The Solutions of Two Dimensional Finite Square Well Potential Problem Using the Finite Difference Time Domain Method

Abstract

We calculate the numerical eigenfunctions and their corresponding energy eigenvalues of the higher excited states for two dimensional finite square well potential, by solving the Schrödinger equation using the finite difference time domain method (FDTD). The iterative procedure involved in this method was improved using symmetric arguments to calculate the lower angular excited states, and we extent this improved method to calculate any excited state directly using suitable initial guess wave function that is close to the desired excited state. This suitable initial guess wave function is calculated analytically using the separation of variables technique. In this paper, our calculations include two essential parts. First, in order to confirm the applicability of the separation of variables technique, we compare the lower states, namely, the ground state, the first angular excited state and the second angular excited state, were calculated by using this technique with their corresponding numerically exact states. Therefore, we can consider the solutions of the separation of variables technique as a semi-analytical approximation. Second, we take advantage of this approach to get any desired excited state directly if it exists.