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.