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Journal of Physics Conference Series

Path integral formulation for position–time-dependent mass systems

Badri Berrabah, Baya Bentag

Published Jan 1, 2021• 0 citations
Publisher

Abstract

Abstract The problem of position-time dependent mass with general time-dependent potential is treated in the framework of the phase space path integral approach. By means of an appropriate point canonical transformation, the problem is converted into that relative to a constant mass with stationary potential. Particular examples are also considered.

DOI
10.1088/1742-6596/1766/1/012034

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Abstract The problem of position-time dependent mass with general time-dependent potential is treated in the framework of the phase space path integral approach. By means of an appropriate point canonical transformation, the problem is converted into that relative to a constant mass with stationary potential. Particular examples are also considered.
In the framework of the position-dependent mass quantum mechanics, the three dimensional Schrödinger equation is studied by applying the Laplace transforms combining with the point canonical transforms. For the potential analogues to Morse potential and via the Pekeris approximation, we introduce the general solutions appropriate for any kind of position dependent mass profile which obeys a key condition. For a specific position-dependent mass profile, the bound state solutions are obtained through an analytical form. The constant mass solutions are also relived.
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.
Some time-dependent physical systems do not admit, in the general case, either an invariant or auxiliary equation. The study of these systems is then in general made easier by space-time transformation of coordinates. This is true for the case for a rectangular well with a moving wall, which generalized canonical transformations reduce, in the path-integral formalism, to the case of a variable-frequency oscillator with fixed walls. The variable-frequency harmonic oscillator is an example where the invariant and the auxiliary equations can be calculated. All calculations are done in phase space.