Matrix algorithm for solving Schroedinger equations with position-dependent mass or complex optical potentials

TL;DR

This paper introduces a matrix-based numerical method to accurately solve low-dimensional quantum Hamiltonians, including those with position-dependent mass and non-Hermitian properties, demonstrated through molecular and physical models.

Contribution

The method extends existing techniques to handle non-Hermitian and PT-symmetric Hamiltonians, and effectively models systems with position-dependent mass in quantum physics.

Findings

Accurately reproduces low-lying bound state energies and wave functions.

Applicable to non-Hermitian and PT-symmetric Hamiltonians.

Demonstrated on molecular and physical models with good agreement to analytical results.

Abstract

We represent low dimensional quantum mechanical Hamiltonians by moderately sized finite matrices that reproduce the lowest O(10) boundstate energies and wave functions to machine precision. The method extends also to Hamiltonians that are neither Hermitian nor PT symmetric and thus allows to investigate whether or not the spectra in such cases are still real. Furthermore, the approach is especially useful for problems in which a position-dependent mass is adopted, for example in effective-mass models in solid-state physics or in the approximate treatment of coupled nuclear motion in molecular physics or quantum chemistry. The performance of the algorithm is demonstrated by considering the inversion motion of different isotopes of ammonia molecules within a position-dependent-mass model and some other examples of one- and two-dimensional Hamiltonians that allow for the comparison to analytical or numerical results in the literature.