More on Numerical Solutions to Simple One-Dimensional Schrödinger Equations

John R. Merrill, Gregory P. Hughes

Wenhua Hai, Xiwen Zhu, Mang Feng +4 more

The most physically interesting systems are not exactly solvable in quantum mechanics. For one-dimensional bound systems without exact solutions, we analytically and numerically find that the Rayleigh-Schrödinger perturbed series sensitively depends on an unsolvable integration, which leads to numerical instability in quantum mechanics. By using an exact formal solution of the non-homogeneous Schrödinger equation, we demonstrate the existence of analytically bound states and propose a simple scheme to truncate infinity so that the instability difficulty is avoided.

Bekir Karaoglu

A remarkable property of a large class of functions is exploited to generate exact solutions to the one-dimensional Schrodinger equation. The method is simple and easy to implement.

Peter Senn

J. Ronald Farias

A simple exact solution of a one-dimensional Schr\"odinger equation which describes the motion of a particle in a continuous potential $V(x)$ and scattering from a finite set of $\ensuremath{\delta}$-function potentials of arbitrary positions and strengths is found.

Patrick J. Cooney, E. P. Kanter, Z. Vager