The Dreams That Stuff is Made of

by Stephen Hawking

  Our integral function U, which alone of the solutions of (7ʹ) is considered for our problem, is therefore not finite for r large, on the above hypothesis. Reserving meanwhile the question of completeness, i.e. the proving that our treatment allows us to find all the linearly independent solutions of the problem, then we may state:

  For negative values of E which do not satisfy condition (15) our variation problem has no solution.

  We have now only to investigate that discrete set of negative E-values which satisfy condition (15). α1 and α2 are then both integers. The first of the integration paths, which previously gave us the fundamental values U1 and U2, must now undoubtedly be modified so as to give a non-vanishing result. For, since α1 − 1 is certainly positive, the point c1 is neither a branch point nor a pole of the integrand, but an ordinary zero. The point c2 can also become regular if α2 − 1 is also not negative. In every case, however, two suitable paths are readily found and the integration effected completely in terms of known functions, so that the behaviour of the solutions can be fully investigated.

  Let

  (15ʹ)

  Then from (14”) we have

  (14”ʹ)

  Two cases have to be distinguished: l ≤ n and l > n.

  (a) l ≤ n. Then c2 and c1 lose every singular character, but instead become starting-points or end-points of the path of integration, in order to fulfil condition (13). A third characteristic point here is at infinity (negative and real). Every path between two of these three points yields a solution, and of these three solutions there are two linearly independent, as is easily confirmed if the integrals are calculated out. In particular, the transcendental integral solution is given by the path from c1 to c2. That this integral remains regular at r = 0 can be seen at once without calculating it. I emphasize this point, as the actual calculation is apt to obscure it. However, the calculation does show that the integral becomes indefinitely great for positive, infinitely great values of r . One of the other two integrals remains finite for r large, but it becomes infinite for r = 0.

  Therefore when l ≤ n we get no solution of the problem.

  (b) l > n. Then from (14”ʹ), c1 isa zero and c2 a pole of the first order at least of the integrand. Two independent integrals are then obtained: one from the path which leads from z = −∞ to the zero, intentionally avoiding the pole; and the other from the residue at the pole. The latter is the integral function. We will give its calculated value, but multiplied by r n , so that we obtain, according to (9) and (10), the solution χ of the original equation

  (7). (The multiplying constant is arbitrary.) We find

  (18)

  It is seen that this is a solution that can be utilised, since it remains finite for all real non-negative values of r . In addition, it satisfies the surface condition (6) because of its vanishing exponentially at infinity. Collecting then the results for E negative:

  For E negative, our variation problem has solutions if, and only if, E satisfies condition (15). Only values smaller than l (and there is always at least one such at our disposal) can be given to the integer n, which denotes the order of the surface harmonic appearing in the equation. The part of the solution depending on r is given by (18).

  Taking into account the constants in the surface harmonic (known to be 2n + 1 in number), it is further found that:

  The discovered solution has exactly 2n + 1 arbitrary constants for any permissible (n, l) combination; and therefore for a prescribed value of l has l2 arbitrary constants.

  We have thus confirmed the main points of the statements originally made about the proper-value spectrum of our variation problem, but there are still deficiencies.

  Firstly, we require information as to the completeness of the collected system of proper functions indicated above, but I will not concern myself with that in this paper. From experience of similar cases, it may be supposed that no proper value has escaped us.

  Secondly, it must be remembered that the proper functions, ascertained for E positive, do not solve the variation problem as originally postulated, because they only tend to zero at infinity as 1/ r , and therefore ∂ψ/∂r only tends to zero on an infinite sphere as 1/r 2. Hence the surface integral (6) is still of the same order as δψ at infinity. If it is desired therefore to obtain the continuous spectrum, another condition must be added to the problem, viz. that δψ is to vanish at infinity, or at least, that it tends to a constant value independent of the direction of proceeding to infinity; in the latter case the surface harmonics cause the surface integral to vanish.

  §2. Condition (15) yields

  (19)

  Therefore the well-known Bohr energy-levels, corresponding to the Balmer terms, are obtained, if to the constant K, introduced into (2) for reasons of dimensions, we give the value

  (20)

  from which comes

  (19ʹ)

  Our l is the principal quantum number. n + 1 is analogous to the azimuthal quantum number. The splitting up of this number through a closer definition of the surface harmonic can be compared with the resolution of the azimuthal quantum into an “equatorial” and a “polar” quantum. These numbers here define the system of node-lines on the sphere. Also the “radial quantum number” l − n − 1 gives exactly the number of the “node-spheres”, for it is easily established that the function f (x) in (18) has exactly l − n − 1 positive real roots. The positive E-values correspond to the continuum of the hyperbolic orbits, to which one may ascribe, in a certain sense, the radial quantum number ∞. The fact corresponding to this is the proceeding to infinity, under continual oscillations, of the functions in question.

