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Physica Scripta

Solvable potentials in pseudo-hermetic Dirac equation with PT symmetry

Fereshte Soliemani, Zahra Bakhshi

Published Aug 17, 2021• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ac1e5b

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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.
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.
By employing an exponential-type approximation scheme to replace the centrifugal term, we have approximately solved the Dirac equation for a spin-1/2 particle subjected to complex parity–time-symmetric scalar and vector Pöschl–Teller (PT) potentials with arbitrary spin–orbit -wave states in view of spin and pseudospin (p-spin) symmetries. The real bound-state energy eigenvalue equation and the corresponding two-spinor components wave function expressible in terms of hypergeometric functions are obtained by means of wave function analysis. The spin- Dirac equation and the spin-0 Klein–Gordon equation with complex PT potentials share the same energy spectrum under the choice of (i.e. exact spin and p-spin symmetries).
We present a new procedure to construct the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the context of the position-dependent effective mass Dirac equation with the vector-coupling scheme in 1+1 dimensions. In the first example, we consider a case for which the mass distribution combines linear and inversely linear forms, the Dirac problem with a PT -symmetric potential is mapped into the exactly solvable Schrödinger-like equation problem with the isotonic oscillator by using the local scaling of the wavefunction. In the second example, we take a mass distribution with smooth step shape, the Dirac problem with a non- PT -symmetric imaginary potential is mapped into the exactly solvable Schrödinger-like equation problem with the Rosen–Morse potential. The real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method.