The Dreams That Stuff is Made of

by Stephen Hawking

  The following seem to be reasonable requirements in our interpretation for a stationary state:(1) The particle energy should be a constant of the motion.

  (2) The quantum-mechanical potential should be independent of time.

  (3) The probability density in our statistical ensemble should be independent of time.

  It is easily verified that these requirements can be satisfied with the assumption that

  (12)

  From the above, we obtain S = Φ(x) − Et. According to the generalized Hamilton-Jacobi Eq. (4), the particle energy is given by

  Thus, we verify that the particle energy is a constant of the motion. Moreover, since P = R2 = |ψ|2, it follows that P (and R) are independent of time. This means that both the probability density in our ensemble and the quantum-mechanical potential are also time independent.

  The reader will readily verify that no other form of solution of Schroedinger’s equation will satisfy all three of our criteria for a stationary state.

  Since ψ is now being regarded as a mathematical representation of an objectively real force field, it follows that (like the electromagnetic field) it should be everywhere finite, continuous, and single valued. These requirements will guarantee in all cases that occur in practice that the allowed values of the energy in a stationary state, and the corresponding eigenfunctions are the same as are obtained from the usual interpretation of the theory.

  In order to show in more detail what a stationary state means in our interpretation, we shall now consider three examples of stationary states.

  CASE I: “s” STATE

  The first case that we shall consider is an “s” state. In an “s” state, the wave function is

  (13)

  where α is an arbitrary constant and r is the radius taken from the center of the atom. We conclude that the Hamilton-Jacobi function is

  The particle velocity is

  The particle is therefore simply standing still, wherever it may happen to be. How can it do this? The absence of motion is possible because the applied force, − ∇V(x), is balanced by the “quantum-mechanical” force, (h2/2m)∇(∇2 R/R), produced by the Schroedinger ψ-field acting on its own particle. There is, however, a statistical ensemble of possible positions of the particle, with a probability density, P(x) = (f(r))2.

  CASE 2: STATE WITH NONZERO ANGULAR MOMENTUM

  In a typical state of nonzero angular momentum, we have

  (14)

  where θ and øare the colatitude and azimuthal polar angles, respectively, Pl m is the associated Legendre polynomial, and β is a constant.

  The Hamilton-Jacobi function is S = β − Et + hmφ. From this result it follows that the z component of the angular momentum is equal to h m. To prove this, we write

  (15)

  Thus, we obtain a statistical ensemble of trajectories which can have different forms, but all have the same “quantized” value of the z component of the angular momentum.

  CASE 3: A SCATTERING PROBLEM

  Let us now consider a scattering problem. Because it is comparatively easy to analyze, we shall discuss a hypothetical experiment, in which an electron is incident in the z direction with an initial momentum, p0, on a system consisting of two slits.ix After the electron passes through the slit system, its position is measured and recorded, for example, on a photographic plate.

  Now, in the usual interpretation of the quantum theory, the electron is described by a wave function. The incident part of the wave function is ψ0 ∼ exp(i p0 z/ h); but when the wave passes through the slit system, it is modified by interference and diffraction effects, so that it will develop a characteristic intensity pattern by the time it reaches the position measuring instrument. The probability that the electron will be detected between x and x + dx is |ψ (x)|2dx. If the experiment is repeated many times under equivalent initial conditions, one eventually obtains a pattern of hits on the photographic plate that is very reminiscent of the interference patterns of optics.

  In the usual interpretation of the quantum theory, the origin of this interference pattern is very difficult to understand. For there may be certain points where the wave function is zero when both slits are open, but not zero when only one slit is open. How can the opening of a second slit prevent the electron from reaching certain points that it could reach if this slit were closed? If the electron acted completely like a classical particle, this phenomenon could not be explained at all. Clearly, then the wave aspects of the electron must have something to do with the production of the interference pattern. Yet, the electron cannot be identical with its associated wave, because the latter spreads out over a wide region. On the other hand, when the electron’s position is measured, it always appears at the detector as if it were a localized particle.

  The usual interpretation of the quantum theory not only makes no attempt to provide a single precisely defined conceptual model for the production of the phenomena described above, but it asserts that no such model is even conceivable. Instead of a single precisely defined conceptual model, it provides, as pointed out in Sec. 2, a pair of complementary models, viz., particle and wave, each of which can be made more precise only under conditions which necessitate a reciprocal decrease in the degree of precision of the other. Thus, while the electron goes through the slit system, its position is said to be inherently ambiguous, so that if we wish to obtain an interference pattern, it is meaningless to ask through which slit an individual electron actually passed. Within the domain of space within which the position of the electron has no meaning we can use the wave model and thus describe the subsequent production of interference. If, however, we tried to define the position of the electron as it passed the slit system more accurately by means of a measurement, the resulting disturbance of its motion produced by the measuring apparatus would destroy the interference pattern. Thus, conditions would be created in which the particle model becomes more precisely defined at the expense of a corresponding decrease in the degree of definition of the wave model. When the position of the electron is measured at the photographic plate, a similar sharpening of the degree of definition of the particle model occurs at the expense of that of the wave model.

