Quasi-exact and asymptotic iterative solutions of Dirac equation in the presence of some scalar potentials
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Quasi-exact and asymptotic iterative solutions of Dirac equation in the presence of some scalar potentials A CHENAGHLOU1,∗ , S AGHAEI2 and R MOKHTARI1 1 Department of Physics, Faculty of Sciences, Sahand University of Technology, P.O. Box 51335-1996, Tabriz, Iran 2 Physics
Department, Faculty of Sciences, Farhangian University, Tehran, Iran author. E-mail: [email protected]
∗ Corresponding
MS received 6 January 2020; revised 30 June 2020; accepted 18 August 2020 Abstract. In this paper, the Dirac equation in the presence of some scalar potentials based on sl(2) Lie algebra is solved by quasi-exact solvability theory. The configuration of the classes III and VI potentials in the Turbiner’s classification is constructed. Then, the Bethe ansatz equations are calculated so that the energy eigenvalues and eigenfunctions are obtained. Also, we study the problem by using asymptotic iteration method. Finally, we compare the results obtained by these two methods. Keywords. Dirac equation; quasi-exact solvability; supersymmetric quantum mechanics; asymptotic iteration method. PACS Nos 03.65.–w; 03.65.Pm; 03.65.Fd; 02.20.–a
1. Introduction In the context of relativistic quantum mechanics and quantum field theory, Lorentz scalar potential has had considerable interest, and has important use in these areas [1–6]. But, some forms of Lorentz scalar potential result in exactly solvable problems. Recently, much efforts were put in to solve the Dirac equation using various methods [7–23]. However, a few potentials can be solved exactly. On the other hand, quasi-exact solvability approach prepares a procedure for deriving a part of the energy spectrum but not the whole spectrum [23]. Also, supersymmetric quantum mechanics has attracted considerable attention in the past years [24–30]. A general method for constructing quasi-exactly solvable problems based on sl(2) Lie algebra has been presented in refs [31–33]. Moreover, construction of all the one-dimensional quasi-exactly solvable models has been done. In fact, factorisability or equivalently supersymmetric structure of the Hamiltonian of the problem plays an important role in this method. At the same time, this method proposes some configurations for the potential that lead to the derivation of a part of the spectrum but not the whole spectrum. Also, possible forms of the Lorentz scalar potential which lead to the quasi-exactly solvable forms of the corresponding Dirac equation are 0123456789().: V,-vol
determined. On the other hand, the asymptotic iteration method was proposed [34,35] for solving eigenvalue problems. So, this method can be used to investigate Dirac equation with various potentials. Turbiner classified the quasi-exactly solvable models based on sl(2) Lie algebra in ten classes [36]. When Dirac equation with the Lorentz scalar potential is factorised, seven classes of quasi-exactly solvable potential can be identified. These classes are classes I to VI and class X in the framework of the Turbiner’s classification. The construct
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