Position-dependent mass Dirac equation and local Fermi velocity

We investigate the (1 + 1)-dimensional position-dependent mass Dirac equation within the confines of so(2, 1) potential algebra by utilizing the character of a spatial varying Fermi velocity. We examine the combined effects of the two when the Dirac equation is equipped with an external pseudoscalar potential. Solutions of the three cases induced by so(2, 1) are explored by profitably making use of a point canonical transformation.

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.

The ordering problem in quantum systems with position-dependent mass (PDM) is treated by inclusion of the classically fictitious similarity transformation into the kinetic term. This provides a generation of supersymmetry with the first order supercharges from the kinetic term alone, while inclusion of the potential term allows also to generate nonlinear supersymmetry with higher order supercharges. A broad class of finite-gap systems with PDM is obtained by different reduction procedures, and general results on supersymmetry generation are applied to them. We show that elliptic finite-gap systems of Lame and Darboux-Treibich-Verdier types can be obtained by reduction to Seiffert's spherical spiral and Bernoulli lemniscate in the presence of Calogero-like or harmonic oscillator potentials, or by angular momentum reduction of a free motion on some AdS_2-related surfaces in the presence of Aharonov-Bohm flux. The limiting cases include the Higgs and Mathews-Lakshmanan oscillator models as well as a reflectionless model with PDM exploited recently in the discussion of cosmological inflationary scenarios.

Dirac equation with spatially or position-dependent mass and an attractive Coulomb-like field is constructed in Hausdorff dimension of order $$0<\alpha \le 1$$. The lower and upper components of the spinor wave function were derived in addition to the corresponding energy eigenvalues of the resulting relativistic equation. It was observed that, in Hausdorff radial dimension, the ground state energy of particles mass with spin half is enhanced which leads to an enhancement of electrons mass in agreement with recent theoretical observations and experimental data.

Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence of bound states in a pure inversely quadratic potential.

It has been suggested that the high symmetries in the Schrödinger equation with the Coulomb or harmonic oscillator potentials may remain in the corresponding relativistic Dirac equation. If the principle is correct, in the Dirac equation the potential should have a form as [Formula: see text] where V(r) is [Formula: see text] for hydrogen atom and κr2 for harmonic oscillator. However, in the case of hydrogen atom, by this combination the spin–orbit coupling term would not exist and it is inconsistent with the observational spectra of hydrogen atom, so that the symmetry of SO(4) must reduce into SU(2). The governing mechanisms QED and QCD which induce potential are vector-like theories, so at the leading order only vector potential exists. However, the higher-order effects may cause a scalar fraction. In this work, we show that for QED, the symmetry restoration is very small and some discussions on the symmetry breaking are made. At the end, we briefly discuss the QCD case and indicate that the situation for QCD is much more complicated and interesting.

One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1]. However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so‐called effective mass that is in general variable and dependent on position. Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position‐dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi‐spaced spectrum solutions, i.e. harmonic oscillator isospectral partners. To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm‐Liouville type, into a Schrödinger‐like standard equation is applied to the PDMSE. In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger‐like standard equation are related through a Riccati‐type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions‐dependent mass distribution (PDMD) m(x). Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.

A Quesne-like ring-shaped spherical harmonic oscillator potential is put foword and studied for spin 1/2 particles based on the Dirac equation， the Dirac Hamiltonian contains a scalar and a vector Quesne-like ring-shaped harmonic oscillator potentials. Setting Σ＝S(r)+V(r)=0,we obtain the bound state solutions and eigenenergies with the two-component approach. The result shows the pseudospin symmetry exists in the Quesne-like ring-shaped harmonic oscillator potential. The general properties of both the ring-shaped spherical harmonic oscillator potential and the ring-shaped non-spherical harmonic oscillator potential are discussed.

