General Quantum Theory

The integer n2 cannot be zero, because this would correspond to a rectilinear motion through the nucleus. We have seen, how ever, that must change by + or -- 1 and in no other way ac cording to the correspondence principle, and therefore the num ber of possible transitions is reduced to three. H« then consists of three lines all close together. Two of the three are so very close to one another that the system is practically a doublet. The separation is approximately that of the two terms corresponding to n= 2 namely How small this is can be judged by comparison with the separation of the D lines of sodium, which is It is easy to see that the correspondence principle requires that each line of Lyman's series is associated with only one kind of transition, and is therefore single.

This theory of the fine structure of spectral lines is in excellent accord with the observations made before and soon of ter its development ; but the improved photometric methods developed in quite recent times have given the problem quite a different character—without impugning the accuracy of the relativistic formulae in Sommerfeld's theory. There is no doubt that the spectrum of hydrogen must be classed with the spectra of the alkali metals, lithium, sodium, etc., and in the present state of the theory for example, has seven components, two pairs of which, however, coincide, so that there are five distinct com ponents instead of the three of the older theory.

Stark Effect.

In the year 1913 Johannes Stark discovered that each of the Balmer lines of hydrogen, which, for the pur poses of this paragraph, may be considered to be single lines, since the effect we are about to study is of a much bigger scale of magnitude than that of the fine structure of the individual lines, splits up into a group of lines symmetrically situated with regard to the original line when the emitting atoms are subjected to a strong external electric field. The complete explanation of this phenomenon is one of the greatest successes of the quantum theory and was given independently by Epstein and Schwarz schild. The mathematical problem involved was actually solved by the German mathematician, Jacobi, in the earlier half of the last century. It is only possible to give the barest outline of it here. We may neglect the motion of the massive nucleus and suppose it provisionally to be at the origin of rectangular co ordinates. The applied electric field we shall take to be in the direction of the X-axis. If p represents the perpendicular dis tance of the electron from the X-axis, the proper coordinates to use in order that the variables may be separated in the sense explained above are n and 4), where and where 4 is the azimuthal angle about the X-axis. Epstein

and Schwarzschild showed that the energy of the system is given by where F is the intensity of the applied field, and the terms involving squares and higher powers of F have been ignored. The quantitizing formulae (4i) give us as the general expression for the energy of a stationary state, where n= nt-l-n,,d-no. The correspondence principle indicates that no changes by +1, — I, or o during a transition from one stationary state to another, and that when Ano= o the emitted radiation is polarized with its vibrations parallel to the direction of the field and in the other cases perpendicular to the field. Kramers has successfully applied the correspondence principle to explain the observed relative intensities of the components. Finally it may be added that Bohr has worked out the problem of the Stark phenomenon by the method of secular perturbations, much used by astronomers. Essentially this consists in making use of the fact that the motion of a planet round the sun or of an electron round the nucleus is to a first approximation elliptical, but we can get a more accurate result by studying the small per turbations of this elliptic motion by the external disturbing bodies or by the applied external field.

Zeeman Effect.

When a source of light such as a sodium flame or a discharge tube is placed in a sufficiently intense magnetic field, we find on observing the spectrum with apparatus of suit able resolving power that each original spectral line is replaced by a number of lines. (See ZEEMAN EFFECT.) The energy equa tion (4o) of a simple Bohr atom has to be modified in the follow ing way when the atom is subjected to a uniform magnetic field of intensity H:— where terms involving IP and higher powers are ignored. In this equation the definitions of the phase integrals J (37) have been extended by replacing the mechanical momentum p by a more general momentum which is the vector sum of p and the product —eA of the charge on the electron and the electromagnetic vector potential. This extension, which was suggested by W. Wilson in 1921, plays an important part in recent developments of the quantum theory. above is the phase integral associated with the azimuthal angle about the axis of the magnetic field. The application of the classical method (42) indicates that the motion of the electron in the presence of the magnetic field is that which would result if we were to superpose on its motion in the absence of a field a rotation or precession about the axis of the field equal to eH revolutions per second (Larmor's theorem).