of a self-closing chain of circular hypotheses, which are in principle unverifiable if true. The only way of avoiding the possibility of such a trap is to study the consequences of postulates that contradict assumptions (1) and (2) at the outset. Thus, we could, for example, postulate that the precise outcome of each individual measurement process is in principle determined by some at present “hidden” elements or variables; and we could then try to find experiments that depended in a unique and reproducible way on the assumed state of these hidden elements or variables. If such predictions are verified, we should then obtain experimental evidence favoring the hypothesis that hidden variables exist. If they are not verified, however, the correctness of the usual interpretation of the quantum theory is not necessarily proved, since it may be necessary instead to alter the specific character of the theory that is supposed to describe the behavior of the assumed hidden variables.
We conclude then that a choice of the present interpretation of the quantum theory involves a real physical limitation on the kinds of theories that we wish to take into consideration. From the arguments given here, however, it would seem that there are no secure experimental or theoretical grounds on which we can base such a choice because this choice follows from hypotheses that cannot conceivably be subjected to an experimental test and because we now have an alternative interpretation.
4. NEW PHYSICAL INTERPRETATION OF SCHROEDINGER’S EQUATION
We shall now give a general description of our suggested physical interpretation of the present mathematical formulation of the quantum theory. We shall carry out a more detailed description in subsequent sections of this paper.
We begin with the one-particle Schroedinger equation, and shall later generalize to an arbitrary number of particles. This wave equation is
(1)
Now ψ is a complex function, which can be expressed as
(2)
where R and S are real. We readily verify that the equations for R and S are
(3)
(4)
It is convenient to write P(x) = R2 (x), or R = P where P(x) is the probability density. We then obtain
(5)
(6)
Now, in the classical limit (h → 0) the above equations are subject to a very simple interpretation. The function S (x) is a solution of the Hamilton-Jacobi equation. If we consider an ensemble of particle trajectories which are solutions of the equations of motion, then it is a well-known theorem of mechanics that if all of these trajectories are normal to any given surface of constant S, then they are normal to all surfaces of constant S, and ∇ S (x)/m will be equal to the velocity vector, v(x), for any particle passing the point x. Equation (5) can therefore be re-expressed as
(7)
This equation indicates that it is consistent to regard P(x) as the probability density for particles in our ensemble. For in that case, we can regard Pv as the mean current of particles in this ensemble, and Eq. (7) then simply expresses the conservation of probability.
Let us now see to what extent this interpretation can be given a meaning even when h ≠ 0. To do this, let us assume that each particle is acted on, not only by a “classical” potential, V(x) but also by a “quantum-mechanical” potential,
(8)
Then Eq. (6) can still be regarded as the Hamilton-Jacobi equation for our ensemble of particles, ∇S(x)/m can still be regarded as the particle velocity, and Eq. (5) can still be regarded as describing conservation of probability in our ensemble. Thus, it would seem that we have here the nucleus of an alternative interpretation for Schroedinger’s equation.
The first step in developing this interpretation in a more explicit way is to associate with each electron a particle having precisely definable and continuously varying values of position and momentum. The solution of the modified Hamilton-Jacobi equation (4) defines an ensemble of possible trajectories for this particle, which can be obtained from the Hamilton-Jacobi function, S(x), by integrating the velocity, v(x) = ∇ S (x)/m. The equation for S implies, however, that the particles moves under the action of a force which is not entirely derivable from the classical potential, V(x), but which also obtains a contribution from the “quantum-mechanical” potential, U (x) = (−h2/2m) × ∇2R/R . The function, R(x), is not completely arbitrary, but is partially determined in terms of S(x) by the differential Eq. (3). Thus R and S can be said to codetermine each other. The most convenient way of obtaining R and S is, in fact, usually to solve Eq. (1) for the Schroedinger wave function, ψ, and then to use the relations,
Since the force on a particle now depends on a function of the absolute value, R(x), of the wave function, ψ (x), evaluated at the actual location of the particle, we have effectively been led to regard the wave function of an individual electron as a mathematical representation of an objectively real field. This field exerts a force on the particle in a way that is analogous to, but not identical with, the way in which an electromagnetic field exerts a force on a charge, and a meson field exerts a force on a nucleon. In the last analysis, there is, of course, no reason why a particle should not be acted on by a ψ -field, as well as by an electromagnetic field, a gravitational field, a set of meson fields, and perhaps by still other fields that have not yet been discovered.
