The restrictions of micro-scale systems are approaching rapidly. In anticipation of this development, nano-scale electronics has become the focus of many researchers and engineers in academia and industry since early 1990s. The basic building blocks of modern integrated circuits have been diodes and transistors with their current-voltage I-V characteristics being of prime significance for the design of complex signal processing and shaping devices and systems. Classical and semi-classical physical principles are no longer powerful enough or even valid to describe the phenomena involved. The application of rich and powerful concepts in quantum theory has become indispensable. These facts have been influential in undertaking this research project. This research is built upon the determination of the Eigenpairs of one and two dimensional positive differential operators with periodic boundary conditions. The Schrödinger equation was solved for positive operators in both one and two dimensions. Fourier series were used to express the derivatives as the summation of Fourier terms. This led to a novel approach for the calculation of the eigenmodels of a perturbed potential well. The perturbation can be done via an electric field applied to the potential well. The research in this thesis includes a thorough understanding of quantum mechanics fundamentals, mastering of different approximation techniques such as the variational technique and results that have been generated and published using the novel techniques.