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CRC Press eBooks

Nonperturbative approach to potentials in impenetrable boxes based on quasi-exact solvability

ALVARO DE SOUZA DUTRA, V. G. C. S. dos Santos, A. M. Stuchi

Published Nov 30, 2003• 0 citations
Publisher

Abstract

In this work we develop an approach to obtain analytical expressions for potentials in an impenetrable box. In this kind of system the expression has the advantage of being valid for arbitrary values of the box length, and respect the correct quantum limits. The similarity of this kind of problem with the quasi exactly solvable potentials is explored in order to accomplish our goals. Problems related to the break of symmetries and simultaneous eigenfunctions of commuting operators are discussed.

DOI
10.1201/9781482269079-101

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In this work we develop an approach to obtain analytical expressions for potentials in an impenetrable box. In this kind of system the expression has the advantage of being valid for arbitrary values of the box length, and respect the correct quantum limits. The similarity of this kind of problem with the quasi exactly solvable potentials is explored in order to accomplish our goals. Problems related to the break of symmetries and simultaneous eigenfunctions of commuting operators are discussed.
Based on an equations-of-motion method in the framework of Heisenberg's matrix mechanics, we investigate the conditions under which a one-dimensional quantum mechanical system becomes exactly solvable. By linearizing the equation of motion which is a double-commutation relation of some appropriately chosen function of the position operator with the Hamiltonian, we obtained a set of nontrivial exactly solvable potentials in one dimension. These potentials not only can be solved analytically in closed forms but also contain both the Morse potential and the Poschl-Teller potential as their limiting cases. They may thus be valuable for some potential model calculations as well as for testing various approximation schemes. We also examine these potentials in the framework of supersymmetric quantum mechanics, which is particularly useful for studying exactly solvable potentials.
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.