The eigenvalue problems in connection with the self-adjoint second order differential equation in Sturm-Liouville form, which occur frequently in many branches of applied mathematics have been reviewed. The classical Rayleigh-Ritz method has been applied to solve approximately the boundary value problems of this kind. In particular, the one-dimensional Schrödinger equation encountered in quantum mechanics has been dealt with for several forms of the potential functions. A scaled Hermite-Weber basis is used in the Rayleigh-Ritz method, and very general recursive relations have been derived in the construction of the variational matrix. Numerical applications have been realised for polynomial and non-polynomial potentials having both single and multi-minima. Numerical experiments show that there exists an optimum value of the scaling parameter in the structure of the basis functions, for all eigenvalue problems considered in this study. Highly accurate numerical results have been presented for the low-lying state eigenvalues of the Schrödinger equation. Furthermore, it is shown also that the Hermite-Weber basis yields extremely accurate results not only for the nearly harmonic potentials but also the purely anharmonic problems. The last chapter of the thesis contains a fairly detailed discussion of the results and the concluding remarks, as usual.

Keywords: Sturm-Liouville Problem, Schrödinger Equation, Rayleigh-Ritz Method, Hermite-Weber Basis, Quantum Mechanical Potentials Having Single and  Multi-minima.