Two Notes on Phase-Integral Methods

I. The connection formulas are derived by a new method. The general approach is that of Zwaan's discussion of Stokes' phenomenon, but no use is made of any assumptions about the reality of the coefficients of the differential equation. Instead of this, the proof is based only on the fact that actual solutions of the differential equation must be single-valued. Both this manner of proof and the form in which the results are obtained are suited to the discussion of certain problems in which a boundary condition consists in the requirement that the field at a great distance contain an out-going wave only. The results serve to establish the validity of Eckersley's phase-integral method for the treatment of problems of wave propagation. II. General arguments used to establish phase-integral methods are asymptotic in nature, and lead to the expectation that the methods will be valid only in a certain limit; in the language of quantum mechanics this is the limit of large quantum numbers. Much of the methods' usefulness, however, comes from the fact that they give in practice surprisingly accurate results even for small quantum numbers. In the case of the energy levels of the anharmonic oscillator, special arguments have been devised by Kemble and by Birkhoff to establish the usual phase-integral formula without assuming large quantum numbers. In this note a special argument is given for calculating normalization factors for the approximate wave functions of the oscillator. It is shown that the usual asymptotic formula for the normalization factor holds for the lowest quantum states, with about the accuracy with which the phase-integral solution approximates the shape of the exact wave function at the point at which the potential energy function has its minimum.