Uniform approximation of wave functions with improved semiclassical transformation amplitudes and Gram-Schmidt orthogonalization

Semiclassical transformation theory implies an integral representation for stationary-state wave functions psi(m)(q) in terms of angle-action variables (theta,J). It is a particular solution of Schrodinger's time-independent equation when terms of order h(2) and higher are omitted, but the preexponential factor A(q,theta) in the integrand of this integral representation does not possess the correct dependence on q. The origin of the problem is identified: the standard unitarity condition invoked in semiclassical transformation theory does not fix adequately in A(q,theta) a factor which is a function of the action J written in terms of q and theta. A prescription for an improved choice of this factor, based on successfully reproducing the leading behavior of wave functions in the vicinity of potential minima, is outlined. Exact evaluation of the modified integral representation via the residue theorem is possible. It yields wave functions which are not, in general, orthogonal. However, closed-form results obtained after Gram-Schmidt orthogonalization bear a striking resemblance to the exact analytical expressions for the stationary-state wave functions of the various potential models considered (namely, a Poschl-Teller oscillator and the Morse oscillator).