The Dreams That Stuff is Made of

by Stephen Hawking

  This example illustrates very clearly how the quantum theory strips even the light waves of the primitive reality which is ascribed to them by the classical theory. The particular solution of the Maxwell equation which represents the emitted radiation depends on the accuracy with which the co-ordinates of the center of mass of the atom are known.

  § 6. THE COMPTON EFFECT AND THE EXPERIMENT OF COMPTON AND SIMON

  There are analogous relations in the theory of the Compton effect, but even though the calculations are the same as those of the preceding paragraph, a summary of the essential results will be given here. It is more interesting to consider bound electrons than free electrons, for then (if one assumes the position of the stationary atomic nucleus as given) there is a certain a priori knowledge concerning the position of the scattering electron. The laws of conservation result in the equations

  (74)

  The unprimed letters refer to variables before the collision, and the primed ones to variables after the collision; p is the linear momentum of the electron, and e and e’ signify unit vectors in the direction of motion of the light quantum; Δp gives the range of momentum of the electron in the atom. If ∼ Δp is small compared with p and hν/c, then (74) enables correspondingly exact conclusions regarding the relation between the directions e’ and p’ to be drawn. If, for example, p’ be measured in a Wilson chamber, then the radiation will have all the properties of needle radiation, since the direction of emission of the light quantum is determined. If pʹ >> Δp, then the translational wave function may be regarded as that of a plane wave, namely, exp 2π i / h · (p’ · r − Eʹt), where r is the vector specifying the position of the electron. Let the wave function of the unperturbed state E, which will be assumed to be the normal state, be ψE (r) exp 2π i / h · Et, where ψE is different from zero in an interval Δl [Δl · Δp ∼ h].

  These wave functions are perturbed by the incident wave of frequency ν, and the perturbation function is a periodic space function of wave-length λ = c/ν. Therefore, as the final result for the perturbed charge distribution, one obtains an expression of the form

  (75)

  Where fE is different from zero only in the interval Δl. If one writes the retarded potentials for points at a great distance from the atom, thencw

  (76)

  In this equation hνʹ = E − Eʹ + hν, r ʹis the vector to the point of integration, R to the point of observation, and R’ = R − r ’. The time factor in equation (76) shows that the frequency of the scattered radiation is ν’ and corresponds to that of equation (74). Furthermore, the integral on the right-hand side of equation (76) vanishes because of interference, if the factor of rʹ is materially greater than the reciprocal atomic diameter. Accordingly, since Δl Δp ∼ h,

  (77)

  in agreement with the second equation of (74). The scattered radiation behaves, therefore, in so far as its coherence properties are concerned, like needle radiation. However, the direction of the light quantum is not exactly prescribed, which may be regarded as a consequence of the indeterminateness of the momentum in the original stationary state. This indeterminateness can be diminished if one experiments with more loosely bound electrons, but then the atomic cross-section will be correspondingly greater. If one applies the considerations to an excited state, then Δl Δp ∼ nh appears in place of Δl Δp ∼ h and in the evaluation of the retarded potentials one must take the number of nodes of ψ (rʹ) into account. Since this involves only nonessential complications, we have confined ourselves to the normal state.

  If one wishes to explain the Geiger-Bothe experiment on the simultaneity of emission of recoil electron and scattered photon, then if the correspondence principle methods sketched here are used, one must deal with charge distributions which radiate only during a definite time interval. The initial state of the electron will be given, by a wave-packet at rest, whose size depends on the experimental arrangement. The final state will be represented by a morning wave-packet, and the charge density, given by the product of the two wave functions, will then be different from zero only during the time the two packets overlap. The radiation produced will then be a finite wave train moving in a definite direction. A more consequent explanation of the Geiger-Bothe experiment, even though it is equivalent in all its essential points, can only be obtained from the quantum theory of radiation. Moreover, as already shown, in this theory the laws of conservation applied to light quanta and electrons hold, so that one can, without any misgivings, use the customary corpuscular theory of this experiment.

  § 7. RADIATION FLUCTUATION PHENOMENA

  The large mean-square fluctuations, which belong to a corpuscular theory, are contained in the mathematical framework of the quantum theory, as shown in the Appendix. It is especially instructive, however, to study the relations between the various physical pictures with which the quantum theory operates by calculating the fluctuation of a radiation field. Let there be given a black cavity, of volume V, containing radiation in temperature equilibrium. The mean energy contained in a small volume element ΔV in the frequency range between ν and ν + Δν is, according to Planck’s formula,

  (78)

  k is the Boltzmann constant and T the temperature. According to general thermodynamic laws,cx the following relation holds for the mean-square fluctuation of :

  Substituting into equation (78), it was shown by Einstein that

  (79)

  This value for the mean-square fluctuation can only be derived partially with the help of the classical theory. The corpuscular viewpoint yields

  (80)

  The classical particle theory thus results only in the first part of formula (79). The classical wave theory of radiation, on the other hand, leads exactly to the second part of (79). The calculations for this will be given later in connection with the quantum theory. Thus, the quantum theory proper is necessary for the derivation of formula (79), in which it is naturally immaterial whether one uses the wave or the corpuscular picture.

