It is true that the problem of classical dynamics also allows itself to be presented in the form of a partial equation, namely, the Hamilton-Jacobi equation. But the manifold of solutions of the problem does not correspond to the manifold of solutions of that equation. An arbitrary “complete” solution of the equation solves the mechanical problem completely; any other complete solution yields the same paths —they are only contained in another way in the manifold of paths.
Whatever the fear expressed about taking equation (18) as the foundation of atomic dynamics comes to, I will not positively assert that no further additional definitions will be required with it. But these will probably no longer be of such a completely strange and incomprehensible nature as the previous “quantum conditions”, but will be of the type that we are accustomed to find in physics with a partial differential equation as initial or boundary conditions. They will be, in no way, analogous to the quantum conditions—because in all cases of classical dynamics, which I have investigated up till now, it turns out that equation (18) carries within itself the quantum conditions. It distinguishes in certain cases, and indeed in those where experience demands it, of itself, certain frequencies or energy levels as those which alone are possible for stationary processes, without any further assumption, other than the almost obvious demand that, as a physical quantity, the function ψ must be single-valued, finite, and continuous throughout configuration space.
Thus the fear expressed is transformed into its contrary, in any case in what concerns the energy levels, or let us say more prudently, the frequencies. (For the question of the “vibrational energy” stands by itself; we must not forget that it is only in the one electron problem that the interpretation as a vibration in real three-dimensional space is immediately suggested.) The definition of the quantum levels no longer takes place in two separated stages: (1) Definition of all paths dynamically possible. (2) Discarding of the greater part of those solutions and the selection of a few by special postulations; on the contrary, the quantum levels are at once defined as the proper values of equation (18), which carries in itself its natural boundary conditions.
As to how far an analytical simplification will be effected in this way in more complicated cases, I have not yet been able to decide. I should, however, expect so. Most of the analytical investigators have the feeling that in the two-stage process, described above, there must be yielded in (1) the solution of a more complicated problem than is really necessary for the final result: energy as a (usually) very simple rational function of the quantum numbers. Already, as is known, the application of the Hamilton-Jacobi method creates a great simplification, as the actual calculation of the mechanical solution is avoided. It is sufficient to evaluate the integrals, which represent the momenta, merely for a closed complex path of integration instead of for a variable upper limit, and this gives much less trouble. Still the complete solution of the Hamilton-Jacobi equation must really be known, i.e. given by quadratures, so that the integration of the mechanical problem must in principle be effected for arbitrary initial values. In seeking for the proper values of a differential equation, we must usually, in practice, proceed thus. We seek the solution, firstly, without regard to boundary or continuity conditions, and from the form of the solution then pick out those values of the parameters, for which the solution satisfies the given conditions. Part I. supplies an example of this. We see by this example also, however—what is typical of proper value problems—that the solution was only given generally in an extremely inaccessible analytical form [equation (12) loc. cit.], but that it is extraordinarily simplified for those proper values belonging to the “natural boundary condition”. I am not well enough informed to say whether direct methods have now been worked out for the calculation of the proper values. This is known to be so for the distribution of proper values of high order. But this limiting case is not of interest here; it corresponds to the classical, macroscopic mechanics. For spectroscopy and atomic physics, in general just the first 5 or 10 proper values will be of interest; even the first alone would be a great result—it defines the ionisation potential. From the idea, definitely outlined, that every problem of proper values allows itself to be treated as one of maxima and minima without direct reference to the differential equation, it appears to me very probable that direct methods will be found for the calculation, at least approximately, of the proper values, as soon as urgent need arises. At least it should be possible to test in individual cases whether the proper values, known numerically to all desired accuracy through spectroscopy, satisfy the problem or not.
I would not like to proceed without mentioning here that at the present time a research is being prosecuted by Heisenberg, Born, Jordan, and other distinguished workers,dw to remove the quantum difficulties, which has already yielded such noteworthy success that it cannot be doubted that it contains at least a part of the truth. In its tendency, Heisenberg’s attempt stands very near the present one, as we have already mentioned. In its method, it is so totally different that I have not yet succeeded in finding the connecting link. I am distinctly hopeful that these two advances will not fight against one another, but on the contrary, just because of the extraordinary difference between the starting-points and between the methods, that they will supplement one another and that the one will make progress where the other fails. The strength of Heisenberg’s programme lies in the fact that it promises to give the line-intensities, a question that we have not approached as yet. The strength of the present attempt—if I may be permitted to pronounce thereon—lies in the guiding, physical point of view, which creates a bridge between the macroscopic and microscopic mechanical processes, and which makes intelligible the outwardly different modes of treatment which they demand. For me, personally, there is a special charm in the conception, mentioned at the end of the previous part, of the emitted frequencies as “beats”, which I believe will lead to an intuitive understanding of the intensity formulae.
