Sewing sound quantum flesh onto classical bones

Semiclassical transformation theory implies an integral representation for stationary-state wave functions ψ m ( q ) in terms of angle-action variables ( θ, J ). It is a particular solution of Schr¨odinger’s time-independent equation when terms of order ¯ h 2 and higher are omitted, but the pre-exponential factor A ( q, θ ) 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, θ ) a factor which is a function of the action J written in terms of q and θ . A prescription for an improved choice of this factor, based on succesfully reproducing the leading behaviour 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 P¨oschl-Teller oscillator and the Morse oscillator).