# Numerical solutions of the time-dependent Schrodinger equation in two dimensions

## TL;DR

This paper presents a numerical approach using an adapted Crank-Nicolson method combined with high-order finite differences to solve the two-dimensional time-dependent Schrödinger equation efficiently and accurately.

## Contribution

It introduces an adapted alternating-direction implicit method with high-order spatial discretization for improved numerical solutions of 2D Schrödinger equations.

## Key findings

- The method achieves high accuracy with controlled step sizes.
- It demonstrates efficiency in solving various 2D quantum systems.
- The approach allows systematic analysis of precision and computational cost.

## Abstract

The generalized Crank-Nicolson method is employed to obtain numerical solutions of the two-dimensional time-dependent Schrodinger equation. An adapted alternating-direction implicit method is used, along with a high-order finite difference scheme in space. Extra care has to be taken for the needed precision of the time development. The method permits a systematic study of the accuracy and efficiency in terms of powers of the spatial and temporal step sizes. To illustrate its utility the method is applied to several two-dimensional systems.