Analytical solution for 3D stationary Schro¨ dinger equation: implementation of Huygens’ principle for matter waves

Abstract

We have found an analytical solution of the 3D stationary Schro¨ dinger equation for any arbitrary potential. The solution has been obtained by transforming the Schro¨ dinger equation to the generalized Ricatti equation, and it has the form of Huygens’ principle for matter waves. We have shown that the solution is more accurate than the JWKB approximation. We discuss possible applications of the solution to molecular dynamics and dynamics of charges in nanostructures.