Corrected Bohr-Sommerfeld quantum conditions for non-separable systems

For a separable or nonseparable system an approximate solution of the Schrodinger equation is constructed of the form Ae”“-‘e. 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-Som-merfeld-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 c Ak exp (ih-%s) In this way we deduce the corrected form of the quantum conditions with the appropriate integer, half-integer or other quantum number (generally a quar-ter 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 nonsepa-rable systems. In addition to yielding this result, the approximate solution of the Schrii-dinger equation-which can be constructed by classical mechanics-may itself prove to be useful