Let it be expressly remarked, however, that the back-coupling term is not necessary for averting a resonance catastrophe! Such can never occur in any circumstances, because according to the theorem of the persistence of normalisation, proved below in § 7, the configuration space integral of ψψ always remains normalised to the same value, even under the influence of arbitrary external forces—and indeed quite automatically, as a consequence of the wave equation (4”). The amplitudes of the ψ -vibrations, therefore, cannot grow indefinitely; they have, “on the average”, always the same value. If one proper vibration waxes, then another must, therefore, wane.
§ 5. GENERALISATION FOR AN ARBITRARY PERTURBATION
If an arbitrary perturbation is in question as was assumed in equation (5) at the beginning of § 2, then we shall expand the perturbation energy r(x, t) as a Fourier series or Fourier integral in terms of the time. The terms of this expansion have, then, the form (6) of the perturbation potential of a light wave. We see immediately that on the right-hand side of equation (11) we then simply get two series (or, possibly, integrals) of imaginary powers of e, instead of merely two terms. If none of the exciting frequencies coincide with a critical frequency, we get the solution in exactly the same way as described in § 2, but, naturally, as Fourier series (or possibly Fourier integrals) of the time. It serves no purpose to write down the formal expansions here, and a more exact working out of separate problems lies outside the scope of the present paper. Yet an important point, already touched upon in § 3, must be mentioned.
Among the critical frequencies of equation (13), the frequency ν = 0, from Ek − Ek = 0, also generally figures. For in this case also one proper value, namely, Ek , appears on the left side as proper value parameter. Thus, if the frequency 0, i.e. a term independent of the time, occurs in the Fourier expansion of the perturbation function r(x, t), we cannot reach our goal by exactly the earlier method. We easily see, however, how it must be modified, for the case of a time-constant perturbation is known from previous work (cf. Part III.). We have then to consider, at the same time, a small alteration and possibly a splitting up of the proper value or values of the excited free vibrations, i.e. in the indices of the powers of e in the first term on the right hand of equation (10) we have to replace Ek by Ek plus a small constant, the perturbation of the proper value. Exactly as described in Part III., § 1 and § 2, this perturbation is defined by the postulation that the right side of the critical Fourier component of our equation (13) is to be orthogonal to uk (or possibly to all the proper functions belonging to Ek).
The number of special problems, which fall under the question formulated in the present paragraph, is extraordinarily great. By superposing the perturbations due to a constant electric or magnetic field and a light wave, we obtain magnetic and electric double refraction, and magnetic rotation of the plane of polarisation. Resonance radiation in a magnetic field also comes under this heading, but for this purpose we must first obtain an exact solution for the resonance case discussed in § 4. Further, we can treat the action of an α-particle or electron flying past the atomgd in this way, if the encounter is not too close for the perturbation of each of the two systems to be calculable from the undisturbed motion of the other. All these questions are mere matters of calculation as soon as the proper values and functions of the unperturbed systems are known. It is, therefore, to be hoped that we will succeed in defining these functions, at least approximately, for heavier atoms also, in analogy with the approximate definition of the Bohr electronic orbits which belong to different types of terms.
§ 6. RELATIVISTIC-MAGNETIC GENERALISATION OF THE FUNDAMENTAL EQUATIONS
As an appendix to the physical problems just mentioned, in which the magnetic field, which has hitherto been completely ignored in this series of papers, plays an important part, I would like to give, briefly, the probable relativistic-magnetic generalisation of the basic equations (4”), although I can only do this meantime for the one electron problem, and only with the greatest possible reserve—the latter for two reasons. Firstly, the generalisation is provisionally based on a purely formal analogy. Secondly, as was mentioned in Part I., though it does formally lead in the Kepler problem to Sommerfeld’s fine-structure formula with, in fact, the “half-integral” azimuthal and radial quantum, which is generally regarded as correct to-day, nevertheless there is still lacking the supplement, which is necessary to secure numerically correct diagrams of the splitting up of the hydrogen lines, and which is given in Bohr’s theory by Goudsmit and Uhlenbeck’s electronic spin.
The Hamilton-Jacobi partial differential equation for the Lorentzian electron can readily be written:
(34)
Here e , m, c are the charge and mass of the electron, and the velocity of light; V, U are the electro-magnetic potentials of the external electro-magnetic field at the position of the electron, and W is the action function.
