We give a simple explanation of the origin of quantum-mechanical correlations of distant objects in the hypothetical experiment of Einstein, Podolsky, and Rosen, which was suggested by these authors as a criticism of the usual interpretation.
Finally, we show that von Neumann’s proof that quantum theory is not consistent with hidden variables does not apply to our interpretation, because the hidden variables contemplated here depend both* Now at Universidade de São Paulo, Faculdade de Filosofia, Ciencias e Letras, São Paulo, Brasil.
Reprinted with permission from the American Physical Society: D. Bohm, Physical Review, Volume 85, Number 2, 1952. © 1952 by the American Physical Society.
on the state of the measuring apparatus and the observed system and therefore go beyond certain of von Neumann’s assumptions.
In two appendixes, we treat the problem of the electromagnetic field in our interpretation and answer certain additional objections which have arisen in the attempt to give a precise description for an individual system at the quantum level.
1. INTRODUCTION
In a previous paper,je to which we shall hereafter refer as I, we have suggested an interpretation of the quantum theory in terms of “hidden” variables. We have shown that although this interpretation provides a conceptual framework that is broader than that of the usual interpretation, it permits of a set of three mutually consistent special assumptions, which lead to the same physical results as are obtained from the usual interpretation of the quantum theory. These three special assumptions are: (1) The ψ-field satisfies Schroedinger’s equation. (2) If we write ψ = R exp (is/h), then the particle momentum is restricted to p = ∇ S (x). (3) We have a statistical ensemble of particle positions, with a probability density, P = |ψ(x)|2. If the above three special assumptions are not made, then one obtains a more general theory that cannot be made consistent with the usual interpretation. It was suggested in Paper I that such generalizations may actually be needed for an understanding of phenomena associated with distances of the order of 10−13 cm or less, but may produce changes of negligible importance in the atomic domain.
In this paper, we shall apply the interpretation of the quantum theory suggested in Paper I to the development of a theory of measurements in order to show that as long as one makes the special assumptions indicated above, one is led to the same predictions for all measurements as are obtained from the usual interpretation. In our interpretation, however, the uncertainty principle is regarded, not as an inherent limitation on the precision with which we can correctly conceive of the simultaneous definition of momentum and position, but rather as a practical limitation on the precision with which these quantities can simultaneously be measured, arising from unpredictable and uncontrollable disturbances of the observed system by the measuring apparatus. If the theory needs to be generalized in the ways suggested in Paper I, Secs. 4 and 9, however, then these disturbances could in principle either be eliminated, or else be made subject to prediction and control, so that their effects could be corrected for. Our interpretation therefore demonstrates that measurements violating the uncertainty principle are at least conceivable.
2. QUANTUM THEORY OF MEASUREMENTS
We shall now show how the quantum theory of measurements is to be expressed in terms of our suggested interpretation of the quantum theory.jf
In general, a measurement of any variable must always be carried out by means of an interaction of the system of interest with a suitable piece of measuring apparatus. The apparatus must be so constructed that any given state of the system of interest will lead to a certain range of states of the apparatus. Thus, the interaction introduces correlations between the state of the observed system and the state of the apparatus. The range of indefiniteness in this correlation may be called the uncertainty, or the error, in the measurement.
Let us now consider an observation designed to measure an arbitrary (hermitian) “observable” Q, associated with an electron. Let x represent the position of the electron, y that of the significant apparatus coordinate (or coordinates if there are more than one). Now, one can showjf that it is enough to consider an impulsive measurement, i.e., a measurement utilizing a very strong interaction between apparatus and system under observation, which lasts for so short a time that the changes of the apparatus and the system under observation that would have taken place in the absence of interaction can be neglected. Thus, at least while the interaction is taking place, we can neglect the parts of the Hamiltonian associated with the apparatus alone and with the observed system alone, and we need retain only the part of the Hamiltonian, H1, representing the interaction. Moreover, if the Hamiltonian operator is chosen to be a function only of quantities that commute with Q, then the interaction process will produce no uncontrollable changes in the observable, Q, but only in observables that do not commute with Q. In order that the apparatus and the system under observation shall be coupled, however, it is necessary that H1 shall also depend on operators involving y.
For the sake of illustration of the principles involved, we shall consider the following interaction Hamiltonian:
(1)
where a is a suitable constant and py is the momentum conjugate to y.
Now, in our interpretation, the system is to be described by a four-dimensional but objectively real wave field that is a function of x and y and by a corresponding four-dimensional representative point, specified by the coordinates, x, of the electron and the coordinate, y, of the apparatus. Since the motion of the representative point is in part determined by forces produced by the ψ-field acting on both electron and apparatus variables, our first step in solving this problem is to calculate the ψ-field. This is done by solving Schroedinger’s equation, with the appropriate boundary conditions on ψ.
