Algebraic Techniques in Quantum Mechanics

Anjana, U (2019) Algebraic Techniques in Quantum Mechanics. Masters thesis, Indian Institute of Science Education and Research Kolkata.

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Abstract

The main motto of my project is to derive a generalized recursion relation between the same matrix elements of different quantum mechanical operators. This was done for expectation values which was then generalised to matrix elements. The derived relation is checked for various systems and the validity of this relation was the same as the previously found result in the literature. The result is significant as it is applicable for all effective 1-D potentials and for any 1- variable function. Futher the fact that matrix elements are important in various quantum mechanical calculation makes the result of considerable importance. The future work of my project work is to apply the derived relation to Quasi-exactly solvable Quantum systems which are those systems with potentials, for which it is possible to obtain a finite part of the eigenvalue spectrum algebraically, while the rest numerically. All exactly solvable QM systems are also quasi-exactly solvable but my intention is to consider the potentials which are only quasi-exactly solvable but not completely solvable. So, as to extract more about the unknown part of the quasi-exactly solvable system from these found out recursion relations.

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