From the classical (relativistic) equation (34) I am now attempting to derive the wave equation for the electron, by the following purely formal procedure, which, we can verify easily, will lead to equations (4”), if it is applied to the Hamiltonian equation of a particle moving in an arbitrary field of force in ordinary (non-relativistic) mechanics. After the squaring, in equation (34), I replace the quantities
(35)
The double linear operator, so obtained, is applied to a wave function ψ and the result put equal to zero, thus:
(36)
(The symbols ∇ 2 and grad have here their elementary three-dimensional Euclidean meaning.) The pair of equations (36) would be the possible relativistic-magnetic generalisation of (4”) for the case of a single electron, and should likewise be understood to mean that the complex wave function has to satisfy either the one or the other equation.
From (36) the fine structure formula of Sommerfeld for the hydrogen atom may be obtained by exactly the same method as is described in Part I., and also we may derive (neglecting the term in U2) the normal Zeeman effect as well as the well-known selection and polarisation rules and intensity formulae. They follow from the integral relations between Legendre functions introduced at the end of Part III.
For the reasons given in the first section of this paragraph, I withhold the detailed reproduction of these calculations meantime, and also in the following final paragraph refer to the “classical”, and not to the still incomplete relativistic-magnetic version of the theory.
§ 7. ON THE PHYSICAL SIGNIFICANCE OF THE FIELD SCALAR
The heuristic hypothesis of the electro-dynamical meaning of the field scalar ψ , previously employed in the one-electron problem, was extended off-hand to an arbitrary system of charged particles in §2, and there a more exhaustive description of the procedure was promised. We had calculated the density of electricity at an arbitrary point in space as follows. We selected one particle, kept the trio of co-ordinates that describes its position in ordinary mechanics fixed; integrated ψ over all the rest of the co-ordinates of the system and multiplied the result by a certain constant, the “charge” of the selected particle ; we did a similar thing for each particle (trio of co-ordinates), in each case giving the selected particle the same position, namely, the position of that point of space at which we desired to know the electric density. The latter is equal to the algebraic sum of the partial results.
This rule is now equivalent to the following conception, which allows the true meaning of ψ to stand out more clearly. ψψ is a kind of weight-function in the system’s configuration space. The wave-mechanical configuration of the system is a superposition of many, strictly speaking of all, point-mechanical configurations kinematically possible. Thus, each point-mechanical configuration contributes to the true wave-mechanical configuration with a certain weight, which is given precisely by ψψ. If we like paradoxes, we may say that the system exists, as it were, simultaneously in all the positions kinematically imaginable, but not “equally strongly” in all. In macroscopic motions, the weight-function is practically concentrated i
n a small region of positions, which are practically indistinguishable. The centre of gravity of this region in configuration space travels over distances which are macroscopically perceptible. In problems of microscopic motions, we are in any case interested also, and in certain cases even mainly, in the varying distribution over the region.
This new interpretation may shock us at first glance, since we have often previously spoken in such an intuitive concrete way of the “ψ - vibrations” as though of something quite real. But there is something tangibly real behind the present conception also, namely, the very real electrodynamically effective fluctuations of the electric spacedensity. The ψ -function is to do no more and no less than permit of the totality of these fluctuations being mastered and surveyed mathematically by a single partial differential equation. We have repeatedly called attentionge to the fact that the ψ -function itself cannot and may not be interpreted directly in terms of three-dimensional space—however much the one-electron problem tends to mislead us on this point—because it is in general a function in configuration space, not real space.
Concerning such a weight-function in the above sense, we would wish its integral over the whole configuration space to remain constantly normalised to the same unchanging value, preferably to unity. We can easily verify that this is necessary if the total charge of the system is to remain constant on the above definitions. Even for non-conservative systems, this condition must obviously be postulated. For, naturally, the charge of a system is not to be altered when, e.g., a light wave falls on it, continues for a certain length of time, and then ceases. (N.B.—This is also valid for ionisation processes. A disrupted particle is still to be included in the system, until the separation is also logically—by decomposition of configuration space—completed.)
The question now arises as to whether the postulated persistence of normalisation is actually guaranteed by equations (4”), to which ψ is subject. If this were not the case, our whole conception would practically break down. Fortunately, it is the case. Let us form
(37)
Now, ψ satisfies one of the two equations (4”), and ψ the other. Therefore, apart from a multiplicative constant, this integral becomes
(38)
where for the moment we putψ = R + iJ .
According to Green’s theorem, integral (38) vanishes identically; the sole necessary condition that functions R and J must satisfy for this—vanishing in sufficient degree at infinity—means physically nothing more than that the system under consideration should practically be confined to a finite region.
