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Physica Scripta

Quantum harmonic oscillator with position-dependent mass in the displacement operator formalism

Martin Tchoffo, M. Vubangsi, Lukong Cornelius Fai

Published Aug 22, 2014• 17 citations
Publisher

Abstract

The position-dependent effective mass quantum harmonic oscillator problem is considered within the displacement operator framework. Using the analytic and the algebraic approaches, exact expressions for quantum mechanical quantities of the system have been obtained. In the limit of no deformation, results of the constant mass oscillator are recovered.

DOI
10.1088/0031-8949/89/10/105201

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The position-dependent effective mass quantum harmonic oscillator problem is considered within the displacement operator framework. Using the analytic and the algebraic approaches, exact expressions for quantum mechanical quantities of the system have been obtained. In the limit of no deformation, results of the constant mass oscillator are recovered.
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.
Abstract We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel–Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to ∞, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.
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 Schrodinger equation with position-dependent mass (PDM) becomes one of interesting subjects in the study of quantum systems, because of its wide applications in many physical problems. Meanwhile, harmonic oscillator becomes important model in most physical problems. In this paper, analytical solutions of the PDM Schrodinger equation, i.e. energy eigenvalues En and eigenfunctions , of a one dimensional harmonic oscillator system using transformation method has been studied. The results showed that there was no difference between the energy eigenvalues, En of the harmonic oscillator systems with and without PDM, whereas it is not for the eigenfunctions,  of these two harmonic oscillator systems. This difference was due to the presence of dimensionless mass function  and the changes of x becomes  variable for the eigenfunctions of the harmonic oscillator system with PDM.  Moreover, the differences in the eigenfunctions will affect the dynamical properties of the quantum systems. It can be concluded that the harmonic oscillator system with PDM is the generalization of the harmonic oscillator system with constant mass.