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Physica Scripta

Mapping between conical space and non-relativistic position-dependent effective mass systems

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

Published Mar 19, 2019• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ab1103

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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.
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.
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.
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.
In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schrödinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schrödinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.