The finite element method (FEM) is reviewed and applied to the one-dimensional eigensystems of the isotropic harmonic oscillator, finite well, infinite well and radial hydrogen atom, and the two-dimensional eigensystems of the isotropic harmonic oscillator and the propagational modes of sound in a rectangular cavity. Computer codes that I developed were introduced and utilized to find accurate results for the FEM eigensolutions. One of the computer codes was modified and applied to the one-dimensional unbound quantum mechanical system of a square barrier potential and also provided accurate results.

I: Quantum information obeys laws that subtly extend those governing classical information, making possible novel effect such as cryptography and quantum computation. Quantum computations are extremely sensitive to disruption by interaction of the computer with its environment, but this problem can be overcome by recently developed quantum versions of classical error-correcting codes and fault-tolerant circuits. Based on these ideas, the purpose of this paper is to provide an approach to quantum information by analyzing and demonstrating Bell's inequality and by discussing the problems related to decoherence and error-correcting. II: The growing need for a better understanding of complex processes has stimulated the development of new and more advanced data analysis techniques. The purpose of this research was to investigate some of the already existing techniques (Hurst's rescaled range and relative dispersion analysis), to develop a software able to process time series with these techniques, and to get familiar with the theory of diffusion processes.

For a spectroscopy purpose, we are interested in producing continuous wave (CW) UV laser light at 243 nm with at least 2 mW power. The theory of nonlinear optics suggests that we should be able to produce a desired 2.9 mW of 243 nm light by second harmonic generation (SHG) from a 50 mW blue laser at 486 nm using a BBO crystal in a build up cavity. The most important physical parameters are calculated. A 10 mm Brewster cut BBO crystal can provide phase matching conditions for coupling two ordinary photons at 486 nm and make a secondary beam at 243 nm. The single pass conversion efficiency is calculated not to be enough to generate 2.9 mW of SH light. My investigation shows that a standing wave build up cavity can provide a buildup factor of 94 and an overall conversion efficiency of 5.9% if one use an input coupler mirror with 1.1% transmission at 486 nm.