The Introduction of more than one Quantum Number.
During the first years after 1913, Bohr was practically alone in working out his theory, at that time still assailed by many, and in showing its application to many problems. In 1916, however, the theorists in other countries, led by the well-known Munich professor, Sommerfeld, began to associate themselves with the Bohr theory, and their investigations gave rise to much essential progress. We shall here mention some of the most important contributions.
In the theory for the hydrogen spectrum propounded above, it was assumed that we had to do with a single series of stationary orbits, each characterized by its quantum number. But as shown by theoretical investigations each of the stationary orbits must, when more detail is asked for, also be indicated by an additional quantum number.
Fig. 26.—A compound electron motion produced
by the very rapid rotation of an elliptical orbit.
This is closely connected with the fact that the motion of the electron is not quite so simple as previously assumed. We have assumed that the electron moves about the nucleus just as a planet moves about the sun (according to Kepler’s Laws), in an ellipse with the sun at one focus, since the electron is influenced by an attraction inversely proportional to the square of the distance, just as the planets are attracted by the sun according to Newton’s Law. We must, however, remember that we are here concerned with the electric attraction which at a given distance is determined, not by mass, but by the electric charges in question. If the latter remain unchanged, while the mass of the electron varies, the motion will be changed, because the same force has less effect upon a greater mass. According to the Einstein principle of relativity, the mass of the electrons, in accordance with ideas expounded long ago by J. J. Thomson, will not be constant, but to a certain extent depend upon the velocity, which will vary from place to place, when the orbit is an ellipse. As a result of this, the motion becomes a central motion of more general nature than a Kepler ellipse. Since the influence of the change of mass is very small, the orbit can still be considered as an approximate Kepler ellipse; but the major axis will slowly rotate in the plane of the orbit. In reality, the orbit will not therefore be closed, but will have the character which is shown in [Fig. 26]; this, however, corresponds to a much more rapid rotation of the major axis than that which actually takes place in the hydrogen atom, where—even in case of the swiftest rotation—the electron will revolve about 40,000 times round the nucleus at the same time as the major axis turns round once.
If the electron moves in a fixed Kepler ellipse, the energy content of the atom will be determined by the major axis of the ellipse only. If these axes for the stationary states with quantum numbers 1, 2, 3 ... are respectively denoted by 2a₁, 2a₂, 2a₃ ... the frequency, for instance, in the transition from No. 3 to No. 2—since it is determined by the loss of energy—will be the same whether the orbits are circles or ellipses. If, on the other hand, the electron moves in an ellipse which itself rotates slowly, the energy content, as can be shown mathematically, will depend not only upon the major axis of the ellipse, but also to a slight degree upon its eccentricity, or, in other words, on its minor axis. Then in the transition 3-2 we shall get different energy losses and consequently different frequencies, according as the ellipse is more or less elongated. If it were the case that the eccentricity of the ellipses for a certain quantum number could take arbitrary values, then in the transition between two numbers we could get frequencies which may take any value within a certain small interval, i.e., a mass of hydrogen with its great quantity of atoms would give diffuse spectral lines, i.e. lines which are broadened over a small continuous spectral interval. This is, however, not the case; but long before the appearance of the Bohr theory it had been discovered that the hydrogen lines, which we hitherto have considered as single, possess what is called a fine structure. With a spectroscopic apparatus of high resolving power each line can be separated into two lying very close to each other. This fine structure can now be explained by the fact that in a stationary state with quantum number 3 and major axis of the orbit 2a₃, for instance, the eccentricity of the orbit has neither one single definite value nor all possible values, but, on the contrary, it has several discrete values of definite magnitude, to which there correspond slightly different but definite values of the energy content of the atom. It is now possible to designate the series of stationary orbits, which have the major axis 2a₃ with the principal quantum number 3, with subscripts giving the auxiliary quantum number for stationary orbits corresponding to the different eccentricities, so that the series is known as 3₁, 3₂, 3₃. Instead of a single line corresponding to the transition 3-2, there are then obtained several spectral lines lying closely together and corresponding to transitions such as 3₃-2₂, 3₂-2₁, etc. By theoretical considerations, requiring considerable mathematical qualifications, but of essentially the same formal nature as those Bohr had originally applied to the determination of the stationary orbits in hydrogen, Sommerfeld was led to certain formal quantum rules which permit the fixing of the stationary states of the hydrogen atom corresponding to such a double set of quantum numbers. The results he obtained as regards the fine structure of the hydrogen lines agree with observation inside the limit of experimental error.
Although Sommerfeld’s methods have also been very fruitful when applied to the spectra of other elements, they were still of a purely formal and rather arbitrary nature; it is, therefore, of great importance that the Leiden professor, Ehrenfest, and Bohr succeeded later in handling the problem from a more fundamental point of view, Bohr making use of the correspondence principle previously mentioned. It should be said here, by way of suggestion, that Bohr used the fact that the motions of the electrons are not simple periodic but “multiple periodic.” We see this most simply if we think of the revolution of the electrons in the elliptical orbit as representing one period, and the rotation for the major axis of the ellipse as representing a second period.
Fig. 27.—The model of the hydrogen atom
with stationary orbits corresponding to principal
quantum numbers and auxiliary quantum numbers.
[Fig. 27] shows a number of the possible stationary orbits in the hydrogen atom according to Sommerfeld’s theory; for the sake of simplicity the orbits are drawn as completely closed ellipses. If we examine, for instance, the orbits with principal quantum number 4, we have here three more or less elongated ellipses, 4₁, 4₂, 4₃, and the circle 4₄; in all of them the major axis has the same length, and the length of the major axis is to that of the minor axis as the principal quantum number is to the auxiliary quantum number (for the circle 4: 4 = 1). On the whole, to a principal quantum number n there correspond the auxiliary quantum numbers 1, 2 ... n, and the orbit for which the auxiliary quantum number equals the principal quantum number is a circle. We see that in the more complicated hydrogen atom model there is possibility for a much greater number of different transitions than in the simple model ([Fig. 25, p. 119]). Some of the transitions are indicated by arrows. Since the energy content of the atom is almost the same for orbits with the same principal quantum number and different auxiliary quantum numbers, three transitions like 3₃-2₂, 3₂-2₁ and 3₁-2₂ will give about the same frequency, and therefore spectral lines which lie very close together. In a transition like 4₄-4₃ the emitted energy quantum hν, and also ν, will be so extremely small that the corresponding line will be too far out in the infra-red for any possibility of observing it.
It must be pointed out that the above considerations only hold if the hydrogen atoms, strictly speaking, are undisturbed. Thus, very small external forces, which may be due to the neighbourhood of other atoms, etc., will be sufficient to cause changes in the eccentricity of the stationary orbits. In such a case the above definition of the auxiliary quantum number becomes obviously illusory, and the original character of the fine structure disappears. This is in agreement with the experiments, since the Sommerfeld fine structure can be found only when the conditions in the discharge tube are especially quiet and favourable.