  In our interpretation of the quantum theory, this experiment is described causally and continuously in terms of a single precisely definable conceptual model. As we have already shown, we must use the same wave function as is used in the usual interpretation; but instead we regard it as a mathematical representation of an objectively real field that determines part of the force acting on the particle. The initial momentum of the particle is obtained from the incident wave function, exp(ip0z/h), as p = ∂s/∂z = p0. We do not in practice, however, control the initial location of the particle, so that although it goes through a definite slit, we cannot predict which slit this will be. The particle is at all times acted on by the “quantum-mechanical” potential, U = (−h2/2m )∇2R/R . While the particle is incident, this potential vanishes because R is then a constant; but after it passes through the slit system, the particle encounters a quantum-mechanical potential that changes rapidly with position. The subsequent motion of the particle may therefore become quite complicated. Nevertheless, the probability that a particle shall enter a given region, dx, is as in the usual interpretation, equal to |ψ(x)|2dx. We therefore deduce that the particle can never reach a point where the wave function vanishes. The reason is that the “quantum-mechanical” potential, U, becomes infinite when R becomes zero. If the approach to infinity happens to be through positive values of U, there will be an infinite force repelling the particle away from the origin. If the approach is through negative values of U, the particle will go through this point with infinite speed, and thus spend no time there. In either case, we obtain a simple and precisely definable conceptual model explaining why particles can never be found at points where the wave function vanishes.

  If one of the slits is closed, the “quantum-mechanical” potential is correspondingly altered, because the ψ-field is changed,
and the particle may then be able to reach certain points which it was unable to reach when both slits were open. The slit is therefore able to affect the motion of the particle only indirectly, through its effect on the Schroedinger ψ-field. Moreover, as we shall see in Paper II, if the position of the electron is measured while it is passing through the slit system, the measuring apparatus will, as in the usual interpretation, create a disturbance that destroys the interference pattern. In our interpretation, however, the necessity for this destruction is not inherent in the conceptual structure; and as we shall see, the destruction of the interference pattern could in principle be avoided by means of other ways of making measurements, ways which are conceivable but not now actually possible.

  6. THE MANY-BODY PROBLEM

  We shall now extend our interpretation of the quantum theory to the problem of many bodies. We begin with the Schroedinger equation for two particles. (For simplicity, we assume that they have equal masses, but the extension of our treatment to arbitrary masses will be obvious.)

  Writing ψ = R(x1, x2) exp[iS(x1, x2)/h ] and R 2 = P, we obtain

  (16)

  (17)

  The above equations are simply a six-dimensional generalization of the similar three-dimensional Eqs. (5) and (6) associated with the one-body problem. In the two-body problem, the system is described therefore by a six-dimensional Schroedinger wave and by a six-dimensional trajectory, specifying the actual location of each of the two particles. The velocity of this trajectory has components, ∇1S/m and ∇2S/m, respectively, in each of the three-dimensional surfaces associated with a given particle. P(x1,x2) then has a dual interpretation. First, it defines a “quantum-mechanical” potential, acting on each particle

  This potential introduces an additional effective interaction between particles over and above that due to the classically inferrable potential V(x). Secondly, the function P(x1,x2) can consistently be regarded as the probability density of representative points (x1, x2) in our six-dimensional ensemble.

  The extension to an arbitrary number of particles is straightforward, and we shall quote only the results here. We introduce the wave function, ψ = R(x1, x2, . . . xn)exp[iS(x1, x2 . . . xn)/h ] and define a 3n-dimensional trajectory, where n is the number of particles, which describes the behavior of every particle in the system. The velocity of the ith particle is vi = ∇iS(x1, x2 . . . xn)/m . The function P(x1, x2 . . . xn = R2 has two interpretations. First, it defines a “quantum-mechanical” potential

  (18)

  Secondly, P(x1,x2 . . . xn) is equal to the density of representative points (x1, x2 . . . xn) in our 3n-dimensional ensemble.

  We see here that the “effective potential,” U(x1, x2 , . . . xn), acting on a particle is equivalent to that produced by a “many-body” force, since the force between any two particles may depend significantly on the location of every other particle in the system. An example of the effects of such a force is given by the exclusion principle. Thus, if the wave function is antisymmetric, we deduce that the “quantum-mechanical” forces will be such as to prevent two particles from ever reaching the same point in space, for in this case, we must have P = 0.

