The Fine Structure of Hydrogen Lines

We would get a system wherein every orbit would be periodic independent of the initial conditions if:

• the relativity modifications were added
• the nucleus repelled the electron proportional to the inverse cube of the distance and equal and opposite to the attraction just mentioned
• small quantities of higher order than the square of the ratio between the velocity of the electron and that of light were neglected

Consequently, the stationary states would be fixed by the single condition I = nh.

The actual hydrogen atom may be considered as a perturbed system formed by this periodic system when it is exposed to a small central field for which the value of ψ is given by (70).

With the approximation mentioned, we get therefore for the total energy in the stationary states of the atom

is the energy in the stationary states of the periodic system just mentioned, and where the last term is obtained by introducing in (70) the value of α2 given by (64) and the value of ωn given by (41), neglecting the small correction due to the finite mass of the nucleus. Remembering that in our notation n1 + n2 = n and n2 = n, it will be seen that, as regards the small differences in the energy of the different stationary states corresponding to the same value of n, formula (71) gives the same result as Sommerfeld’s formula (68). In fact, comparing (68) and (71), we get

which is seen to be a function of n only. This expression might also have been deduced directly from the condition I = nh by considering, for instance, a circular orbit, in which case the calculation can be very simply performed.

The fixation of the stationary states, leading to the formulæ (68) or (71), is based on the assumptions:

• the electron’s motion is that of a mass point which moves in a conservative field of force according to the laws of ordinary relativistic mechanics
• we have looked apart from all such forces which, according to electrodynamic theory, would act on an accellerated charged particle, and which constitute the reaction from the radiation which on this theory would accompany the motion of the electron.

This procedure is a radical departure from the ordinary theory of electrodynamics. This is necessary in the quantum theory in order to avoid dissipation of energy in the stationary states.

We are entirely ignorant on the mechanism of radiation.

Yet the above treatment will let us determine the motion in the stationary states, only with an approximation which looks apart from small quantities of the same order as the ratio between the radiation forces in ordinary electrodynamics and the main forces on the electron due to the attraction from the nucleus.

1. Now it is1) Compare Part I, page 6. It may in this connection be noted that the degree of approximation, involved in the determination of the frequencies of an atomic system by means of relation (1) if in the fixation of the stationary states we look apart from small forces of the same order of magnitude as the radiation forces in ordinary electrodynamics, would appear to be intimately connected with the limit of sharpness of the spectral lines, which depends on the total number of waves contained in the radiation emitted during the transition between two stationary states. In fact, from a consideration based on the general connection between the quantum theory and the ordinary theory of radiation, it easily shown that this ratio will be a small quantity of the same order of magnitude as

and it would therefore beforehand seem justified in the expression for the total energy in the stationary states to retain small terms of the same order as the second term in (71), while at the same time it might appear highly questionable, whether, in the complete expression for the total energy in the stationary states deduced by Sommerfeld and Debye on the basis of the conditions (16), it has a physical meaning to etain terms of higher order than those retained in formula (68);

Unless N is a large number, as in the theory of the R¨ontgenspectra to be discussed in Part III.

While the preceding considerations, which deal with the determination of the energy in the stationary states of the hydrogen atom, allow to determine the frequency of the radiation which would be emitted during a transition between two such states, they leave quite untouched the problem of the actual occurrence of these transitions in the luminous gas, and therefore give no direct information about the numseems natural to assume that the rate, at which radiation is emitted during a transition between two stationary states, is of the same order of magnitude as the rate, at which radiation would be emitted from the system in these states according to ordinary electrodynamics.

But this will be seen to imply that the total number of waves in question will just be of the same order as the ratio between the main forces acting on the particles of the system and the reaction from the radiation in ordinary electrodynamics.

ber and relative intensities of the components into which the hydrogen lines may be expected to split up as a consequence of the relativity modifications. This problem has recently been discussed by Sommerfeld,

1. who in this connection emphasises the importance of the different a-priori probabilities of the stationary states, characterised by different sets of values of the n’s in the conditions (16). Thus Sommerfeld attempts to obtain a measure for the relative intensities of the components of the fine structure of a given line, by comparing the intensities observed with the products of the values of the a-priori probabilities of the two states, involved in the emission of the components under consideration; and he tries in this connection to test different expressions for these a-priori probabilities (See Part I, page 47).