  It is interesting to note that the range, inside which the functions of (18) differ sensibly from zero, and outside which their oscillations die away, is of the general order of magnitude of the major axis of the ellipse in each case. The factor, multiplied by which the radius vector enters as the argument of the constant-free function f, is—naturally—the reciprocal of a length, and this length is

  (21)

  where al = the semi-axis of the lth elliptic orbit. (The equations follow from (19) plus the known relation E l = −2ael 2 )

  The quantity (21) gives the order of magnitude of the range of the roots when l and n are small; for then it may be assumed that the roots of f(x) are of the order of unity. That is naturally no longer the case if the coefficients of the polynomial are large numbers. At present I will not enter into a more exact evaluation of the roots, though I believe it would confirm the above assertion pretty thoroughly.

  § 3. It is, of course, strongly suggested that we should try to connect the function ψ with some vibration process in the atom, which would more nearly approach reality than the electronic orbits, the real existence of which is being very much questioned to-day. I originally intended to found the new quantum conditions in this more intuitive manner, but finally gave them the above neutral mathematical form, because it brings more clearly to light what is really essential. The essential thing seems to me to be, that the postulation of “whole numbers” no longer enters into the quantum rules mysteriously, but that we have traced the matter a step further back, and found the “integralness” to have its origin in the finiteness and single-valuedness of a certain space function.

  I do not wish to discuss further the possible representations of the vibration process, before more complicated cases have been calculated successfully from the new stand-point. It is not decided that the results will merely re-echo those of the usual quantum theory. For example, if the relativistic Kepler problem be worked out, it is found to lead in a remarkable manner to half-integral partial quanta (radial and azimuthal).

  Still, a few remarks on the representation of the vibration may be permitted. Above all, I wish to mention that I was led to these deliberations in the first place by the suggestive papers of M. Louis de Broglie,dh and by reflecting over the space distribution of those “phase waves”, of which he has shown that there is always a whole number, measured alo
ng the path, present on each period or quasi-period of the electron. The main difference is that de Broglie thinks of progressive waves, while we are led to stationary proper vibrations if we interpret our formulae as representing vibrations. I have lately showndi that the Einstein gas theory can be based on the consideration of such stationary proper vibrations, to which the dispersion law of de Broglie’s phase waves has been applied. The above reflections on the atom could have been represented as a generalisation from those on the gas model.

  If we take the separate functions (18), multiplied by a surface harmonic of order n, as the description of proper vibration processes, then the quantity. E must have something to do with the related frequency. Now in vibration problems we are accustomed to the “parameter” (usually called λ) being proportional to the square of the frequency. However, in the first place, such a statement in our case would lead to imaginary frequencies for the negative E-values, and, secondly, instinct leads us to believe that the energy must be proportional to the frequency itself and not to its square.

  The contradiction is explained thus. There has been no natural zero level laid down for the “parameter” E of the variation equation (5), especially as the unknown function ψ appears multiplied by a function of r, which can be changed by a constant to meet a corresponding change in the zero level of E. Consequently, we have to correct our anticipations, in that not E itself—continuing to use the same terminology—but E increased by a certain constant is to be expected to be proportional to the square of the frequency. Let this constant be now very great compared with all the admissible negative E-values (which are already limited by (15)). Then firstly, the frequencies will become real, and secondly, since our E-values correspond to only relatively small frequency differences, they will actually be very approximately proportional to these frequency differences. This, again, is all that our “quantum-instinct” can require, as long as the zero level of energy is not fixed.

  The view that the frequency of the vibration process is given by

  (22)

  where C is a constant very great compared with all the E’s, has still another very appreciable advantage. It permits an understanding of the Bohr frequency condition. According to the latter the emission frequencies are proportional to the E-differences, and therefore from (22) also to the differences of the proper frequencies v of those hypothetical vibration processes. But these proper frequencies are all very great compared with the emission frequencies, and they agree very closely among themselves. The emission frequencies appear therefore as deep “difference tones” of the proper vibrations themselves. It is quite conceivable that on the transition of energy from one to another of the normal vibrations, something—I mean the light wave—with a frequency allied to each frequency difference, should make its appearance. One only needs to imagine that the light wave is causally related to the beats, which necessarily arise at each point of space during the transition; and that the frequency of the light is defined by the number of times per second the intensity maximum of the beat-process repeats itself.

