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Research Paper: Solution of the Dirac Equation for Pseudo-Hermitian Hamiltonian and Energy-levels Crossing

Zahra Bakhshi, Fereshte Soleimani, Sare Khoshdooni

Published Jun 1, 2021• 0 citations

Abstract

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.

DOI: 10.22051/ijap.2021.35731.1203

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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.

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.

\mathcal {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.

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.

We introduce the Dirac Inverted Oscillator, a new exactly solvable relativistic quantum system obtained through a Hermitian modification of the generalized momentum operator in the Dirac equation. In contrast to the conventional Dirac oscillator, where the generalized momentum is intrinsically non-Hermitian while the Hamiltonian remains Hermitian, the present construction reverses these Hermiticity properties: the generalized momentum becomes Hermitian, whereas the corresponding Dirac Hamiltonian is intrinsically non-Hermitian. This inversion leads to a new class of relativistic Hamiltonians within the framework of non-Hermitian quantum mechanics. Starting from the stationary Dirac equation, we derive the corresponding second-order wave equation and establish the generalized symmetry properties of the model, including pseudo-Hermiticity and pseudo- symmetry. We then prove that the Dirac Inverted Oscillator is exactly related to the conventional Dirac oscillator through a Hermitian but non-unitary similarity transformation generated by a dilation operator. This transformation is bijective and invertible, mapping the original non-Hermitian relativistic problem onto an equivalent Hermitian Schrodinger eigenvalue problem while preserving its complete spectral structure. The transformed Hermitian problem is solved analytically, allowing the exact relativistic spectrum and the corresponding eigenfunctions of the original non-Hermitian Hamiltonian to be determined explicitly. The present work therefore establishes an exact correspondence between a non-Hermitian relativistic Dirac system and a Hermitian Schrodinger problem, providing a new exactly solvable model and illustrating the effectiveness of Hermitian non-unitary similarity transformations for constructing analytically solvable relativistic Hamiltonians beyond the conventional Hermitian framework.