In this work, the consequences of several modifications to quantum mechanics are examined. These modifications, motivated by string theory, fall into two categories: ones in which the canonical commutation relations between position and momentum are deformed and ones in which the space of states used are vector spaces over Galois fields instead of complex Hilbert spaces. The particular deformation of the canonical commutation relations used leads to a minimum value of the uncertainty in position which is interpreted as a minimum length scale. Both harmonic and anharmonic oscillators are studied in this framework with normalizable, positive energy eigenstates found in both cases. The quantum uncertainty relations and classical counterparts to these states are discussed. Creating modified quantum theories by replacing the Hilbert spaces of canonical quantum mechanics with vector spaces defined over several finite, Galois fields is accomplished. Correlation functions are calculated in these theories and the maximum values are shown to not behave as would be expected by the standard, Bell-like, bounding inequality theorems. The interpretations and implications of these theories are discussed. / Ph. D.

In Part I the nature of the electron correlation problem is briefly discussed and some approaches to its solution and analysis are described. In addition, some previous work concerning the 2p2 3P state of helium-like systems is reviewed. In Part II, position space electron correlation effects in the 2p2 3P state of H-, He, Li+ and Be++ are investigated. This study is conducted by examining the effects of correlation on a variety of radial, angular and interparticle distribution functions and expectation values. A number of different correlated and uncorrelated wavefunctions are employed for this purpose, allowing the merits of different approaches to be assessed. In particular, a natural orbital analysis performed on a configuration interaction wave- function is used to gauge the relative importance of the angular and radial components of correlation, which are found to act in unison. It is seen that the correlation effects are similar in kind to those found in the ground state systems, though they are of a greater magnitude in this doubly excited state. The variation of the correlation effects with the atomic number, Z, is also examined. To complement the results for position space, a parallel study of electron correlation effects in momentum space is presented in Part III. In contrast to the position space investigation, the radial and angular components of correlation are found to have opposing effects, producing a rich and informative total correlation effect. As in position space, the Z-dependent correlation behaviour is studied and it is seen to be more informative in momentum space. The correlation effects in momentum space are similar to the effects in ground state systems, but are of a greater magnitude. Some techniques used in this research are described in three appendices in Part IV.

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The phenomenon of superconductivity has so for defied all attempts of explanation since it was first discovered in 1911. Although two phenomenological theories have been put forward and proved very successful, yet no atomic theories based on quantum mechanics have proved adequate. In the present thesis a new theory is proposed which will be shown to account for most of the important properties of a superconductor. In this part we briefly review the experimental facts and the earlier theories respectively ~1 and ~2.

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