  If, in particular, one treats the problem by means of the configuration space of the particles (although it is true that this has not been done in a detailed manner for light quanta), then one must note that the whole term system of the problem can be subdivided into non-combining partial systems, from which a definite one can be chosen as a solution. Because of the exchange relations (84), which become apparent from the corresponding uncertainty relations, that term system must be taken whose characteristic functions are symmetric in the co-ordinates of the light quanta. This choice leads to the Bose statistics for the light quanta and also, as Bosecy has shown, to equation (78).

  If the wave picture be used, then one obtains the number of light quanta corresponding to the vibration concerned from the amplitudes of the characteristic vibrations, and therefore the same mathematical scheme. In order to avoid unnecessary complications in the calculations, let us treat a vibrating string of length l instead of the black radiation cavity. Let ϕ(x, t) be its lateral displacement, and c the velocity of sound in the string. The Lagrangian function becomes

  (81)

  whence (A § 9)

  (82)

  and

  (83)

  The following exchange relations are to be used:

  (84)

  With the introduction of

  goes over into

  (85)

  On introducing the momenta associated to qk ,

  (86)

  equation (84) becomes

  (87)

  or

  (88)

  The characteristic frequencies of the string are νk = k(c/2l), and therefore

  (89)

  For the energy in a small section (0, a) of the string, one obtains, however,

  (90)

  If the terms of this sum with j = k be singled out, then under the explicit hypothesis that the wave-lengths to be considered are all small with respect to a, one obtains the value

  One thus finds the fluctuation by neglecting the terms with j = k in (90). The integr
ation results in

  (91)

  where

  (92)

  Accordingly, the mean-square fluctuation is given by

  The sums over j and k may be replaced by an integral over the frequencies ν j and νk , respectively, if it be assumed that the string l is very long, so that its characteristic frequencies are close together. In addition, one finally assumes that a is large and uses the relation

  (93)

  if ν1 > 0, ν2 > 0. The double integral then becomes a simple integral and one finds that

  (94)

  Because of the exchange relations (84),

  (95)

  so that

  (96)

  where Zν dν denotes the number of characteristic frequencies in the interval dν, or, in this case, Zν = 2l/c . If the integral be taken over the frequency interval Δν, one obtains

  (97)

  (98)

  One then subdivides into the thermal energy F* and the zero point energy:

  and finds

  (99)

  This value corresponds exactly to formula (79). The corresponding relation in the classical wave theory may be obtained by passing to the limit h = 0 in (99). The classical wave theory thus leads only to the second term of equation (99). The quantum theory, which one can interpret as a particle theory or as a wave theory as one sees fit, leads to the complete fluctuation formula.

  § 8. RELATIVISTIC FORMULATION OF THE QUANTUM THEORY

  The conditions imposed on all physical theories by the principle of relativity have been neglected in most of the foregoing discussions, and consequently the results obtained are applicable only under those conditions in which the velocity of light may be regarded as infinite. The reason for this neglect is that all relativistic effects belong to the terra incognita of quantum theory; the physical principles which have been elucidated in this book must be valid in this region also and thus it seemed proper not to obscure them with questions that cannot be answered definitely at the present time. None the less, this book would be incomplete without a brief discussion of the attempts to construct theories which shall embody both sets of principles, and the difficulties which have arisen in these attempts.

  Diraccz has set up a wave equation which is valid for one electron and is invariant under the Lorentz transformation. It fulfils all requirements of the quantum theory, and is able to give a good account of the phenomena of the “spinning” electron, which could previously only be treated by ad hoc assumptions. The essential difficulty which arises with all relativistic quantum theories is not eliminated however. This arises from the relation

  (100)

  between the energy and momentum of a free electron. According to this equation there are two values of E which differ in sign associated with each set of values of px, py, pz. The classical theory could eliminate this by arbitrarily excluding the one sign, but this is not possible according to the principles of quantum theory. Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong. Under these conditions it is very remarkable that the positive energy-levels (at least in the case of one electron) coincide with those actually observed.