  It may be objected that these conclusions are based on the relation (22), in its approximate form (after expansion of the square root), from which the Bohr frequency condition itself seems to obtain the nature of an approximation. This, however, is merely apparently so, and it is wholly avoided when the relativistic theory is developed and makes a profounder insight possible. The large constant C is naturally very intimately connected with the rest-energy of the electron (mc2). Also the seemingly new and independent introduction of the constant h (already brought in by (20)), into the frequency condition, is cleared up, or rather avoided, by the relativistic theory. But unfortunately the correct establishment of the latter meets right away with certain difficulties, which have been already alluded to.

  It is hardly necessary to emphasize how much more congenial it would be to imagine that at a quantum transition the energy changes over from one form of vibration to another, than to think of a jumping electron. The changing of the vibration form can take place continuously in space and time, and it can readily last as long as the emission process lasts empirically (experiments on canal rays by W. Wien); nevertheless, if during this transition the atom is placed for a comparatively short time in an electric field which alters the proper frequencies, then the beat frequencies are immediately changed sympathetically, and for just as long as the field operates. It is known that this experimentally established fact has hitherto presented the greatest difficulties. See the well-known attempt at a solution by Bohr, Kramers, and Slater.

  Let us not forget, however, in our gratification over our progress in these matters, that the idea of only one proper vibration being excited whenever the atom does not radiate—if we must hold fast to this idea—is very far removed from the natural picture of a vibrating system. We know that a macroscopic system does not behave like that, but yields in general a pot-pourri of its proper vibrations. But we should not make up our minds too quickly on this point. A pot-pourri of proper vibrations would also be permissible for a single atom, since thereby no beat frequencies could arise other than those which, according to experience, the atom is capable of emitting occasionally. The actual sending out of many of these spectral lines simultaneously by the same atom does not contradict experience. It is thus conceivable that only in the normal state (and approximately in certain “metastable” states) the atom vibrates with one proper frequency and just for this reason does not radiate, namely, because no beats arise. The stimulation may consist of a simultaneous excitation of one or of several other proper frequencies, whereby beats originate and evoke emission of light.

  Under all circumstances, I believe, the proper functions, which belong to the same frequency, are in general all simultaneously stimulated. Multipleness of the proper values corresponds, namely, in the language of the previous theory to degeneration. To the reduction of the quantisation of degenerate systems probably corresponds the arbitrary partition of the energy among the functions belonging to one proper value.

  ADDITION AT THE PROOF CORRECTION ON 28. 2. 1926.

  In the case of conservative systems in classical mechanics, the variation problem can be formulated in a neater way than was previously shown, and without express reference to the Hamilton-Jacobi differential equation. Thus, let T (q, p) be the kinetic energy, expressees as a function of the co-ordinates and momenta, V the potential energy and dτ the volume element of the space, “measured rationally”, i.e. it is not simply the product dq1 dq2 dq3 . . . dqn , but this divided by the square root of the discriminant of the quadratic form T (q, p) (Cf. Gibbs’ Statistical Mechanics.) Then let ψ be such as to make the “Hamilton integral”

  (23)

  stationary, while fulfilling the normalising, accessory condition

  (24)

  The proper values of this variation problem are then the stationary values of integral (23) and yield, according to our thesis, the quantum-levels of the energy.

  It is to be remarked that in the quantity α2of (14”) we have essentially the well-known Sommerfeld expression .(Cf. Atombau, 4th (German) ed., p. 775.)

  Physical Institute of the University of Zürich.

  (Received January 27, 1926.)

  QUANTISATION AS A PROBLEM OF PROPER VALUES (PART II)

  (Annalen der Physik (4), vol. 79, 1926)

  § 1. THE HAMILTONIAN ANALOGY BETWEEN MECHANICS AND OPTICS

  BEFORE we go on to consider the problem of proper values for further special systems, let us throw more light on the general correspondence which exists between the Hamilton-Jacobi differential equation of a mechanical problem and the “allied” wave equation, i.e. equation (5) of Part I in the case of the Kepler problem. So far we have only briefly described this correspondence on its external analytical side by the transformation (2), which is in itself unintelligible, and by the equally incomprehensible transition from the equating to zero of a certain expression to the postulation that the space integral of the said expression shall
be stationary.dj

  The inner connection between Hamilton’s theory and the process of wave propagation is anything but a new idea. It was not only well known to Hamilton, but it also served him as the starting-point for his theory of mechanics, which grewdk out of his Optics of Non-homogeneous Media. Hamilton’s variation principle can be shown to correspond to Fermat’s Principle for a wave propagation in configuration space (q-space), and the Hamilton-Jacobi equation expresses Huygens’ Principle for this wave propagation. Unfortunately this powerful and momentous conception of Hamilton is deprived, in most modern reproductions, of its beautiful raiment as a superfluous accessory, in favour of a more colourless representation of the analytical correspondence.dl