Abstract

In quantum mechanics, one of the most studied problems is that of solving the Schrödinger equation to find its discrete spectrum. This problem cannot always be solved in an exact form, and so comes the need of approximations. This thesis is based on the theory of the Schrödinger operators and Sturm-Liouville problems. We use the Rayleigh-Ritz variational method (mix-max theory) to find eigenvalues for these operators. The variational analysis we present in this thesis relies on the sine-basis, which we obtain from the solutions of the particle-in-a-box problem. Using this basis we approximate the eigenvalues of a variety of potentials using computational implementations. The potentials studied here include problems such as the harmonic oscillator in d dimensions, the quartic anharmonic oscillator, the hydrogen atom, a confined hydrogenic system, and a highly singular potential. When possible the results are compared either with those obtained in exact form or results from the literature