The optimal dimensions of the domain for solving the single-band Schrödinger equation by the finite-difference and finite-element methods

The finite-difference and finite-element methods are employed to solve the one-dimensional single-band Schr?dinger equation in the planar and cylindrical geometries. The analyzed geometries correspond to semiconductor quantum wells and cylindrical quantum wires. As a typical example, the GaAs/AlGaAs system is considered. The approximation of the lowest order is employed in the finite-difference method and linear shape functions are employed in the finite-element calculations. Deviations of the computed ground state electron energy in a rectangular quantum well of finite depth, and for the linear harmonic oscillator are determined as function of the grid size. For the planar geometry, the modified P?schl-Teller potential is also considered. Even for small grids, having more than 20 points, the finite-element method is found to offer better accuracy than the finite-difference method. Furthermore, the energy levels are found to converge faster towards the accurate value when the finite-element method is employed for calculation. The optimal dimensions of the domain employed for solving the Schr?dinger equation are determined as they vary with the grid size and the ground-state energy.