The JWKB Method in Central-Field Problems. Planar Radial Wave Equation and Resolution of Kramers' Dilemma

Abstract

It is well-known that applications of the JWKB method to central-field problems in three dimensions require half-integral quantization of the angular momentum for their success. Thus, the square of the angular momentum must be represented by the term (/ + ½) 2 h 2 rather than /(/ + 1)h 2 . This was first shown by Kramers in 1926 and has subsequently been discussed by several authors including, in particular, Langer (1937). While Kramers based his discussion on the ordinary radial variable r, Langer switched to the variable x defined by r = e x . In this new representation of the central-field problem, the expression (/ + ½) 2 h 2 emerges naturally. The ad hoc character of the Langer transformation has, however, often been emphasized. In the present communication, we choose a different entry to the problem. We keep the variable r and focus on physically equivalent forms of the radial Schrodinger equation in this variable. This leads to a smoother emergence of the (/ + ½) 2 h 2 term. Our analysis is carried out for a general dimension D. For a given D, there are D physically equivalent radial equations, corresponding to the subdimensions d = 1, 2..., D. We show that it is only the d = 2 equation that can be satisfactorily treated by the JWKB approximation. In the past, the focus was always on the d = I equation, and this was the reason behind the problems encountered by Kramers and Langer. As to the d = 2 equation, we finally show that this equation also is the most convenient starting point for determining the exact solutions of a central-field problem for general values of D and angular-momentum quantum number L.