Numerical Solutions of Finite Well in Two Dimensions Using the Finite Difference Time Domain Method

The higher excited states for two dimensional finite rectangular well potential are calculated numerically, by solving the Schrödinger equation using the finite difference time domain method. Although, this method is suitable to calculate the ground state of the quantum systems, it has been improved to calculate the higher excited states directly. The improvement is based on modifying the iterative process involved in this method to include two procedures. The first is known as cooling steps and the second is known as a heating step. By determining the required length of the cooling iteration steps using suitable excitation energy estimate, and repeating these two procedures using suitable initial guess function for sufficient times. This modified iteration will lead automatically to the desired excited state. In the two dimensional finite rectangular well potential problem both of the suitable excitation energy and the suitable initial guess wave function are calculated analytically using the separation of variables technique.