We investigate the influence of the temporal variations of various medium parameters on the propagation of Dirac-type waves in materials where the quasiparticles are described by a generalized version of the pseudospin-1/2 Dirac equation. Our considerations also include the propagation of electromagnetic waves in metamaterials with the Dirac-type dispersion. We focus on the variations of the scalar and vector potentials, mass, Fermi velocity, and tilt velocity describing the Dirac cone tilt. We derive the scattering coefficients associated with the temporal interfaces and slabs analytically and find that the temporal scattering is caused by the changes of the mass, Fermi velocity, and vector potential, but does not arise from the changes of the scalar potential and tilt velocity. We also explore the conditions under which the temporal Brewster effect and total interband transition occur and calculate the change in total wave energy. We examine bilayer Dirac temporal crystals where parameters switch between two different sets of values periodically and prove that these systems do not have momentum gaps. Finally, we assess the potential for observing these temporal scattering effects in experiments.

We describe a procedure to solve an up to problem where the particles are separated topologically in N groups with at most two particles in each. Arbitrary interactions are allowed between the (two) particles within one group. All other interactions are approximated by harmonic oscillator potentials. The problem is first reduced to an analytically solvable N-body problem and N independent two-body problems. We calculate analytically spectra, wave functions, and normal modes for both the inverse square and delta-function two-body interactions. In particular, we calculate separation energies between two strings of particles. We find that the string separation energy increases with N and interaction strength.

The effects of position-dependent effective mass on the absorption coefficients (ACs) of spherical quantum dot (QD) and quantum antidot (QAD) have been investigated. We show that the radial position-dependent effective mass Schrodinger equation can be reduced into ordinary one-dimensional Schrodinger equation with some suitable transformations. Our numerical calculations are performed using both a constant effective mass and the position-dependent effective mass. The comparative approach is used for presenting the results of both QD and QAD models. It is found that not only for the effective constant mass but also for the position-dependent effective mass, the ACs curves corresponding to QD and QAD are considerably different in shapes and behaviors. Our obtained results show that the spatial variation of the electron effective mass can not be neglected in the optical ACs of both QD and QAD nanostructures.

One-dimensional effective mass Dirac equation is solved in asymmetric Hulthén potential. Scattering and bound state solutions are obtained in terms of hypergeometric functions. Transmission and reflection coefficients are calculated by matching conditions on the wave function. The necessary conditions for the existence of transmission resonances and supercriticality are derived.

We propose improved ring shaped like potential of the form, V(r, θ) = V(r) + (ħ2/2Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2 and its exact solutions are presented via the Nikiforov–Uvarov method. The angle dependent part V(θ) = (ħ2 / 2 Mr2)[(β sin2 θ + γ cos2 θ + λ) / sin θ cos θ]2, which is reported for the first time embodied the novel angle dependent (NAD) potential and harmonic novel angle dependent potential (HNAD) as special cases. We discuss in detail the effects of the improved ring shaped like potential on the radial parts of the spherical harmonic and Coulomb potentials.

The eigenvalue problems corresponding to one and two particles confined in harmonic oscillator potentials are analyzed. In particular Klein-Gordon and Dirac equations, which in the non-relativistic limit transform, respectively, into LBvyLeblond and Schrodinger equations for harmonic oscillators, are constructed. Differences between the equations describing spin-0 and spin-3 particles are briefly discussed. In particular the eigenvalue spectra of different kinds of harmonic oscillators are compared. Besides, some properties of a system of two identical particles interacting with a Coulomb force and bound by a Hook’s. law potential (harmonium) are discussed. Usefulness of the harmonic-oscillator-type potentials in modeling of spatially confined systems is emphasized.

It is observed that the Feynman path summation for the one-dimensional Dirac equation can be projected into three spatial dimensions to yield a path-summation formula for physical spin-\textonehalf{} particles of nonzero mass. Since the three-space projection matrix is independent of time and does not involve the particle's mass, relativistic motion governed by the Dirac equation has an underlying one-dimensional aspect.