5. “OBSERVABLES” OF USUAL INTERPRETATION ARE NOT A COMPLETE DESCRIPTION OF SYSTEM IN OUR INTERPRETATION
We have seen in Sec. 3 that in the measurement of an “observable,” Q, we cannot obtain enough information to provide a complete specification of the state of an electron, because we cannot infer the precisely defined values of the particle momentum and position, which are, for example, needed if we wish to make precise predictions about the future behavior of the electron. Moreover, the process of measuring an observable does not provide any unambiguous information about the state that existed before the measurement took place; for in such a measurement, the ψ-field is transformed into an in practice unpredictable and uncontrollable eigenfunction, ψq(x), of the measured “observable” Q. This means that the measurement of an “observable” is not really a measurement of any physical property belonging to the observed system alone. Instead, the value of an “observable” measures only an incompletely predictable and controllable potentiality belonging just as much to the measuring apparatus as to the observed system itself.jh At best, such a measurement provides unambiguous information only at a classical level of accuracy, where the disturbance of the ψ-field by the measuring apparatus can be neglected. The usual “observables” are therefore not what we ought to try to measure at a quantum level of accuracy. In Sec. 6, we shall see that it is conceivable that we may be able to carry out new kinds of measurements, providing information not about “observables” having a very ambiguous significance, but rather about physically significant properties of a system, such as the actual values of the particle position and momentum.
As an example of the rather indirect and ambiguous significance of the “observable,” we may consider the problem of measuring the momentum of an electron. Now, in the usual interpretation, it is stated that one can always measure the momentum “observable” without changing the value of the momentum. The result is said, for example, to be obtainable with the aid of an impulsive interaction involving only operators which commute with the momentum operator, px. To represent such a measurement, we could choose HI = −a px py in Eq. (1). In our interpretation, however, we cannot in general conclude that such an interaction will enable us to measure the actual particle momentum without changing its value. In fact, in our interpretation, a measurement of particle momentum that does not change the value of this momentum is possible only if the ψ-field initially takes the special form, exp(ip · x/h). If, however, ψ initially takes its most general possible form,
(15)
then as we have seen in Secs. 2 and 3, the process of measuring the “observable” px will effectively transform the ψ-field of the electron into
(16)
with a probability |ap|2 that a given value of px will be obtained. When the ψ-field is altered in this way, large quantities of momentum can be transferred to the particle by the changing ψ-field, even though the interaction Hamiltonian, H1, commutes with the momentum operator, p.
As an example, we may consider a stationary state of an atom, of zero angular momentum. As shown in Paper I, Sec. 5, the ψ-field for such a state is real, so that we obtainp = ∇ S = 0.
Thus, the particle is at rest. Nevertheless, we see from Eqs. (14) and (15) that if the momentum “observable” is measured, a large value of this “observable” may be obtained if the ψ-field happens to have a large fourier coefficient, ap, for a high value of p. The reason is that in the process of interaction with the measuring apparatus, the ψ-field is altered in such a way that it can give the electron particle a correspondingly large momentum, thus transforming some of the potential energy of interaction of the particle with its ψ-field into kinetic energy.
A more striking illustration of the points discussed above is afforded by the problem of a “free” particle contained between two impenetrable and perfectly reflecting walls, separated by a distance L. For this case, the spatial part of the ψ-field is
where n is an integer and the energy of the electron isE = (1/2m) (nh/L)2.
Because the ψ-field is real, we deduce that the particle is at rest.
Now, at first sight, it may seem puzzling that a particle having a high energy should be at rest in the empty space between two walls. Let us recall, however, that the space is not really empty, but contains an objectively real ψ-field that can act on the particle. Such an action is analogous to (but of course not identical with) the action of an electromagnetic field, which could create non-uniform motion of the particle in this apparently “empty” enclosure. We observe that in our problem, the ψ-field is able to bring the particle to rest and to transform the entire kinetic energy into potential energy of interaction with the ψ-field. To prove this, we evaluate the “quantum-mechanical potential” for this ψ-field
and note that it is precisely equal to the total energy, E.
Now, as we have seen, any measurement of the momentum “observable” must change the ψ-field in such a way that in general some (and in our case, all) of this potential energy is transformed into kinetic energy. We may use as an illustration of this general result a very simple specific method of measuring the momentum “observable,” namely, to remove the confining walls suddenly and then to measure the distance moved by the particle after a fairly long time. We can compute the momentum by dividing this distance by the time of transit. If (as in the usual interpretation of the quantum theory) we assume that the electron is “free,” then we conclude that the process of removing the walls should not appreciably change the momentum if we do it fast enough, for the probability that the particle is near a wall when this happens can then in principle be made arbitrarily small. In our interpretation, however, the removal of the walls alters the particle momentum indirectly, because of its effect on the ψ-field, which acts on the particle. Thus, after the walls are removed, two wave packets moving in opposite directions begin to form, and ultimately they become completely separate in space. Because the probability density is |ψ|2, we deduce that the particle must end up in one packet or the other. Moreover, the reader will readily convince himself that the particle momentum will be very close to ±nh/L , the sign depending on which packet the particle actually enters. As in Sec. (2), the packet not containing the particle can subsequently be ignored. In principle, the final particle momentum is determined by the initial form of the ψ-field and by the initial particle position. Since we do not in practice know the latter, we can at best predict a probability of that the particle ends up in either packet. We conclude then that this measurement of the momentum “observable” leads to the same result as is predicted in the usual interpretation. However, the actual particle momentum existing before the measurement took place is quite different from the numerical value obtained for the momentum “observable,” which, in the usual interpretation, is called the “momentum.”
