Quantum anharmonic potentials with operator and factorization methods - Tan, Min Yang (2023)
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.