Analytically Solvable 2D Potentials: Bound States

TL;DR

This paper develops an analytical method using Laguerre basis sets to solve for bound states in certain 2D potentials, providing exact solutions and validating them against numerical methods.

Contribution

It introduces a novel analytical approach for solving 2D potentials with a tridiagonal matrix representation, enabling exact bound state calculations.

Findings

Exact bound state spectra obtained analytically.

Results agree well with numerical Gauss quadrature.

Method simplifies solving specific 2D quantum systems.

Abstract

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix representation of the associated Hamiltonian for few exactly solvable 2D potentials. This enabled us to treat analytically the full Hamiltonian and compute the associated bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we compared our results satisfactorily with those obtained using the Gauss quadrature numerical integration approach.