In this way, however, it was not found possible to account in a satisfactory manner for the observations; and the difficulty in obtaining an explanation of the intensities on this basis was also strikingly brought out by the fact, that the number and relative intensities of the components observed varied in a remarkable way with the experimental conditions under which the lines were excited. Thus Paschen found a greater number of components in the fine structure of the helium lines, mentioned above, when the gas was subject to a condensed interrupted discharge, than when a continuous voltage was applied. It would seem, however, that all the facts observed obtain a simple interpretation on the basis of the general con1 ) A. Sommerfeld, Ber. Akad. M¨unchen, 1917, p. 83. 133 siderations about the relation between the quantum theory of line spectra and the ordinary theory of radiation discussed in Part I. According to this relation, we shall assume that the probability, for a transition between two given stationary states to take place, will depend not only on the a-priori probability of these states, which is determining for their occurrence in a distribution of statistical equilibrium, but will also depend essentially on the motion of the particles in these states, characterised by the harmonic vibrations in which this motion can be resolved. Now, in the absence of external forces, the motion of the electron in the hydrogen atom forms a special simple case of the motion of a conditionally periodic system possessing an axis of symmetry, and may therefore be represented by trigonometric series of the type deduced for such motions in Part I. Taking a line through the nucleus perpendicular to the plane of the orbit as z-axis, we get from the calculations on page 59 z = const. and x = XCτ cos 2π  (τω1 + ω2)t + cτ

, ±y = XCτ sin 2π  (τω1 + ω2)t + cτ

, (73) where ω1 is the frequency of the radial motion and ω2 is the mean frequency of revolution, and where the summation is to be extended over all positive and negative entire values of τ . It will thus be seen that the motion may be considered as a 134 superposition of a number of circular harmonic vibrations, for which the direction of rotation is the same as, or the opposite of, that of the revolution of the electron round the nucleus, according as the expression τω1 + ω2 is positive or negative respectively. From the relation just mentioned between the quantum theory of line spectra and the ordinary theory of radiation, we shall therefore in the present case expect that, if the atom is not disturbed by external forces, only such transitions between stationary states will be possible, in which the plane of the orbit remains unaltered, and in which the number n2 in the conditions (16) decreases or increases by one unit; i. e. where the angular momentum of the electron round the nucleus decreases or increases by h/2π. From the relation under consideration, we shall further expect that there will be an intimate connection between the probability of a spontaneous transition of this type between two stationary states, for which n1 is equal to n 0 1 and n 00 1 respectively, and the intensity of the radiation of frequency (n 0 1 − n 00 1 )ω1 ± ω2, which on ordinary electrodynamics would be emitted by the atom in these states, and which would depend on the value Cτ of the amplitude of the harmonic rotation, corresponding to τ = ±(n 0 1 −n 00 1 ), which appears in the motion of the electron. Without entering upon a closer examination of the numerical values of these amplitudes, it will directly be seen that the amplitudes of the harmonic rotations, which have the same direction as the revolution of the electron, in general, are considerably larger than the amplitudes of the rotations in the opposite direction, and 135 we shall accordingly expect that the probability of spontaneous transition will in general be much larger for transitions, in which the angular momentum decreases, than for transitions in which it increases. This expectation is verified by Paschen’s observations of the fine structure of the helium lines, which show that, for a given line, the components corresponding to the transitions of the former kind are by far the strongest. On Paschen’s photographs, however, especially in the case of the application of a condensed discharge to the vacuum tube containing the gas, there appear, in addition to the main components corresponding to transitions for which the angular momentum changes by h/2π, a number of weaker components, corresponding to transitions for which the angular momentum remains unchanged or changes by higher multipla of h/2π. This fact obtains a simple interpretation on the considerations in Part I on page 64 about the influence of small external forces on the spectrum of a conditionally periodic system. Thus, in the presence of small perturbing forces, the motion will generally not remain in a plane, and in the trigonometric series representing the displacement of the electron in space, there will occur small terms corresponding to frequencies (τ1ω1+τ2ω2), where τ2 may be different from one. In the presence of such forces, we shall therefore expect that, in addition to the regular probabilities of the above mentioned main transitions, there will appear small probabilities for other transitions.1 ) 1 ) Note added during the proof. As remarked in Part I, this con- 136 A detailed discussion of these problems will be given in a later paper by Mr. H. A. Kramers, who on my proposal has kindly undertaken to examine the resolution of the motion of the electron in its constituent harmonic vibrations more closely, and who has deduced explicit expressions for the amplitudes of these vibrations, not only for the motion of the electron in the undisturbed atom, but also for the perturbed motion in the presence of a small external homogeneous electric field. As it will be shown by Kramers, these calculations allow to account in particulars for the observations of the relative intensities of the components of the fine structure of the hydrogen lines and the analogous helium lines, as well as for the characteristic way in which this phenomenon is influenced by the variation of the experimental conditions.