Rapid Numerical Solution of the One-Dimensional Schrödinger Equation

Abstract

It is shown that, contrary to common experience and opinion, the exact solutions to Schrödinger's equation in one dimension (or any similar ordinary linear second-order differential equation) can be numerically computed at a speed characterized by the variations of the potential function, i.e. at effectively the speed of solving Hamilton's equations. The method of solution depends upon the existence and calculation of certain "fundamental" and unique special solutions, closely associated with the JWKB series. These solutions have exponential form with exponents which are everywhere finite, nonoscillatory functions varying smoothly with the potential ("quantal action"); their existence does not seem to have previously been known. They are generated when a novel iterative technique is employed to solve certain Riccati equations. The expressions obtained appear to be asymptotic representations, but are numerically essentially convergent (error < 1 × 10−6) whenever the JWKB approximation is reasonably valid. Because the exponent function and its derivative ("quantal momentum") are smooth and slowly varying, this iteration can be used to generate initial values in asymptotic domains and rapid numerical integration of the Riccati equation for the quantal momentum then performed to compute the exact solutions everywhere. Because of the simple character of the solutions, a number of elegant and useful results in perfect analogy with the JWKB approximations for eigenvalue conditions (Bohr–Sommer-feid rule) and scattering phase shifts can be obtained. The method can be employed whenever the desired solution possesses more than two or three de Broglie oscillations over a region of significant variation of the potential. We have investigated the method for a number of representative potential functions; we find that it is always at least one order of magnitude faster than standard methods based on direct integration of the Schrödinger equation, and for those cases where it can be employed we believe it will render them obsolete.