The Generalization of the WKB Method to Radial Wave Equations

Abstract

There are two difficulties which arise in connection with the usual generalizations of the WKB method to the Schrödinger and Dirac radial wave equations. One difficulty is that, in deriving the approximations, the Dirac radial equation, the Schrödinger radial equation, and the one-dimensional Schrödinger equation are treated nonuniformly so that the approximate solutions are not simply related to each other in the same way that the original equations are. The other difficulty is that the Dirac radial WKB approximation cannot be applied to a certain type of bound state. In this paper it is shown that these difficulties can be avoided by using as the basis for the WKB method, instead of the customary exponential function, the functions which arise in solving the free-particle problem. In the radial problems these are Bessel functions of order integer-plus-one-half. As well as avoiding the difficulties mentioned above, this technique has the following additional advantages. In the radial problems connection formulas across the turning points are not required. The eigenvalue condition for the bound states arises from a comparison of asymptotic expressions in such a way that one would expect the approximate eigenvalues to be close to the exact ones. For a free particle the approximate wave functions become identical with the exact wave functions, so the accuracy of the approximation increases with increasing energy. On the other hand, the approximations based on Bessel functions have the disadvantages that they are clumsier to apply than the usual approximations and that they give especially poor results near the turning points.