Numerical solution of the Schrödinger equation in polar coordinates using the finite-difference time-domain method

Abstract

In quantum mechanics, many concepts, equations, and interactions are expressed as functions of the radius and angles and are therefore best understood and handled directly in polar or spherical coordinates. A finite-difference time-domain (FDTD) method for solving the two-dimensional Schrodinger equation in polar coordinates is proposed herein. In this method and through a subgridding approach, new nodes are added on rings far from the origin to retain the precision of the mesh grids; then a trigonometric interpolation is used to calculate the derivatives at these nodes. A comparison with analytic solutions for a two-dimensional (2D) harmonic oscillator is carried out to verify the performance of the code. A simple method based on the spatial Fourier transform is presented for the separation of degenerate eigenstates. A 2D quantum dot is also simulated and analyzed. When using this polar FDTD method along with proposed subgridding approach, the resolution of the solutions and Hamiltonian terms are conserved in the whole space of a plane of polar coordinates, and all operators or results expressed in polar coordinates can be easily implemented or obtained.