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arXiv:1909.04030

Position dependent effective mass and pseudo-hermetic generator in Hamiltonians with PT symmetry

Fereshte Soliemani, Zahra Bakhshi

Published Sep 8, 2019• 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 introduction of two first-order operator against η-pseudo-hermetic(η-weak-pseudo-hermiticity) with position dependent effective mass and applying it to Dirac equation with spinor wave function and non-hermitic potentials with real energy spectrum considering position dependent effective mass. Also Pöschl-teller complex and Eckart potentials will be discussed as examples.

DOI
10.48550/arxiv.1909.04030

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In this paper we present a general method to solve non hermetic potentials with PT symmetry using the introduction of two first-order operator against η-pseudo-hermetic(η-weak-pseudo-hermiticity) with position dependent effective mass and applying it to Dirac equation with spinor wave function and non-hermitic potentials with real energy spectrum considering position dependent effective mass. Also Pöschl-teller complex and Eckart potentials will be discussed as examples.
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The ηηη-pseudo-hermitic generator in the deformed Woods Saxons potential
In this paper, we present a general method to solve non-hermetic potentials with PT symmetry using the definition of two ηηη-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 ηηη-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 ηηη-pseudo-hermitic.
In this paper, the relativistic Dirac equation in one dimension is investigated for a particle in an external electromagnetic field, with the property of position-dependent effective mass approximation (PDEM), in the absence of vector potential. By removing the lower spinor component and combining the pair of equations, a Schrödinger-like equation is obtained for the upper spinor component. Using canonical transformations and introducing two first-order Hermitian and anti-Hermitian differential operators, a formalism for pseudo-hermitic Hamiltonians with parity-time reversal symmetry (PT) has been obtained. Comparing the equation derived from pseudo-Hermitian Hamiltonian with the non-relativistic Schrödinger equation leads to a general formalism for one-dimensional solvable imaginary non-Hermitian potentials with real energy spectra. Also, using this process, the complex potentials of Pöschl-Teller and Scarf II with real energy spectra in Dirac equation with PDEM approximationand PT symmetry have been investigated and their application has been expressed. For some particular parameters we will see the phenomenon of energy-levels crossing. In fact, it means that energy levels disappear from the spectrum. Also, for the mentioned examples, potential figures are drawn.
We study the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the framework of the position-dependent effective mass Dirac equation. The Dirac equation is mapped into the exactly solvable Schrödinger-like equation endowed with position-dependent effective mass that we present a new procedure to solve it. The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function, the imaginary potential can be obtained. The spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials. We also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials.
The Dirac-modified Poschl-Teller problem with position dependent mass is investigated within the framework of the asymptotic iteration method in N dimensions. Any -state solutions are obtained by using the exponential approximation for the centrifugal term. The energy eigenvalues and two-component spinor corresponding eigenfunctions are determined for different  screening parameters explicitly. The corresponding eigenfunctions obtained in the form of hypergeometric functions are normalized and plotted for different quantum states. Effects on modified Poschl-Teller potential, bound state energy eigenvalues and normalized corresponding eigenfunctions of the  screening parameters are investigated and its results are discussed.    Key words: Dirac equation, position-dependent mass (PDM), Poschl-Teller (PT) potential, asymptotic iteration method, bound state.