Semiclassical Expansion for the Angular Momentum

The semiclassical method of torus quantization is just the first term of a certain h̄-expansion, usually called the WKB expansion. The method goes back to the early days of quantum mechanics and was developed by Bohr and Sommerfeld for onefreedom systems and separable systems, it was then generalized for integrable (but not necessarily separable) systems by Einstein [1], which is called EBK or torus quantization. In fact, Einstein’s result was corrected for the phase changes due to caustics by Maslov [2,3], but the torus quantization formula thus obtained is still just a first term in the WKB expansion, whose higher terms can be calculated with a recursion formula in one degree systems, but are generally unknown in systems with more than one degree of freedom. Our goal in the present paper is to generalize the WKB expansion of the Schrödinger equation to the angular momentum operator. To the best of our knowledge a detailed analysis of this problem has not been undertaken in the literature so far. Thus our present work is the first systematic semiclassical expansion of the angular momentum problem. In section 2 we treat the one–dimensional stationary Schrödinger equation by analyzing the corrections to the leading torus quantization term, and in the section 3 we study the solutions of the angular momentum operator by calculating the corrections to the leading torus quantization term. In section 4 we discuss the results and draw some general conclusions.