Exact solution of Schroedinger equation in the case of reduction to Riccati type of ODE

TL;DR

This paper presents an exact solution to the 3+1 dimensional Schrödinger equation by reducing it to a Riccati ODE through separation of variables in spherical coordinates, revealing new solution types and approximations.

Contribution

The paper introduces a novel exact solution method for the Schrödinger equation by reducing it to a Riccati ODE using separation of variables in spherical coordinates.

Findings

Exact solutions in spherical coordinates with separated variables.

Existence of time-dependent 2-angles solutions aligned with wave propagation.

A paraxial approximation for small polar angles.

Abstract

A new type of solution for the full 3+1 dimensional space-time Schroedinger equation is presented here. We consider elegant presentation of the exact solution in a spherical coordinate system, along with the assuming of separation of the two angular co-ordinates from the radial and time variables. The separation of variables follows from an assumed product form of the full potential function, which should allow us to reduce the Schroedinger Eq. to Riccati ODE in stationary case. If the angular dependence is constant, then the azimuthal dependence is linear and the polar dependence is logarithmic. With this reduction the remaining partial differential equation links the radial and time dependence which is, in effect, the standard time-dependent spherical radial Schroedinger equation. Besides, we obtain that the time-depended 2-angles solutions exist if the axis of preferential direction of wave propagation is similar to the time arrow in mechanical processes. The other possibility is the appropriate phase-shifting of the time-depended solutions. A paraxial type approximation is obtained for the solution in the limit in which the polar angle approaches zero.