Gauge invariant grid discretization of Schr\"odinger equation

TL;DR

This paper introduces a gauge invariant grid discretization method for the Schrödinger equation using Wilson formulation, improving the accuracy of energy spectra and probability densities in magnetic fields compared to naive discretizations.

Contribution

A novel gauge invariant discretization of the Schrödinger equation using Wilson formulation, ensuring accurate physical results in electromagnetic fields.

Findings

Naive discretization breaks gauge invariance and causes large errors.

Proposed method accurately computes energy spectra and probability densities.

Reliable estimation of physical quantities in magnetic fields.

Abstract

Using the Wilson formulation of lattice gauge theories, a gauge invariant grid discretization of a one-particle Hamiltonian in the presence of an external electromagnetic field is proposed. This Hamiltonian is compared both with that obtained by a straightforward discretization of the continuous Hamiltonian by means of balanced difference methods, and with a tight-binding Hamiltonian. The proposed Hamiltonian and the balanced difference one are used to compute the energy spectrum of a charged particle in a two-dimensional parabolic potential in the presence of a perpendicular, constant magnetic field. With this example we point out how a "naive" discretization gives rise to an explicit breaking of the gauge invariance and to large errors in the computed eigenvalues and corresponding probability densities; in particular, the error on the eigenfunctions may lead to very poor estimates of the mean values of some relevant physical quantities on the corresponding states. On the contrary, the proposed discretized Hamiltonian allows a reliable computation of both the energy spectrum and the probability densities.