# The variational-relaxation algorithm for finding quantum bound states

## TL;DR

This paper introduces a simple variational-relaxation algorithm for efficiently computing ground and excited states of 2D quantum systems with complex potentials, leveraging a relaxation method adapted from solving Poisson's equation.

## Contribution

It presents a novel, straightforward algorithm that extends relaxation techniques to quantum bound state problems, especially effective for nonseparable potentials in two dimensions.

## Key findings

- Efficiently finds quantum bound states in complex 2D systems.
- Applicable to nonseparable potentials where simpler methods fail.
- Minimal computation time for targeted states.

## Abstract

I describe a simple algorithm for numerically finding the ground state and low-lying excited states of a quantum system. The algorithm is an adaptation of the relaxation method for solving Poisson's equation, and is fundamentally based on the variational principle. It is especially useful for two-dimensional systems with nonseparable potentials, for which simpler techniques are inapplicable yet the computation time is minimal.