The analogy with the electromagnetic (and other) field goes quite far. For just as the electromagnetic field obeys Maxwell’s equations, the ψ -field obeys Schroedinger’s equation. In both cases, a complete specification of the fields at a given instant over every point in space determines the values of the fields for all times. In both cases, once we know the field functions, we can calculate force on a particle, so that, if we also know the initial position and momentum of the particle, we can calculate its entire trajectory.
In this connection, it is worth while to recall that the use of the Hamilton-Jacobi equation in solving for the motion of a particle is only a matter of convenience and that, in principle, we can always solve directl
y by using Newton’s laws of motion and the correct boundary conditions. The equation of motion of a particle acted on by the classical potential, V (x), and the “quantum-mechanical” potential, Eq. (8), is
(8a)
It is in connection with the boundary conditions appearing in the equations of motion that we find the only fundamental difference between the ψ -field and other fields, such as the electromagnetic field. For in order to obtain results that are equivalent to those of the usual interpretation of the quantum theory, we are required to restrict the value of the initial particle momentum to p = ∇ S (x). From the application of Hamilton-Jacobi theory to Eq. (6), it follows that this restriction is consistent, in the sense that if it holds initially, it will hold for all time. Our suggested new interpretation of the quantum theory implies, however, that this restriction is not inherent in the conceptual structure. We shall see in Sec. 9, for example, that it is quite consistent in our interpretation to contemplate modifications in the theory, which permit an arbitrary relation between p and ∇ S (x). The law of force on the particle can, however, be so chosen that in the atomic domain, p turns out to be very nearly equal to ∇ S (x)/m, while in processes involving very small distances, these two quantities may be very different. In this way, we can improve the analogy between the ψ -field and the electromagnetic field (as well as between quantum mechanics and classical mechanics).
Another important difference between the ψ -field and the electromagnetic field is that, whereas Schroedinger’s equation is homogeneous in ψ , Maxwell’s equations are inhomogeneous in the electric and magnetic fields. Since inhomogeneities are needed to give rise to radiation, this means that our present equations imply that the ψ -field is not radiated or absorbed, but simply changes its form while its integrated intensity remains constant. This restriction to a homogeneous equation is, however, like the restriction to a homogeneous equation is, however, like the restriction to p = ∇ S (x), not inherent in the conceptual structure of our new interpretation. Thus, in Sec. 9, we shall show that one can consistently postulate inhomogeneities in the equation governing ψ , which produce important effects only at very small distances, and negligible effects in the atomic domain. If such inhomogeneities are actually present, then the ψ -field will be subject to being emitted and absorbed, but only in connection with processes associated with very small distances. Once the ψ -field has been emitted, however, it will in all atomic processes simply obey Schroedinger’s equation as a very good approximation. Nevertheless, at very small distances, the value of the ψ -field would, as in the case of the electromagnetic field, depend to some extent on the actual location of the particle.
Let us now consider the meaning of the assumption of a statistical ensemble of particles with a probability density equal to P (x) = R2 (x) = |ψ (x)|2. From Eq. (5), it follows that this assumption is consistent, provided that ψ satisfies Schroedinger’s equation, and v = ∇ S (x)/m. This probability density is numerically equal to the probability density of particles obtained in the usual interpretation. In the usual interpretation, however, the need for a probability description is regarded as inherent in the very structure of matter (see Sec. 2), whereas in our interpretation, it arises, as we shall see in Paper II, because from one measurement to the next, we cannot in practice predict or control the precise location of a particle, as a result of corresponding unpredictable and uncontrollable disturbances introduced by the measuring apparatus. Thus, in our interpretation, the use of a statistical ensemble is (as in the case of classical statistical mechanics) only a practical necessity, and not a reflection of an inherent limitation on the precision with which it is correct for us to conceive of the variables defining the state of the system. Moreover, it is clear that if in connection with very small distances we are ultimately required to give up the special assumptions that ψ satisfies Schroedinger’s equation and that v = ∇ S (x)/m, then |ψ|2 will cease to satisfy a conservation equation and will therefore also cease to be able to represent the probability density of particles. Nevertheless, there would still be a true probability density of particles which is conserved. Thus, it would become possible in principle to find experiments in which |ψ |2 could be distinguished from the probability density, and therefore to prove that the usual interpretation, which gives |ψ |2 only a probability interpretation must be inadequate. Moreover, we shall see in Paper II that with the aid of such modifications in the theory, we could in principle measure the particle positions and momenta precisely, and thus violate the uncertainty principle. As long as we restrict ourselves to conditions in which Schroedinger’s equation is satisfied, and in which v = ∇ S (x)/m, however, the uncertainty principle will remain an effective practical limitation on the possible precision of measurements. This means that at present, the particle positions and momenta should be regarded as “hidden” variables, since as we shall see in Paper II, we are not now able to obtain experiments that localize them to a region smaller than that in which the intensity of the ψ-field is appreciable. Thus, we cannot yet find clear-cut experimental proof that the assumption of these variables is necessary, although it is entirely possible that, in the domain of very small distances, new modifications in the theory may have to be introduced, which would permit a proof of the existence of the definite particle position and momentum to be obtained.
