Automatic computation of quantum-mechanical bound states and wavefunctions

TL;DR

This paper presents an automated method for solving the multichannel Schrödinger equation using a continuous product (CP) approach, which efficiently handles oscillations and refines eigenvalue determination through a new Pr"ufer-type mechanism.

Contribution

It introduces a novel CP-based approach and a Pr"ufer-type scheme for automatic, accurate computation of quantum bound states and wavefunctions, improving eigenvalue shooting.

Findings

Efficient handling of oscillatory solutions without step size restrictions.

Refined eigenvalue shooting process with user-specified eigenvalues.

Improved accuracy in computing quantum bound states and wavefunctions.

Abstract

We discuss the automatic solution of the multichannel Schr\"odinger equation. The proposed approach is based on the use of a CP method for which the step size is not restricted by the oscillations in the solution. Moreover, this CP method turns out to form a natural scheme for the integration of the Riccati differential equation which arises when introducing the (inverse) logarithmic derivative. A new Pr\"ufer type mechanism which derives all the required information from the propagation of the inverse of the log-derivative, is introduced. It improves and refines the eigenvalue shooting process and implies that the user may specify the required eigenvalue by its index.