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arXiv:2007.15718

The $η$-pseudo-hermitic generator in the deformed Woods Saxons potential

Zahra Bakhshi, Mohsen Hafezghoran

Published Jul 29, 2020• 0 citations
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Abstract

In this paper, we present a general method to solve non-hermetic potentials with PT symmetry using the definition of two $η$-pseudo-hermetic and first-order operators. This generator applies to the Dirac equation which consists of two spinor wave functions and non-hermetic potentials, with position that mass is considered a constant value and also Hamiltonian hierarchy method and the shape invariance property are used to perform calculations. Furthermore, we show the correlation between the potential parameters with transmission probabilities that $η$-pseudo-hermetic using the change of focal points on Hamiltonian can be formalized based on Schrödinger-like equation. By using this method for some solvable potentials such as deformed Woods Saxon's potential, it can be shown that these real potentials can be decomposed into complex potentials consisting of eigenvalues of a class of $η$-pseudo-hermitic.

DOI
10.48550/arxiv.2007.15718

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Abstract This paper presents a general method to solve non-hermeticity potentials with PT (Parity-Time) symmetry using two first-order operators against η -weak-pseudo-hermiticity having position-dependent effective mass. It futher suggests a way to apply this method to Dirac equation with spinor wave functions and non-hermetic potentials with real energy spectrum considering position-dependent effective mass. To show the utilities this method was apply to some superpotentials such as Eckart, Scarf-II, Rosen-Mörse-II and Pöchl-Teller. Finally, it was shown that the real potentials can be converted to complex potentials with eigenvalues of a class of η -pseudo-hermeticity.
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PT\mathcal {PT}PT symmetric Hamiltonian model and Dirac equation in 1+1 dimensions
In this paper, we have introduced a symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first-order differential operators. The Hermitian form of the Hamiltonian is obtained by suitable mappings and it is interrelated to the time-independent one-dimensional Dirac equation in the presence of position-dependent mass. Then, Dirac equation is reduced to a Schrödinger-like equation and two new complex non- symmetric vector potentials are generated. We have obtained a real spectrum for these new complex vector potentials using the shape invariance method. We have searched the real energy values using numerical methods for the specific values of the parameters.
The one-dimensional Dirac equation with position dependent mass in the generalized asymmetric Woods-Saxon potential is solved in terms of the hypergeometric functions. The transmission and reflection coefficients are obtained by considering the one-dimensional electric current density for the Dirac particle and the equation describing the bound states is found by utilizing the continuity conditions of the obtained wave function. Also, by using the generalized asymmetric Woods-Saxon potential solutions, the scattering states are found out without making calculation for the Woods-Saxon, Hulthen, cusp potentials, and so forth, which are derived from the generalized asymmetric Woods-Saxon potential and the conditions describing transmission resonances and supercriticality are achieved. At the same time, the data obtained in this work are compared with the results achieved in earlier studies and are observed to be consistent.