Matrix and Grid-Based Methods for Quantum Bound States

We investigate matrix and grid-based methods for finding the bound states of multidimensional quantum systems. In the matrix method, we expand the wave functions in terms of products of sine-wave basis functions, then diagonalize the resulting Hamiltonian matrix. In the grid-based method, we solve a discretized version of the Schr\"{o}dinger equation using a relaxation algorithm based on the variational principle. We find that the matrix method is preferred if high accuracy is needed, while the grid method is easier to code. Although these methods can be used for arbitrary trapping potentials, we focus on a system consisting of a particle in a two-dimensional double triangular well. This potential models a pair of vertically coupled quantum dots embedded within an AlGaAs barrier material. Quantum dots are of current research interest because their optoelectronic properties can be precisely tuned through the modification of their composition and geometry.