§ 3. APPLICATION TO EXAMPLES
We will now add a few more examples to the Kepler problem treated in Part I., but they will only be of the very simplest nature, since we have provisionally confined ourselves to classical mechanics, with no magnetic field.dx
1. The Planck Oscillator. The Question of Degeneracy
Firstly we will consider the one-dimensional oscillator. Let the coordinate q be the displacement multiplied by the square root of the mass. The two forms of the kinetic energy then are
(20)
The potential energy will be
(21)
where ν0 is the proper frequency in the mechanical sense. Then equation (18) reads in this case
(22)
For brevity write
(23)
Therefore
(22ʹ)
Introduce as independent variable
(24)
and obtain
(22”)
The proper values and functions of this equation are known.dy The proper values are, with the notation used here,
(25)
The functions are the orthogonal functions of Hermite,
(26)
Hn (x) means the nth Hermite polynomial, which can be defined as
(27)
or explicitly by
(27ʹ)
The first of these polynomials are
(27”)
Considering next the proper values, we get from (25) and (23)
(25ʹ)
Thus as quantum levels appear so-called “half-integral” multiples of the “quantum of energy” peculiar to the oscillator, i.e. the odd multiples of . The intervals between the levels, which alone are important for the radiation, are the same as in the former theory. It is remarkable that our quantum levels are exactly those of Heisenberg’s theory. In the theory of specific heat this deviation from the previous theory is not without significance. It becomes important first when the proper frequency ν0 varies owing to the dissipation of heat. Formally it has to do with the old question of the “zero
-point energy”, which was raised in connection with the choice between the first and second forms of Planck’s Theory. By the way, the additional term also influences the law of the band-edges.
The proper functions (26) become, if we reintroduce the original q from (24) and (23),
(26ʹ)
Consideration of (27”) shows that the first function is a Gaussian Error-curve ; the second vanishes at the origin and for x positive corresponds to a “Maxwell distribution of velocities” in two dimensions, and is continued in the manner of an odd function for x negative. The third function is even, is negative at the origin, and has two symmetrical zeros at ± , etc. The curves can easily be sketched roughly and it is seen that the roots of consecutive polynomials separate one another. From (26ʹ) it is also seen that the characteristic points of the proper functions, such as half-breadth (for n = 0), zeros, and maxima, are, as regards order of magnitude, within the range of the classical vibration of the oscillator. For the classical amplitude of the nth vibration is readily found to be given by
(28)
Yet there is in general, as far as I see, no definite meaning that can be attached to the exact abscissa of the classical turning points in the graph of the proper function. It may, however, be conjectured, because the turning points have this significance for the phase space wave, that, at them, the square of the velocity of propagation becomes infinite and at greater distances becomes negative. In the differential equation (22), however, this only means the vanishing of the coefficient of ψ and gives rise to no singularities.
I would not like to suppress the remark here (and it is valid quite generally, not merely for the oscillator), that nevertheless this vanishing and becoming imaginary of the velocity of propagation is something which is very characteristic. It is the analytical reason for the selection of definite proper values, merely through the condition that the function should remain finite. I would like to illustrate this further. A wave equation with a real velocity of propagation means just this: there is an accelerated increase in the value of the function at all those points where its value is lower than the average of the values at neighbouring points, and vice versa. Such an equation, if not immediately and lastingly as in case of the equation for the conduction of heat, yet in the course of time, causes a levelling of extreme values and does not permit at any point an excessive growth of the function. A wave equation with an imaginary velocity of propagation means the exact opposite: values of the function above the average of surrounding values experience an accelerated increase (or retarded decrease), and vice versa. We see, therefore, that a function represented by such an equation is in the greatest danger of growing beyond all bounds, and we must order matters skilfully to preserve it from this danger. The sharply defined proper values are just what makes this possible. Indeed, we can see in the example treated in Part I. that the demand for sharply defined proper values immediately ceases as soon as we choose the quantity E to be positive, as this makes the wave velocity real throughout all space.
