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Algebraic Techniques in Quantum Mechanics

U Anjana

Published May 1, 2019• 0 citations
OpenAlex

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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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.
We give a systematic construction of new quasi-exactly solvable systems via Bethe ansatz and supersymmetric quantum mechanics (SUSYQM). Methods based on the intertwining of supercharges have been extensively used in the literature for exactly solvable systems. We generalize the state-deleting (Krein-Adler) supersymmetric transformations to quasi-exactly exactly solvable (QES) systems building on the Bethe ansatz method and related Bethe roots. This enables us to construct superpartners for a wide class of known QES systems classified previously through a hidden sl(2)sl(2)sl(2) algebra. We present our constructions of factorizations and intertwining relations related to 1st-order SUSYQM and the n=1n=1n=1 state for 10 nonequivalent types, denoted I,...,X. In order to have a unified treatment we rely on their ODE standard form as this is also the appropriate setting to obtain the Bethe ansatz equations which constrain the polynomial solutions. This setting also allows one to deal with systems with nnn states in a unified manner, using analysis based on the Bethe ansatz equations to build the supersymmetric transformations in terms of the Bethe ansatz roots. We derive the Schrödinger potentials for the n=1n=1n=1 superpartners of the 10 QES cases and give closed-form solutions for the spectra and wavefunctions of the corresponding QES SUSYQM systems. Furthermore, we present numerical results for higher excited states up to the n=10n=10n=10 level. The results obtained may have wider applicability as our framework is built on ODEs with polynomial coefficients.
Heisenberg's form of the correspondence principle for non-relativistic matrix elements has been used to evaluate matrix elements for various potentials. These are compared with available quantum mechanical results to check the general validity of the use of classical mechanics and correspondence principles in the derivation of quantum mechanical expressions. Using a one dimensional harmonic oscillator potential matrix elements of the form (n mod qm mod n+s) and (n mod pm mod n+s) are worked out. A Morse potential is then considered and the matrix elements for position, momentum, and kinetic energy are evaluated. Using a Coulomb potential a similar procedure is used for functions of various position coordinates, from which dipole and quadrupole moments from transitions between n, l states are calculated. The agreement with quantum mechanics is found to be generally good, and in some cases identical results are obtained.
The validity of the Jarzynski equation for a very simple, exactly solvable quantum system is analyzed. The implications of two different definitions of work proposed in the literature are investigated. The first one derives from measurements of the system energy at the beginning and at the end of the process under consideration making the work a classical stochastic variable with transition probabilities derived from quantum-mechanics. In the second definition an operator of work is introduced and the average in the Jarzynski equation is a quantum expectation value. For the first definition a general quantum-mechanical version of the Jarzynski equation is known to hold. For the second one the Jarzynski equation fails to yield the free energy difference at low temperature.