  7. TRANSITIONS BETWEEN STATIONARY STATES—THE FRANCK-HERTZ EXPERIMENT

  Our interpretation of the quantum theory describes all processes as basically causal and continuous. How then can it lead to a correct description of processes such as the Franck-Hertz experiment, the photoelectric effect, and the Compton effect, which seem to call most strikingly for an interpretation in terms of discontinuous and incompletely determined transfers of energy and momentum? In this section, we shall answer this question by applying our suggested interpretation of the quantum theory in the analysis of the Franck-Hertz experiment. Here, we shall see that the apparently discontinuous nature of the process of transfer of energy from the bombarding particle to the atomic electron is brought about by the “quantum-mechanical” potential, U = (−h2/2m)∇2 R/R, which does not necessarily become small when the wave intensity becomes small. Thus, even if the force of interaction between two particles is very weak, so that a correspondingly small disturbance of the Schroedinger wave function is produced by the interaction of these particles, this disturbance is capable of bringing about very large transfers of energy and momentum between the particles in a very short time. This means that if we view only the end results, this process presents the aspect of being discontinuous. Moveover, we shall see that the precise value of the energy transfer is in principle determined by the initial position of each particle and by the initial form of the wave function. Since we cannot in practice predict or control the initial particle positions with complete precision, we are also unable to predict or control the final outcome of such an experiment, and can, in practice, predict only the probability of a given outcome. Because the probability that the particles will enter a region with coordinates, x1, x2, is proportional to R 2(x1, x2), we conclude that although a Schroedinger wave of low intensity can bring about large transfers of energy, such a process is (as in the usual interpretation) highly improbable.

  In Appendix A of Paper II, we shall see that similar possibilities arise in connection with the interaction of the electromagnetic field with charged matter, so that electromagnetic waves can very rapidly transfer a full quantum of energy (and momentum) to an electron, even after they have spread out and fallen to a very low intensity. In this way, we shall explain the photoelectric effect and the Compton effect. Thus, we are able in our interpretation to understand by means of a causal and continuous model just those properties of matter and light which seem most convincingly to require the assumption of discontinuity and incomplete determinism.

  Before we discuss the process of interaction between two particles, we shall find it convenient to analyze the problem of an isolated single particle that happens to be in a nonstationary state. Because the field function ψ is a solution of Schroedinger’s equation, we can linearly suppose stationary-state solutions of this equation and in this way obtain new solutions. As an illustration, let us consider a superposition of two solutions

  where C 1 , C 2 , ψ1 , and ψ 2 are real. Thus we write ψ1 = R 1 , ψ2 = R 2 , and

  Writing ψ = R exp(iS/h ), we obtain

  (19)

  (20)

  We see immediately that the particle experiences a “quantum-mechanical” potential, U (x) = (−h /2m)∇ 2 R / R , which fluctuates with angular frequency, w = ( E 1 − E 2)/ h , and that the energy of this particle, E = −∂ S /∂ t , and its momentum p = ∇ S , fluctuate with the same angular frequency. If the particle happens to enter a region of space where R is small, these fluctuations can become quite violent. We see then that, in general, the orbit of a particle in a nonstationary state is very irregular and complicated, resembling Brownian motion more closely than it resembles the smooth track of a planet around the sun.

  If the system is isolated, these fluctuations will continue forever. The result is quite reasonable, since as is well known, a system can make a transition from one stationary state to another only if it can exchange energy with some other system. In order to treat the problem of transition between stationary states, we must therefore introduce another system capable of exchanging energy with the system of interest. In this section, we shall discuss the Franck-Hertz experiment, in which this other system consists of a bombarding particle. For the sake of illustration, let us suppose that we have hydrogen atoms of energy E0 and wave function, ψ0(x), which are bombarded by particles that can be scattered inelastically, leaving the atom with energy En and wave function, ψn (x).

  We begin by writing down the initial wave function, Ψi (x, y, t ). The incident particle, whose coordinates are represented by y must be associated with a wave packet, which can be written as

  (21)

  The center of this packet occurs where the phase has an extremum as a function of k, or where y = h k0t/m.

  Now, as in the usual interpretation, we begin by writing the incident wave function for the combined
system as a product

  (22)

  Let us now see how this wave function is to be understood in our interpretation of the theory. As pointed out in Sec. 6, the wave function is to be regarded as a mathematical representation of a six-dimensional but objectively real field, capable of producing forces that act on the particles. We also assume a six-dimensional representative point, described by the coordinates of the two particles, x and y. We shall now see that when the combined wave function takes the form (22) involving a product of a function of x and a function of y, the six-dimensional system can correctly be regarded as being made up of two independent three-dimentional subsystems. To prove this, we writeψ0(x) = R0(x) exp[i S0 (x)/ h] and

  f 0 (y, t ) = M0 (y, t ) exp[i N0(y, t )/ h ].

  We then obtain for the particle velocities

  (23)

  and for the “quantum-mechanical” potential

  (24)