Phase-integral formulas for level densities, normalization factors, and quantal expectation values, not involving wave functions

Abstract

Relations, important in some physical applications of the phase-integral method developed by Fr\"oman and Fr\"oman, are derived. For the case of a single well potential these relations are then used for obtaining general phase-integral formulas for level densities, normalization factors, and quantal expectation values, under differing assumptions concerning the behavior of the potential. The derivation of those formulas is based on the phase-integral quantization conditions pertinent to the potentials considered. The wave functions themselves do not occur in the final simple formulas obtained, which can, furthermore, in general be expected to be more accurate than corresponding formulas involving wave functions. Some of the results have been obtained in earlier papers, but the present treatment covers more general situations. To illuminate the application of the phase-integral formula for expectation values, the radial Schr\"odinger equation is considered, and certain exact relations are discussed within the framework of the phase-integral formulas.