We conclude that our suggested interpretation of the quantum theory provides a much broader conceptual framework than that provided by the usual interpretation, for all of the results of the usual interpretation are obtained from our interpretation if we make the following three special assumptions which are mutually consistent:(1) That the ψ-field satisfies Schroedinger’s equation.
(2) That the particle momentum is restricted to p = ∇ S (x).
(3) That we do not predict or control the precise location of the particle, but have, in practice, a statistical ensemble with probability density P(x) = |ψ (x)|2. The use of statistics is, however, not inherent in the conceptual structure, but merely a consequence of our ignorance of the precise initial conditions of the particle.
As we shall see in Sec. 9, it is entirely possible that a better theory of phenomena involving distances of the order of 10−13 cm or less would require us to go beyond the limitations of these special assumptions. Our principal purpose in this paper (and in Paper II) is to show, however, that if one makes these special assumptions, our interpretation leads in all possible experiments to the same predictions as are obtained from the usual interpretation.9
It is now easy to understand why the adoption of the usual interpretation of the quantum theory would tend to lead us away from the direction of our suggested alternative interpretation. For in a theory involving hidden variables, one would normally expect that the behavior of an individual system should not depend on the statistical ensemble of which it is a member, because this ensemble refers to a series of similar but disconnected experiments carried out under equivalent initial conditions. In our interpretation, however, the “quantum-mechanical” potential, U (x), acting on an individual particle depends on a wave intensity, P(x), that is also numerically equal to a probability density in our ensemble. In the terminology of the usual interpretation of the quantum theory, in which one tacitly assumes that the wave function has only one interpretation; namely, in terms of a probability, our suggested new interpretation would look like a mysterious dependence of the individual on the statistical ensemble of which it is a member. In our interpretation, such a dependence is perfectly rational, because the wave function can consistently be interpreted both as a force and as a probability density.iw
It is instructive to carry our analogy between the Schroedinger field and other kinds of fields a bit further. To do this, we can derive the wave Eqs. (5) and (6) from a Hamiltonian functional. We begin by writing down the expression for the mean energy as it is expressed in the usual quantum theory:
Writing ψ = P exp(iS/h), we obtain
(9)
We shall now reinterpret P (x) as a fi
eld coordinate, defined at each point, x, and we shall tentatively assume that S (x) is the momentum, canonically conjugate to P(x). That such an assumption is appropriate can be verified by finding the Hamiltonian equations of motion for P (x) and S(x), under the assumption that the Hamiltonian functional is equal to H (See Eq. (9)). These equations of motion are
These are, however, the same as the correct wave Eqs. (5) and (6).
We can now show that the mean particle energy averaged over our ensemble is equal to the usual quantum mechanical mean value of the Hamiltonian, H. To do this, we note that according to Eqs. (3) and (6), the energy of a particle is
(10)
The mean particle energy is found by averaging E (x) with the weighting function, P (x). We obtain
A little integration by parts yields
(11)
5. THE STATIONARY STATE
We shall now show how the problem of stationary states is to be treated in our interpretation of the quantum theory.
The Dreams That Stuff is Made of Page 47