A Note on Shape Invariant Potentials for Discretized Hamiltonians

TL;DR

This paper introduces a discretized version of N=2 Supersymmetric Quantum Mechanics using exact discretization of the Schrödinger equation, enabling precise calculation of spectra and wavefunctions for discretized systems.

Contribution

It develops a novel discretization method for supersymmetric quantum systems that preserves shape invariance, allowing exact solutions similar to continuous models.

Findings

Derived energy spectrum for discretized Coulomb potential

Obtained ground state wavefunction for discretized Coulomb system

Validated the method's effectiveness for discretized quantum systems

Abstract

Using the method of the "exact discretization" of the Schr\"odinger equation, we propose a particular discretized version of the N=2 Supersymmetric Quantum Mechanics. After defining the corresponding shape invariance condition, we show that the energy spectra and wavefunctions for discretized Quantum Mechanical systems can be found using the technique of N=2 Supersymmetric Quantum Mechanics exactly the same way as it is done for their continuous counterparts. As a demonstration of the present method, we find the energy spectrum for a discretized Coulomb potential and its ground state wave function.