After this digression, let us return to the oscillator and ask ourselves if anything is altered when we allow it two or more degrees of freedom (space oscillator, rigid body). If different mechanical proper frequencies (ν0-values) belong to the separate co-ordinates, then nothing is changed. ψ is taken as the product of functions, each of a single co-ordinate, and the problem splits up into just as many separate problems of the type treated above as there are co-ordinates present. The proper functions are products of Hermite orthogonal functions, and the proper values of the whole problem appear as sums of those of the separate problems, taken in every possible combination. No proper value (for the whole system) is multiple, if we presume that there is no rational relation between the ν0-values.
If, however, there is such a relation, then the same manner of treatment is still possible, but it is certainly not unique. Multiple proper values appear and the “separation” can certainly be effected in other co-ordinates, e.g. in the case of the isotropic space oscillator in spherical polars.dz
The proper values that we get, however, are certainly in each case exactly the same, at least in so far as we are able to prove the “completeness” of a system of proper functions, obtained in one way. We recognise here a complete parallel to the well-known relations which the method of the previous quantisation meets with in the case of degeneracy. Only in one point there is a not unwelcome formal difference. If we applied the Sommerfeld-Epstein quantum conditions without regard to a possible degeneracy then we always got the same energy levels, but reached different conclusions as to the paths permitted, according to the choice of co-ordinates.
Now that is not the case here. Indeed we come to a completely different system of proper functions, if we, for example, treat the vibration problem corresponding to unperturbed Kepler motion in parabolic co-ordinates instead of the polars used in Part I. However, it is not just the single proper vibration that furnishes a possible state of vibration, but an arbitrary, finite or infinite, linear aggregate of such vibrations. And as such the proper functions found in any second way may always be represented; namely, they may be represented as linear aggregates of the proper functions found in an arbitrary way, provided the latter form a complete system.
The question of how the energy is really distributed among the proper vibrations, which has not been taken into account here up till now, will, of course, have to be faced some time. Relying on the former quantum theory, we will be disposed to assume that in the degenerate case only the energy of the set of vibrations belonging to one definite proper value must have a certain prescribed value, which in the non-degenerate case belongs to one single proper vibration. I would like to leave this question still quite open—and also the question whether the discovered “energy levels” are really energy steps of the vibration process or whether they merely have the significance of its frequency. If we accept the beat theory, then the meaning of energy levels is no longer necessary for the explanation of sharp emission frequencies.
2. Rotator with Fixed Axis
On account of the lack of potential energy and because of the Euclidean line element, this is the simplest conceivable example of vibration theory. Let A be the moment of inertia and φ the angle of rotation, then we clearly obtain as the vibration equation
(29)
which has the solution
(30)
Here the argument must be an integral multiple of φ, simply because otherwise ψ would neither be single-valued nor continuous throughout the range of the co-ordinate φ, as we know φ + 2π has the same significance as φ. This condition gives the well-known result
(31)
in complete agreement with the former quantisation.
No meaning, however, can be attached to the result of the application to band spectra. For, as we shall learn in a moment, it is a peculiar fact that our theory gives another result for the rotator with free axis. And this is true in general. It is not allowable in the applications of wave mechanics, to think of the freedom of movement of the system as being more strictly limited, in order to simplify calculation, than it actually is, even when we know from the integrals of the mechanical equations that in a single movement certain definite freedoms are not made use of. For micro-mechanics, the fundamental system of mechanical equations is absolutely incompetent; the single paths with which it deals have now no separate existence. A wave process fills the whole of the phase space. It is well known that even the number of the dimensions in which a wave process takes place is very significant.
3. Rigid Rotator with Free Axis
If we introduce as co-ordinates the polar angles θ, φ of the radius from the nucleus, then for the kinetic energy as a function of the momenta we get
(32)
According to its form this is the kinetic energy of a particle constrained to move on a spherical surface. The Laplacian operator is thus simply that part of the spatial Laplacian operator which depends on the polar angles, and the vibration equation (18”) takes
the following form,
(33)
The postulation that ψ should be single-valued and continuous on the spherical surface leads to the proper value condition
(34)
The proper functions are known to be spherical surface harmonics. The energy levels are, therefore,
The Dreams That Stuff is Made of Page 28