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Journal of Mathematical Physics

Exact mapping between charge-monopole and position-dependent effective mass systems via Pauli equation

Anderson L. de Jesus, Alexandre G. M. Schmidt

Published Dec 1, 2019• 8 citations
Publisher

Abstract

The main purpose of this work is to reproduce a quantum system charge-monopole utilizing position-dependent effective mass (PDM) system in the nonrelativistic regime via the Pauli equation. In this case, we substitute the exact charge-monopole wavefunction into the free PDM Pauli equation and then solve it for the mass distribution considering a radial dependence only, i.e., M = M(r). The resulting equations are nonlinear, and in such cases, we were able to numerically solve them, fixing θ0 and considering specific values of μ and m satisfying a given condition. The mapping was studied for eigenvalues starting from the minimal value j = μ − 1/2.

DOI
10.1063/1.5114812

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The main purpose of this work is to reproduce a quantum system charge-monopole utilizing position-dependent effective mass (PDM) system in the nonrelativistic regime via the Pauli equation. In this case, we substitute the exact charge-monopole wavefunction into the free PDM Pauli equation and then solve it for the mass distribution considering a radial dependence only, i.e., M = M(r). The resulting equations are nonlinear, and in such cases, we were able to numerically solve them, fixing θ0 and considering specific values of μ and m satisfying a given condition. The mapping was studied for eigenvalues starting from the minimal value j = μ − 1/2.
Abstract The purpose of this work is to reproduce the dynamic of a neutral quantum particle bounded in a conical surface utilizing a position-dependent effective mass background in the non-relativistic regime. We start this task with the construction of the position-dependent effective mass Hamiltonian, containing the effective potential in the Zhou–Kroemer parametrization. We obtain a differential equation for the mass distribution which could reproduce the exact wavefunctions of a scalar quantum particle constrained to a conical space.
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). The explicit expressions for the potentials, energy eigenvalues and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term VmV_mVm​ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of VmV_mVm​ for several mass functions are discussed.
Abstract In this work, we study the influence of the Aharonov–Bohm–Coulomb (ABC) system on the relativistic and non-relativistic quantum dynamics of a charged Dirac particle with position-dependent mass (PDM). In order to solve exactly our system, we use both the left - handed and right - handed projection operators. Next, we determine the Dirac spinor and the relativistic energy spectrum, where we observe that this spinor is written in terms of the generalized Laguerre polynomials and this spectrum depends explicitly on the quantum numbers n and m l , parameters Z and that characterize the ABC system, and on the parameter that characterizes the PDM. In particular, we analyse the behavior of the energies as a function of for different values of Z . Finally, we consider the non-relativistic limit and we compared our problem with others works in the literature, where we verify that our results turn out to generalize some particular cases.
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.
The properties of the s -wave for a quasi-free particle with position-dependent mass (PDM) have been discussed in details. Differed from the system with constant mass in which the localization of the s -wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in D dimensions except D = 1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction, which may be attractive in some cases. The analytical expressions of the eigenfunctions and the corresponding probability densities for the s -waves of the two- and three-dimensional systems with a special PDM are given, and the existences of localization around the origin for these systems are shown.