6. ON THE POSSIBILITY OF MEASUREMENTS OF UNLIMITED PRECISION
We have seen that the so-called “observables” do not measure any very readily interpretable properties of a system. For example, the momentum “observable” has in general no simple relation to the actual particle momentum. It may therefore be fruitful to consider how we might try to measure properties which, according to our interpretation, are (along with the ψ-field) the physically significant properties of an electron, namely, the actual particle position and momentum. In connection with this problem, we shall show that if, as suggested in Paper I, Secs. 4 and 9, we give up the three mutually consistent special assumptions leading to the same results as those of the usual interpretation of the quantum theory, then in our interpretation, the particle position and momentum can in principle be measured simultaneously with unlimited precision.
Now, for our purposes, it will be adequate to show that precise predictions of the future behavior of a system are in principle possible. In our interpretation, a sufficient condition for precise predictions is as we have seen that we shall be able to prepare a system in a state in which the ψ-field and the initial particle position and momentum are pr
ecisely known. We have shown that it is possible, by measuring the “observable,” Q, with the aid of methods that are now available, to prepare a state in which the ψ-field is effectively transformed into a known form, ψq(x); but we cannot in general predict or control the precise position and momentum of the particle. If we could now measure the position and momentum of the particle without altering the ψ-field, then precise predictions would be possible. However, the results of Secs. 2, 3, and 4 prove that as long as the three special assumptions indicated above are valid, we cannot measure the particle position more accurately without effectively transforming the ψ-function into an incompletely predictable and controllable packet that is much more localized than ψq (x). Thus, efforts to obtain more precise definition of the state of the system will be defeated. But it is clear that the difficulty originates in the circumstance that the potential energy of interaction between electron and apparatus, V(x,y), plays two roles. For it not only introduces a direct interaction between the two particles, proportional in strength to V(x,y) itself, but it introduces an indirect interaction between these particles, because this potential also appears in the equation governing the ψ-field. This indirect interaction may involve rapid and violent fluctuations, even when V(x,y) is small. Thus, we are led to lose control of the effects of this interaction, because no matter how small V(x,y) is, very large and chaotically complicated disturbances in the particle motion may occur.
If, however, we give up the three special assumptions mentioned previously, then it is not inherent in our conceptual structure that every interaction between particles must inevitably also produce large and uncontrollable changes in the ψ-field. Thus, in Paper I, Eq. (31), we give an example in which we postulate a force acting on a particle that is not necessarily accompanied by a corresponding change in the ψ-field. Equation (31), Paper I, is concerned only with a one-particle system, but similar assumptions can be made for systems of two or more particles. In the absence of any specific theory, our interpretation permits an infinite number of kinds of such modifications, which can be chosen to be important at small distances but negligible in the atomic domain. For the sake of illustration, suppose that it should turn out that in certain processes connected with very small distances, the force acting on the apparatus variable isFy = ax ,
where a is a constant. Now if “a” is made large enough so that the interaction is impulsive, we can neglect all changes in y that are brought about by the forces that would have been present in the absence of this interaction. Moreover, for the sake of illustration of the principles involved, we are permitted to make the assumption, consistent with our interpretation, that the force on the electron is zero. The equation of motion of y is thenÿ = ax/m.
The solution isy − y0 = (axt2/2m)+ ẏ0t ,
where ẏ0 is the initial velocity of the apparatus variable and y0 its initial position. Now, if the product, at2, is large enough, then y − y0 can be made much larger than the uncertainty in y arising from the uncertainty of y0, and the uncertainty of ẏ0. Thus, y − y0 will be determined primarily by the particle position, x. In this way, it is conceivable that we could obtain a measurement of x that does not significantly change x, ẋ, or the ψ-function. The particle momentum can then be obtained from the relation, p = ∇ S(x), where S/h is the phase of the ψ-function. Thus, precise predictions would in principle be possible.
