Quantum anharmonic potentials with operator and factorization methods

Anharmonic potential is one of the main focuses of this research. This research is also carried out to deepen our understanding of the mathematical tools in nonrelativistic quantum mechanics. Specifically, the mathematical tools, namely factorization method and supersymmetry are interesting. Fact...

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Main Author:	Tan, Min Yang
Format:	Thesis
Language:	English English
Published:	2023
Subjects:	Quantum theory Factorization method (Quantum theory)
Online Access:	http://psasir.upm.edu.my/id/eprint/112189/1/FS%202023%205%20-%20IR.pdf
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id	my-upm-ir.112189
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spelling	my-upm-ir.1121892024-09-26T08:44:05Z Quantum anharmonic potentials with operator and factorization methods 2023-03 Tan, Min Yang Anharmonic potential is one of the main focuses of this research. This research is also carried out to deepen our understanding of the mathematical tools in nonrelativistic quantum mechanics. Specifically, the mathematical tools, namely factorization method and supersymmetry are interesting. Factorization method is effective in determining the energy spectra of one-dimensional exactly solvable potentials. On the other hand, given any potential, supersymmetry allows us to build a partner potential with an identical energy spectrum except for the ground state. This research can be divided into two distinct parts. In the first part of this research, the mathematical structure of SU(2) group, that is, the commutation relation of ladder operators of the Morse oscillator, which is anharmonic, is examined. The concept of ladder operator often appears in the context of factorization method. It is realized that the mathematical structure of the ladder operators of the Morse oscillator depends on some Morse oscillator’s parameters. The commutation relation is analysed analytically by considering the effect of parameters on the operators. The parameter space of Morse oscillator is visualised to scrutinise the mathematical relations that are related to Morse oscillator. This parameter space is the space of all possible parameter values depending on the depth of Morse potential well and other molecular constants. The equality of eigenvalues calculated in two different perspectives is investigated. It is possible for the algorithm in this work to be also applicable to other one-dimensional quantum systems with certain modifications. The second part of this research is more focusing on the connection between deductive method and supersymmetric quantum mechanics. The traditional factorization method and supersymmetric quantum mechanics are immensely explored in the literature. However, the so-called deductive method proposed by Green in 1965 is less being considered by researchers. This deductive method can be reinterpreted as a different formulation of factorization method. It is shown to be related to the supersymmetric quantum mechanics. An alternative way to obtain the superpotential in terms of supersymmetric quantum mechanics is deduced. Finally, our reasoning is successfully demonstrated with two anharmonic systems, namely Deng-Fan and generalized Mobius square potentials. Quantum theory Factorization method (Quantum theory) 2023-03 Thesis http://psasir.upm.edu.my/id/eprint/112189/ http://psasir.upm.edu.my/id/eprint/112189/1/FS%202023%205%20-%20IR.pdf text en public masters Universiti Putra Malaysia Quantum theory Factorization method (Quantum theory) Mohd Shah, Nurisya English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
advisor	Mohd Shah, Nurisya
topic	Quantum theory Factorization method (Quantum theory)
spellingShingle	Quantum theory Factorization method (Quantum theory) Tan, Min Yang Quantum anharmonic potentials with operator and factorization methods
description	Anharmonic potential is one of the main focuses of this research. This research is also carried out to deepen our understanding of the mathematical tools in nonrelativistic quantum mechanics. Specifically, the mathematical tools, namely factorization method and supersymmetry are interesting. Factorization method is effective in determining the energy spectra of one-dimensional exactly solvable potentials. On the other hand, given any potential, supersymmetry allows us to build a partner potential with an identical energy spectrum except for the ground state. This research can be divided into two distinct parts. In the first part of this research, the mathematical structure of SU(2) group, that is, the commutation relation of ladder operators of the Morse oscillator, which is anharmonic, is examined. The concept of ladder operator often appears in the context of factorization method. It is realized that the mathematical structure of the ladder operators of the Morse oscillator depends on some Morse oscillator’s parameters. The commutation relation is analysed analytically by considering the effect of parameters on the operators. The parameter space of Morse oscillator is visualised to scrutinise the mathematical relations that are related to Morse oscillator. This parameter space is the space of all possible parameter values depending on the depth of Morse potential well and other molecular constants. The equality of eigenvalues calculated in two different perspectives is investigated. It is possible for the algorithm in this work to be also applicable to other one-dimensional quantum systems with certain modifications. The second part of this research is more focusing on the connection between deductive method and supersymmetric quantum mechanics. The traditional factorization method and supersymmetric quantum mechanics are immensely explored in the literature. However, the so-called deductive method proposed by Green in 1965 is less being considered by researchers. This deductive method can be reinterpreted as a different formulation of factorization method. It is shown to be related to the supersymmetric quantum mechanics. An alternative way to obtain the superpotential in terms of supersymmetric quantum mechanics is deduced. Finally, our reasoning is successfully demonstrated with two anharmonic systems, namely Deng-Fan and generalized Mobius square potentials.
format	Thesis
qualification_level	Master's degree
author	Tan, Min Yang
author_facet	Tan, Min Yang
author_sort	Tan, Min Yang
title	Quantum anharmonic potentials with operator and factorization methods
title_short	Quantum anharmonic potentials with operator and factorization methods
title_full	Quantum anharmonic potentials with operator and factorization methods
title_fullStr	Quantum anharmonic potentials with operator and factorization methods
title_full_unstemmed	Quantum anharmonic potentials with operator and factorization methods
title_sort	quantum anharmonic potentials with operator and factorization methods
granting_institution	Universiti Putra Malaysia
publishDate	2023
url	http://psasir.upm.edu.my/id/eprint/112189/1/FS%202023%205%20-%20IR.pdf
_version_	1811767784675213312

Quantum anharmonic potentials with operator and factorization methods

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