β Scribed by Joseph B. Keller
No coin nor oath required. For personal study only.
For a separable or nonseparable system an approximate solution of the Schrodinger equation is constructed of the form Ae""-'e.
From the singlevaluedness 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 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 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 Schriidinger equation-which can be constructed by classical mechanics-may itself prove to be useful.
π SIMILAR VOLUMES
We present a detailed study, in the semiclassical regime h β 0, of microlocal properties of systems of two commuting h-pseudodifferential operators P 1 (h) and P 2 (h) such that the joint principal symbol p = (p 1 , p 2 ) has a special kind of singularity called a focus-focus singularity. Typical ex