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Physica Scripta

Position-dependent effective mass system in a variable potential: displacement operator method

M. Vubangsi, Martin Tchoffo, Lukong Cornelius Fai

Published Jan 21, 2014• 25 citations
Publisher

Abstract

Within the framework of the translation operator for a quantum system with position-dependent mass, we examine the quantum state of a position-dependent mass system in a variable potential. By imposing conditions of resolvability, we arrive at a potential with a quartic and a quadratic term. It emerges naturally that the energy eigen states of the system are negative. We have found the quantum mechanical quantities: energy spectrum, eigen functions and uncertainty relation. These quantities depend on the parameters of the potential.

DOI
10.1088/0031-8949/89/02/025101

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Within the framework of the translation operator for a quantum system with position-dependent mass, we examine the quantum state of a position-dependent mass system in a variable potential. By imposing conditions of resolvability, we arrive at a potential with a quartic and a quadratic term. It emerges naturally that the energy eigen states of the system are negative. We have found the quantum mechanical quantities: energy spectrum, eigen functions and uncertainty relation. These quantities depend on the parameters of the potential.
A translation operator is introduced to describe the quantum dynamics of a position-dependent mass particle in a null or constant potential. From this operator, we obtain a generalized form of the momentum operator as well as a unique commutation relation for $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{\ensuremath{\gamma}}$. Such a formalism naturally leads to a Schr\"odinger-like equation that is reminiscent of wave equations typically used to model electrons with position-dependent (effective) masses propagating through abrupt interfaces in semiconductor heterostructures. The distinctive features of our approach are demonstrated through analytical solutions calculated for particles under null and constant potentials like infinite wells in one and two dimensions and potential barriers.
Recently Costa Filho et al. [Phys. Rev. A 84, 050102(R) (2011)] have introduced a position-dependent infinitesimal translation operator which corresponds to a position-dependent linear momentum and consequently to a quantum particle with position-dependent effective mass. Although there is no doubt about the novelty of the idea and the formalism, we believe that some aspects of the quantum mechanics in their original work could be enhanced. Here in this Brief Report first we address those points and then an alternative is introduced. Finally we apply the formalism for a quantum particle under a null potential confined in a square well, and the results are compared with those in the paper mentioned above.
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.
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.
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.