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arXiv:1203.2799 · Physical Review A

Revisiting the displacement operator for quantum systems with position-dependent mass

S. Habib Mazharimousavi

Published Mar 20, 2012• 66 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.85.034102

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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.
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.
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.
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.
An algebraic treatment of shape-invariant quantum-mechanical position-dependent effective mass systems is discussed. Using shape invariance, a general recipe for construction of ladder operators and associated algebraic structure of the pertaining system, is obtained. These operators are used to find exact solutions of general one-dimensional systems with spatially varying mass. We apply our formalism to specific translationally shape-invariant potentials having position-dependent effective mass.