Now, during interaction, Schroedinger’s equation is approximated by
(2)
It is now convenient to expand Ψ in terms of the complete set ψq (x) of eigenfunctions of the operator, Q, where q denotes an eigenvalue of Q. For the sake of simplicity, we assume that the spectrum of Q is discrete, although the results are easily generalized to a continuous spectrum. Denoting the expansion coefficients by fq (y,t), we obtain
(3)
Noting that Qψq (x) = q ψq (x), we readily verify that Eq. (2) can now be reduced to the following series of equations for fq (y,t):
(4)
If the initial value of fq (y,t) and (y) we obtain as a solution
(5)
and
(6)
Now, Initially the apparatus and the electron were independent. As shown in Paper I, Sec. 7, in our interpretation (as in the usual interpretation), independent systems must have wave fields Ψ (x,y,t) that are equal to a product of a function of x and a function of y . Initially, we therefore have
(7)
where the cq are the (unknown) expansion coefficients of ψq (x), and g0 (y) is the initial wave function of the apparatus coordinate, y. The function g0 (y) will take the form of a packet. For the sake of convenience, we assume that this packet is centered at y = 0 and that its width is Δy. Normally, because the apparatus is classically describable, the definition of this packet is far less precise than that allowed by the limits of precision set by the uncertainty principle.
From Eqs. (7) and (3), we shall readily deduce that (y) = cqg0 (y) When this value of fq0 (y) is inserted into Eq. (6), we obtain
(8)
Equation (8) indicates already that the interaction has introduced a correlation between q and the apparatus coordinate, y. In order to show what this correlation means in our interpretation of the quantum theory, we shall use some arguments that have been developed in more detail in Paper I, Sec. 7, in connection with a similar problem involving the interaction of two particles in a scattering process. First we note that while the electron and the apparatus are interacting, the wave function (8) becomes very complicated, so that if it is expressed asΨ (x,y,t) = R (x,y,t) exp[i S (x, y,t)/h ],
then R and S undergo rapi
d oscillations both as a function of position and of time. From this we deduce that the “quantum-mechanical” potential,
undergoes violent fluctuations, especially where R is small, and that the particle momenta, p = ∇x S (x,y,t) and py = ∂ S (x,y,t)/∂y, also undergo corresponding violent and extremely complicated fluctuations. Eventually, however, if the interaction continues long enough, the behavior of the system will become simpler because the packets g0 (y − aqt/h2), corresponding to different values of q, will cease to overlap in y space. To prove this, we note that the center of the qth packet in y space is at
(9)
If we denote the separation of adjacent values of q by δq , we then obtain for the separation of the centers of adjacent packets in y space
(10)
It is clear that if the product of the strength of interaction a, and the duration of Interaction, t, is large enough, then δy can be made much larger than the width Δy of the packet. Then packets corresponding to different values of q will cease to overlap in y space and will, in fact, obtain separations large enough to be classically describable.
Because the probability density is equal to |Ψ|2, we deduce that the apparatus variable, y , must finally enter one of the packets and remain with that packet thereafter (since it does not enter the intermediate space between packets in which the probability density is practically zero). Now, the packet entered by the apparatus variable y determines the actual result of the measurement, which the observer will obtain when he looks at the apparatus. The other packets can (as shown in Paper I, Sec. 7) be ignored, because they affect neither the quantum-mechanical potential acting on the particle coordinates x and y, nor the particle momenta, px = ∇x S and p y = ∂S/∂y . Moreover, the wave function can also be renormalized without affecting any of the above quantities. Thus, for all practical purposes, we can replace the complete wave function, Eq. (8), by a new renormalized wave function
(11)
where q now corresponds to the packet actually containing the apparatus variable, y. From this wave function, we can deduce, as shownin Paper I, Sec. 7, that the apparatus and the electron will subsequently behave independently. Moreover, by observing the approximate value of the apparatus coordinate within an error Δy « δy , we can deduce with the aid of Eq. (9) that since the electron wave function can for all practical purposes be regarded as ψq (x), the observable, Q, must have the definite value, q. However, if the product, “a t δq / h2,” appearing in Eqs. (8), (9), (10), and (11), had been less than Δy , then no clear measurement of Q would have been possible, because packets corresponding to different q would have overlapped, and the measurement would not have had the requisite accuracy.jg
Finally, we note that even if the apparatus packets are subsequently caused to overlap, none of those conclusions will be altered. For the apparatus variable y will inevitably be coupled to a whole host of internal thermodynamic degrees of freedom, y1, y2, ...ys,asaresult of effects such as friction and brownian motion. As shown in Paper I, Sec. 7, interference between packets corresponding to different values of q would be possible only if the packets overlapped in the space of y1, y2, ...ys, as well as in y space. Such an overlap, however, is so improbable that for all practical purposes, we can ignore the possibility that it will ever occur.
