New Method for Rapid Numerical Solution of the One-Dimensional Schrödinger Equation

One can generate certain special solutions to the one-dimensional Schrödinger equation, $\exp [\pm i{S}_{\pm }\mathrm{}(x\left)\right]$, such that $S_{\pm }\mathrm{}\left(x\right)\mathrm{}$ and derivatives are computationally unique, slowly varying, and do not have oscillatory or steplike behavior associated with the de Broglie wavelength. This leads to a method for accurate numerical solution which has all the advantages of high computational speed and conceptual simplicity of the JWKB approximation—of which it is the extension to an exact solution.