  The difficulty inherent in formula (100) is also shown by a calculation of O. Klein,da who proves that if the electron is governed by any equation based on this relation it will be able to pass unhindered through regions in which its potential energy is greater than 2mc2.

  If only motion in the x-direction be considered the formulas (31a ) (31c) become

  whence

  while the wave function has the form

  For very small values of V, pʹx ’ is real and there are transmitted waves, just as in chapter ii, § 2f. For larger values, pʹx becomes a pure imaginary, so that the wave is totally reflected at the discontinuity and decreases exponentially in region II. But for very large values of V, px again becomes real, i.e., the electron wave again penetrates into the region II with constant amplitude. A more exact calculation verifies this result.

  A difficulty of a somewhat different character arises in the calculation of the energy of the field of the electron according to the relativistic theory. For a point electron (one of zero radius) even the classical theory yields an infinite value of the energy, as is well known, so that it becomes necessary to introduce a universal constant of the dimension of a length—the “radius of the electron.” It is remarkable that in the non-relativistic theory this difficulty can be avoided in another way—by a suitable choice of the order of non-commutative factors in the Hamiltonian function. This has hitherto not been possible in the relativistic quantum theory.

  The hope is often expressed that after these problems have been solved the quantum theory will be seen to be based, in a large measure at least, on classical concepts. But even a superficial survey of the trend of the evolution of physics in the past thirty years shows that it is far more likely that the solution will result in further limitations on the applicability of classical concepts than that it will result in a removal of those already discovered. The list of modifications and limitations of our ideal world—which now contains those required by the relativity theory (for which c is characteristic) and the uncertainty relations (symbolized by Planck’s constant h)—will be extended by others which correspond to e, µ, M. But the character of these is as yet not to be anticipated.

  THE DEVELOPMENT OF QUANTUM MECHANICS

  BY

  WERNER HEISENBERG

  Nobel lecture, December 11, 1933

  Quantum mechanics, on which I am to speak here, arose, in its formal content, from the endeavour to expand Bohr’s principle of correspondence to a complete mathematical scheme by refining his assertions. The physically new viewpoints that distinguish quantum mechanics from classical physics were prepared by the researches of various investigators engaged in analysing the difficulties posed in Bohr’s theory of atomic structure and in the radiation theory of light.

  In 1900, through studying the law of black-body radiation which he had discovered, Planck had detected in optical phenomena a discontinuous phenomenon totally unknown to classical physics which, a few years later, was most precisely expressed in Einstein’s hypothesis of light quanta. The impossibility of harmonizing the Maxwellian theory with the pronouncedly visual concepts expressed in the hypothesis of light quanta subsequently compelled research workers to the conclusion that radiation phenomena can only be understood by largely renouncing their immediate visualization. The fact, already found by Planck and used by Einstein, Debye, and others, that the element of discontinuity detected in radiation phenomena also plays an important part in material processes, was expressed systematically in Bohr’s basic postulates of the quantum theory which, together with the Bohr-Sommerfeld quantum conditions of atomic structure, led to a qualitative interpretation of the chemical and optical properties of atoms. The acceptance of these basic postulates of the quantum theory contrasted uncompromisingly with the application of classical mechanics to atomic systems, which, however, at least in its qualitative affirmations, appeared indispensable for understanding the properties of atoms. This circumstance was a fresh argument in support of the assumption that the natural phenomena in which Planck’s constant plays an important part can be understood only by largely foregoing a visual description of them. Classical physics seemed the limiting case of visualization of a fundamentally unvisualizable microphysics, the more accurately realizable the more Planck’s constant vanishes relative to the parameters of the system. This view of classical mechanics as a limiting case of quantum mechanics also gave rise to Bohr’s principle of correspondence which, at least in qualitative terms, transferred a number of conclusions formulated in classical mechanics to quantum mechanics. In connection with the principle of correspondence there was also discussion whether the quantum-mechanical laws could in principle be of a statistical nature; the possibility became particularly apparent in Einstein’s derivation of Planck’s law of radiation. Finally, the a
nalysis of the relation between radiation theory and atomic theory by Bohr, Kramers, and Slater resulted in the following scientific situation:

  According to the basic postulates of the quantum theory, an atomic system is capable of assuming discrete, stationary states, and therefore discrete energy values; in terms of the energy of the atom the emission and absorption of light by such a system occurs abruptly, in the form of impulses. On the other hand, the visualizable properties of the emitted radiation are described by a wave field, the frequency of which is associated with the difference in energy between the initial and final states of the atom by the relationE1 − E 2 = hν