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Factorization approach to generalized Dirac oscillators

Jafar Sadeghi, Seithuti Philemon Moshokoa, M. Rostami, F A Larijani

Published Jan 15, 2015• 0 citations
Publisher

Abstract

We study generalized Dirac oscillators with complex interactions in (1+1) dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are  pseudo Hermitian with respect to certain metric operators  . Exact solutions of the generalized Dirac Oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with Anti Jaynes Cummings type model and by spin flip it can also be identified with Jaynes Cummings type model.

DOI
10.5281/zenodo.3980835

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We consider a couple of examples to study the pseudo-Hermitian interaction in relativistic quantum mechanics. Rasbha interaction, commonly used to study the spin Hall effect, is considered with imaginary coupling. The corresponding Dirac Hamiltonian is shown to be parity pseudo-Hermitian. In the other example we consider parity pseudo-Hermitian scalar interaction with arbitrary parameter in Dirac theory. In both cases we show that the energy spectrum is real and all the other features of nonrelativistic pseudo-Hermitian formulation are present. Using the spectral method, the positive definite metric operator (η) has been calculated explicitly for both the models to ensure positive definite norms for the state vectors.
We propose a nonlocal extension of the generalized Dirac oscillator (GDO) in (1+1)(1+1)(1+1) dimensions by replacing the multiplicative interaction f(x)f(x)f(x) with an integral operator F^\hat FF^ with kernel f(x,x′)f(x,x')f(x,x′). The resulting Dirac equation preserves an operator factorization and decouples into two nonlocal Schrödinger-type (Sturm--Liouville) equations for the spinor components. We derive explicit expressions for the associated supersymmetric partner kernels in terms of fff and its derivatives, and we show that a complex-translation metric η=e−θpxη=e^{-θp_x}η=e−θpx​ leads to a simple sufficient \emph{kernel-level} pseudo-Hermiticity constraint, $f(x+\ii\hbarθ,x'+\ii\hbarθ)=f^*(x',x)$, extending the familiar local complex-shift criteria. To provide a transparent \emph{nonlocal-to-local} interpretation, we adapt the Coz--Arnold--MacKellar current-based localization to each component equation, obtaining energy-dependent equivalent local potentials and multiplicative Perey (damping) factors. The mapping breaks down precisely at current zeros, thereby diagnosing the spurious solutions of the corresponding nonlocal Schrödinger problem. Finally, we illustrate the formalism with analytically tractable benchmarks (the local Dirac oscillator and a translation-invariant kernel) and with a finite-rank separable model (Gaussian form factor) that reduces the integro-differential problem to a small set of coupled ordinary differential equations and algebraic constraints.
The Dirac oscillator is a relativistic quantum system, characterized by its linearity in both position and momentum. Moreover, considering (1+1)(1{+}1)(1+1) and (2+1)(2{+}1)(2+1) dimensions, the system can be mapped onto the Jaynes-Cummings and anti-Jaynes-Cummings models, as illustrated in an exact manner by Bermudez \textit{et al.} [\href{ https://doi.org/10.1103/PhysRevA.76.041801}{Phys. Rev. A 76, 041801(R) (2007)}]. Using the optical counterparts of the Dirac oscillator, we analyze an extension of the model that incorporates a time-dependent frequency. We focus on the consequences of these time modulations on the angular momentum observables and spin-orbit entanglement. Noticeable changes in the \emph{Zitterbewegung} are found. We show that a specific choice of time dependence yields aperiodic evolution of the observables, whereas an alternative choice allows analytical solutions.
The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We point out all properties of both of the original Mandal and the original Jaynes-Cummings Hamitonians. It is shown that these Hamiltonians are respectively pseudo-hermitian and hermitian REF _Ref536606452 \r \h \* MERGEFORMAT [1] REF _Ref536606454 \r \h [2]. Like the direct approach to invariant vector spaces used in Refs. REF _Ref536606456 \r \h [3] REF _Ref536606457 \r \h [4], we reveal the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator.
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.