On the Wentzel-Brillouin-Kramers Approximate Solution of the Wave Equation

A discussion of the Wentzel-Brillouin-Kramers method of obtaining an approximate solution of the wave equation is given from the point of view that it forms a link between the quantum theory of Bohr and the new quantum mechanics. This becomes clear when one compares the probability distributions (see Figs. 2 and 3) and calculates the mean values $\overline{r^k}$ (see Table I). It is shown that for high quantum numbers these become the classical values. The method leads also, as shown by Zwaan, to quantization by the classical phase integrals with the use of half-integer quantum numbers. In the last section the method is applied to the charged shell atom model. It is shown that the condition of smooth joining of the wave function is practically equivalent to half-integer quantization of the sum of the inner and outer phase integrals. Of course there is no longer a sharp distinction between penetrating and non-penetrating orbits. The Landé formula for the doublet separations is derived. The value of ${\psi }^2\mathrm{}\left(0\right)\mathrm{}$ (0), which occurs in the calculation of the hyperfine separations in ${}_{}{}^2{}_{}{}^{}S$ states, is also given.