Selection Rules and Intensities for the Hydrogen Atom

The selection rule for l, discussed in Section 40d, allows only transitions with Al = +1 for the hydrogen atom. The lines of the Lyman series, with lower state that with n = 1 and l — 0, are in consequence due to transitions from upper states with 1 = 1. The radial electric dipole moment integral 

1 For a more complete discussion of this subject the reader is referred to the papers of Condon and to the discussions in Condon and Morse, Quantum Mechanics, Chap. V, and Ruark and Urey, Atoms, Molecules and Quanta, Chap. XII. 

This has a non-vanishing value for all values of n greater than 1. Hence there is no selection rule for n for the Lyman series, all transitions being allowed. It is similarly found that there is no selection rule for n for spectral series in general. 

For the Balmer series, with lower state that with n = 2 and l = 0 or 1, the selection rule for l permits the transitions 0 — 1, 1 — 0, and 2 — 1. The total intensity corresponding to these transitions from the level n = n to n = 2 is, except for a constant factor, 

The operation of the selection rule for l for hydrogen and hydrogenlike ions can be seen by the study of the fine structure of the lines. The phenomena are complicated, however, by the influence of electron spin.1 In alkali atoms the levels with given n and varying l are widely separated, and the selection rule for l plays an important part in determining the nature of their spectra. Theoretical calculations have also been made of the intensities of lines in these spectra with the use of wave functions such as those described in Chapter IX, leading to results in approximate agreement with experiment. 

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