Corrected bohr-sommerfeld quantum conditions for nonseparable systems

Abstract

For a separable or nonseparable system an approximate solution of the Schrödinger equation is constructed of the form Ae i h ̷ −1S . From the single-valuedness of the solution, assuming that A is single-valued, a condition on S is obtained from which follows A. Einstein's generalized form of the Bohr-Sommerfeld-Wilson quantum conditions. This derivation, essentially due to L. Brillouin, yields only integer quantum numbers. We extend the considerations to multiple valued functions A and to approximate solutions of the form σ A k exp (i h ̷ −1S k). In this way we deduce the corrected form of the quantum conditions with the appropriate integer, half-integer or other quantum number (generally a quarter integer). Our result yields a classical mechanical principle for determining the type of quantum number to be used in any particular instance. This fills a gap in the formulation of the “quantum theory”, since the only other method for deciding upon the type of quantum number—that of Kramers—applies only to separable systems, whereas the present result also applies to nonseparable systems. In addition to yielding this result, the approximate solution of the Schrödinger equation—which can be constructed by classical mechanics—may itself prove to be useful.