We can put this in a somewhat different way, by not immediately integrating over the whole configuration space, but by merely changing the time-derivative of the weight-function into a divergence by Green’s transformation. Through this we get an insight into the question of the flow of the weight-function, and thus of electricity. The two equations
(4”)
are multiplied by ρψ and ρψ respectively, and added. Hence
(39)
To carry out in extenso the transformation of the right-hand side, we must remember the explicit form of our many-dimensional, non-Euclidean, Laplacian operator:gf
(40)
By a small transformation we readily obtain
(41)
The right-hand side appears as the divergence of a many-dimensional real vector, which is evidently to be interpreted as the current density of the weight-function in configuration space. Equation (41) is the continuity equation of the weight-function.
From it we can obtain the equation of continuity of electricity, and, indeed, a separate equation of this sort is valid for the charge density “originating from each separate particle”. Let us fix on the αth particle, say. Let its “charge” be eα, its mass mα, and let its co-ordinate space be described by Cartesians xα , yα , zα, for the sake of simplicity. We denote the product of the differentials of the remaining co-ordinates shortly by dxʹ. Over the latter, we integrate equation (41), keeping xα, yα, zα, fixed. As the result, all terms except three disappear from the right-hand side, and we obtain
(42)
In this equation, div and grad have the usual three-dimensional Euclidean meaning, and xα , yα , zα are to be interpreted as Cartesian co-ordinates of real space. The equation is the continuity equation of that charge density which “originates from the αth particle”. If we form all the others in an analogous fashion, and add them together, we obtain the total equation of continuity. Of course, we must emphasize that the interpretation of the integrals on the right-hand side as components of the current density, is, as in all such, cases, not absolutely compulsory, because a divergence-free vector could be added thereto.
To give an example, in the conservative one-electron problem, if ψ is given by
(43)
we get for the current density J
(44)
We see, and this is valid for conservative systems generally, that, if only a single proper vibration is excited, the current components disappear and the distribution of electricity is constant in time. The latter is also immediately evident from the fact that ψ becomes constant with respect to the time. This is still the case even when several proper vibrations are excited, if they all belong to the same proper value. On the other hand, the current density then no longer needs to vanish, but there may be present, and generally is, a stationary current distribution. Since the one or the other occurs in the unperturbed normal state at any rate, we may in a certain sense speak of a return to electrostatic and magnetostatic atomic models. In this way the lack of radiation in the normal state would, indeed, find a startingly simple explanation.
I hope and believe that the present statements will prove useful in the elucidation of the magnetic properties of atoms and molecules, and further for explaining the flow of electricity in solid bodies.
Meantime, there is no doubt a certain crudeness in the use of a complex wave function. If it were unavoidable in principle, and not merely a facilitation of the calculation, this would mean that there are in principle two wave functions, which must be used together in order to obtain information on the state of the system. This somewhat unacceptable inference admits, I believe, of the very much more congenial interpretation that the state of the system is given by a real function and its time-derivative. Our inability to give more accurate information about this is intimately connected with the fact that, in the pair of equations (4”), we have before us only the substitute—extraordinarily convenient for the calculation, to be sure—for a real wave equation of probably the fourth order, which, however, I have not succeeded in forming for the non-conservative case.
Zürich, Physical Institute of the University.
(Received June 23, 1926.)
Chapter Four
Both Werner Heisenberg’s and Erwin Schrodinger’s formulations of quantum mechanics were non-relativistic, meaning that they did not include Einstein’s special theory of relativity. In order to describe very fast moving quantum particles, a relativistic quantum theory would need to be developed. In 1928, a brilliant paper by Paul Dirac entitled “The Quantum Theory of the Electron” presented a relativistic quantum theory and a relativistic replacement to the Schrodinger Equation which is now known as the Dirac Equation. Remarkably, Dirac’s theory required the existence of anti-particles even though no such thing had yet been supposed. Thus he was able to predict the existence of the positron (which is the electron’s anti-particle) before it had even been experimentally detected! Equally astounding, Dirac was also able to show that by including relativity in quantum mechanics he could explain the intrinsic angular momentum or “spin” of electrons which had been an unsolved problem since its discovery.
It had been found that spin was quantized in units of ½ ħ—that is, all particles have spin with an integer times ½ ħ (where ħ is Planck’s constant divided by 2π ). In his paper “The Connection Between Spin and Statistics,” Wolfgang Pauli further developed the theory of spin by showing that particles
with half-integer spin must obey Fermi-Dirac statistics. Those particles are now known as fermions. Pauli also showed that particles with integral spin obey Bose-Einstein statistics, so are known as bosons.
The Dreams That Stuff is Made of Page 36