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arXiv:1103.5440 · Journal of Physics A Mathematical and Theoretical

Motion of position-dependent effective mass as a damping–antidamping process: application to the Fermi gas and to the Morse potential

Y Hamdouni

Published Sep 1, 2011• 13 citations
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Abstract

The object of this paper is to investigate, classically and quantum mechanically, the relation existing between the position-dependent effective mass and damping–antidamping dynamics. The quantization of the equations of motion is carried out using the geometric interpretation of the motion, and we compare it with the one based on the ordering ambiguity scheme. Furthermore, we apply the obtained results to a Fermi gas of damped–antidamped particles, and solve the Schrödinger equation for an exponentially increasing (decreasing) mass in the presence of the Morse potential.

DOI
10.1088/1751-8113/44/38/385301

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The object of this paper is to investigate, classically and quantum mechanically, the relation existing between the position-dependent effective mass and damping–antidamping dynamics. The quantization of the equations of motion is carried out using the geometric interpretation of the motion, and we compare it with the one based on the ordering ambiguity scheme. Furthermore, we apply the obtained results to a Fermi gas of damped–antidamped particles, and solve the Schrödinger equation for an exponentially increasing (decreasing) mass in the presence of the Morse potential.
We solve the parametric generalized effective Schrödinger equation with a specific choice of position-dependent mass function and Morse oscillator potential by means of the Nikiforov–Uvarov method combined with the Pekeris approximation scheme. All bound-state energies are found explicitly and all corresponding radial wave functions are built analytically. We choose the Weyl or Li and Kuhn ordering for the ambiguity parameters in our numerical work to calculate the energy spectrum for a few (H2, LiH, HCl and CO) diatomic molecules with arbitrary vibration n and rotation l quantum numbers and different position-dependent mass functions. Two special cases including the constant mass and the vibration s-wave (l = 0) are also investigated.
A systematic procedure to study one-dimensional Schrodinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals an interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrodinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. Many exact results are derived in this way.
The position dependent mass Schrödinger equation (PDMSE) has a wide range of quantum applications such as the study of semiconductors, quantum wells, quantum dots and impurities in crystals, among many others. On the other hand, the Morse potential is one of the most important potential models used to study the electronic properties of diatomic molecules. In this work, the solution of the effective mass one-dimensional Schrödinger equation for the Morse potential is presented. This is done by means of the canonical transformation method in algebraic form. The PDMSE is solved for any model of the proposed kinetic energy operators as for example the BenDaniel-Duke, Gora-Williams, Zhu-Kroemer or Li-Kuhn. Also, in order to solve the PDMSE with Morse potential, we consider a superpotential leading to a special form of the exactly solvable Schrödinger equation of constant mass for a class of multiparameter exponential-type potential along with a proper mass distribution. The proposed approach is general and can be applied in the search of new potentials suitable on science of materials by looking into the viable choices of the mass function.
Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation. PACS numbers: 03.65.-w; 03.65.Ge; 12.39.Fd Keywords: Position-dependent mass, Effective mass Schrödinger equation, Morse potential, Nikiforov-Uvarov method
A systematic procedure to study one-dimensional Schrödinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals a new and interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrödinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. This provides a unified approach to many exact results known in the literature, as well as to a lot of new ones.