3. THE ROLE OF PROBABILITY IN MEASUREMENTS—THE UNCERTAINTY PRINCIPLE
In principle, the final result of a measurement is determined by the initial form of the wave function of the combined system, Ψ0(x,y), and by the initial position of the electron particle, x0, and the apparatus variable, y0. In practice, however, as we have seen, the orbit fluctuates violently while interaction takes place, and is very sensitive to the precise initial values of x and y, which we can neither predict nor control. All that we can predict in practice is that in an ensemble of similar experiments performed under equivalent initial conditions, the probability density is |Ψ(x,y)|2. From this information, however, we are able to calculate only the probability that in an individual experiment, the result of a measurement of Q will be a specific number q. To obtain the probability of a given value of q, we need only integrate the above probability density over all x and over all values of y in the neighborhood of the qth packet. Because the packets do not overlap, the Ψ-field in this region is equal to cqψq(x)g0(y − aqt/h2) [see Eq. (8)]. Since, by definition, ψq(x) and g0(y) are normalized, the total probability that a particle is in the qth packet is
(12)
The above is, however, just what is obtained from the usual interpretation. We conclude then that our interpretation is capable of leading in all possible experiments to identical predictions with those obtained from the usual interpretation (provided, or course, that we make the special assumptions indicated in the introduction).
Let us now see what a measurement of the observable, Q, implies with regard to the state of the electron particle and its Ψ-field. First, we note that the process of interaction with an apparatus designed to measure the observable, Q, effectively transforms the electron ψ-field from whatever it was before the measurement took place into an eigenfunction ψq(x) of the operator Q. The precise value of q that comes out of this process is as we have seen, not, in general, completely predictable or controllable. If, however, the same measurement is repeated after the ψ-field has been transformed into ψq(x), we can then predict that (as in the usual interpretation), the same value of q, and therefore the same wave function, ψq(x), will be obtained again. If, however, we measure an observable “P” that does not commute with Q, then the results of this measurement are not, in practice, predictable or controllable. For as shown in Eq. (8), the Ψ-field after interaction with the measuring apparatus is now transformed into
(13)
where φp(x) is an eigenfunction of the operator, P, belonging to an eigenvalue, p, and where ap,q is an expansion coefficient defined by
(14)
Since the packets corresponding to different p ultimately become completely separate in z space, we deduce, as in the case of the measurement of Q, that for all practical purposes, this wave function may be replaced by
where p now represents the packet actually entered by the apparatus coordinate, y . As in the case of measurement of Q, we readily show that the precise value of p that comes out of this experiment cannot be predicted or controlled and that the probability of a given value of p is equal to |apq|2 . This is, however, just what is obtained in the usual interpretation of this process.
It is clear that if two “observables,” P and Q, do not commute, one cannot carry out a measurement of both simultaneously on the same system. The reason is that each measurement disturbs the system in a way that is incompatible with carrying out the process necessary for the measurement of the other. Thus, a measurement of P requires that wave field, ψ, shall become an eigenfunction of P, while a measurement of Q requires that it shall become an eigenfunction of Q. If P and Q do not commute, then by definition, no ψ-function can be simultaneously an eigenfunction of both. In this way, we understand in our interpretation why measurements, of complementary quantities, must (as in the usual interpretation) necessarily be limited in their precision by the uncertainty principle.
4. PARTICLE POSITIONS AND MOMENTA AS “HIDDEN VARIABLES”
We have seen that in measurements that can now be carried out, we cannot make precise inferences about the particle position, but can say only that the particle must be somewhere in the region in which |ψ| is appreciable. Similarly, the momentum of a particle that happens to be at the point, x, is given by p = ∇ S(x), so that since x is not known, the precise value of p is also not, in general, inferrable. Hence, as long as we are restricted to making observations of this kind, the precise values of the particle position and momentum must, in general, be regarded as “hidden,” since we cannot at present measure them. They are, however, connected with real and already observable properties of matter because (along with the ψ-field) they determine in principle the actual result of each individual measurement. B
y way of contrast, we recall here that in the usual interpretation of the theory, it is stated that although each measurement admittedly leads to a definite number, nothing determines the actual value of this number. The result of each measurement is assumed to arise somehow in an inherently indescribable way that is not subject to a detailed analysis. Only the statistical results are said to be predictable. In our interpretation, however, we assert that the at present “hidden” precisely definable particle positions and momenta determine the results of each individual measurement process, but in a way whose precise details are so complicated and uncontrollable, and so little known, that one must for all practical purposes restrict oneself to a statistical description of the connection between the values of these variables and the directly observable results of measurements. Thus, we are unable at present to obtain direct experimental evidence for the existence of precisely definable particle positions and momenta.
The Dreams That Stuff is Made of Page 50