7. THE ORIGIN OF THE STATISTICAL ENSEMBLE IN THE QUANTUM THEORY
We shall now see that even if, because of a failure of the three special assumptions mentioned in Secs. 1 and 6, we are able to determine the particle positions and momenta precisely, we shall nevertheless ultimately obtain a statistical ensemble again at the atomic level, with a probability density equal to |ψ|2. The need for such an ensemble arises from the chaotically complicated character of the coupling between the electron and classical systems, such as volumes of gas, walls of containers, pieces of measuring apparatus, etc., with which this particle must inevitably in practice interact. For as we have seen in Sec. 2, and in Paper I, Sec. 7, during the course of such an interaction, the “quantum-mechanical” potential undergoes violent and rapid fluctuations, which tend to make the particle orbit wander over the whole region in which the ψ-field is appreciable. Moreover, these fluctuations are further complicated by the effects of molecular chaos in the very large number of internal thermodynamic degrees of freedom of these classically describable systems, which are inevitably excited in any interaction process. Thus, even if the initial particle variables were well defined, we should soon in practice lose all possibility of following the particle motion and would be forced to have recourse to some kind of statistical theory. The only question that remains is to show why the probability density that ultimately comes about should be equal to |ψ|2 and not to some other quantity.
To answer this question, we first note that a statistical ensemble with a probability density |ψ(x)|2 has the property that under the action of forces which prevail at the atomic level, where our three special assumptions are satisfied, it will be preserved by the equations of motion of the particles, once it comes into existence. There remains only the problem of showing that an arbitrary deviation from this ensemble tends, under the action of the chaotically complicated forces described in the previous paragraph, to decay into an ensemble with a probability density of |ψ(x)|2. This problem is very similar to that of proving Boltzmann’s H theorem, which shows in connection with a different but analogous problem that an arbitrary ensemble tends as a result of molecular chaos to decay into an equilibrium Gibbs ensemble. We shall not carry out a detailed proof here, but we merely suggest that it seems plausible that one could along similar lines prove that in our problem, an arbitrary ensemble tends to decay into an ensemble with a density of |ψ(x)|2. These arguments indicate that in our interpretation, quantum fluctuations and classical fluctuations (such as the Brownian motion) have basically the same origin; viz., the chaotically complicated character of motion at the microscopic level.
8. THE HYPOTHETICAL EXPERIMENT OF EINSTEIN, PODOLSKY, AND ROSEN
The hypothetical experiment of Einstein, Podolsky, and Rosenji is based on the fact that if we have two particles, the sum of their momenta, p = p 1 + p 2, commutes with the difference of their positions, ξ = x1 − x2. We can therefore define a wave function in which p is zero, while ξ has a given value, a . Such a wave function is
(17)
In the usual interpretation of the quantum theory, p1 − p2 and x1 + x2 are completely undetermined in a system having the above wave function.
The whole experiment centers on the fact that an observer has a choice of measuring either the momentum or the position of any one of the two particles. Whichever of these quantities he measures, he will be able to infer a definite value of the corresponding variable in the other particle, because of the fact that the above wave function implies correlations between variables belonging to each particle. Thus, if he obtains a position x1 for the first particle, he can infer a position of x2 = a − x1 for the second particle; but he loses all possibility of making any inferences about the momenta of either particle. On the other hand, if he measures the momentum of the first particle and obtains a value of p 1 , he can infer a value of p 2 = − p 1 for the momentum of the second particle; but he loses all possibility of making any inferences about the position of either particle. Now, Einstein, Podolsky, and Rosen believe that this result is itself probably correct, but they do not believe that quantum theory as usually interpreted can give a complete description of how these correlations are propagated. Thus, if these were classical particles, we could easily understand the propagation of correlations because each particle would then simply move with a velocity opposite to that of the other. But in the usual interpretation of quantum theory, there is no similar conceptual model showing in detail how the second particle, which is not in any way supposed to interact with the first particle, is nevertheless able to obtain either an uncontrollable disturbance of its position or an uncontrollab
le disturbance of its momentum depending on what kind of measurement the observer decided to carry out on the first particle. Bohr’s point of view is, however, that no such model should be sought and that we should merely accept the fact that these correlations somehow manage to appear. We must note, of course, that the quantum-mechanical description of these processes will always be consistent, even though it gives us no precisely definable means of describing and analyzing the relationships between the classically describable phenomena appearing in various pieces of measuring apparatus.
In our suggested new interpretation of the quantum theory, however, we can describe this experiment in terms of a single precisely definable conceptual model, for we now describe the system in terms of a combination of a six-dimensional wave field and a precisely definable trajectory in a six-dimensional space (see Paper I, Sec. 6). If the wave function is initially equal to Eq. (17), then since the phase vanishes, the particles are both at rest. Their possible positions are, however, described by an ensemble, in which x 1 − x 2 = a. Now, if we measure the position of the first particle, we introduce uncontrollable fluctuations in the wave function for the entire system, which, through the “quantum-mechanical” forces, bring about corresponding uncontrollable fluctuations in the momentum of each particle. Similarly, if we measure the momentum of the first particle, uncontrollable fluctuations in the wave function for the system bring about, through the “quantum-mechanical” forces, corresponding uncontrollable changes in the position of each particle. Thus, the “quantum-mechanical” forces may be said to transmit uncontrollable disturbances instantaneously from one particle to another through the medium of the ψ-field.
The Dreams That Stuff is Made of Page 51