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arXiv:1611.09798

Notes on the phase space formulation of the propagator of Hamiltonians with spatially-dependent kinetic energy

Yamen Hamdouni

Published Nov 29, 2016• 0 citations
arXivHTMLPDF

Abstract

These short notes present to the reader (students, in particular) a concise approach to the derivation of the propagator of Hamiltonians with position-dependent kinetic energy. The formalism is applied to the von Roos Hamiltonian with arbitrary ordering ambiguity parameters, and a simple scheme to convert the problem to a constant-mass motion is presented. The motion in curved spaces is treated along the same lines, where a phase space formulation is used to derive the propagator for arbitrary discretization choices.

DOI
10.48550/arxiv.1611.09798

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These short notes present to the reader (students, in particular) a concise approach to the derivation of the propagator of Hamiltonians with position-dependent kinetic energy. The formalism is applied to the von Roos Hamiltonian with arbitrary ordering ambiguity parameters, and a simple scheme to convert the problem to a constant-mass motion is presented. The motion in curved spaces is treated along the same lines, where a phase space formulation is used to derive the propagator for arbitrary discretization choices.
In this paper we present a complete classification scheme for kinetic energy operators (KEO's) describing a particle endowed with position-dependent mass (PDM). We first present a generalized formulation of KEO's with PDM and show that it is equivalent to three-parameter linear formulation. This reduces further to two independent parameters under Hermiticity condition. Based on this linear formulation we prove that, contrary to what was widely believed, von Roos family is not the most general ordering. We found an entire new family of Hermitian KEO's that does not fit into von Roos ordering. We were able to construct all the Hermitian KEO's and classify them into two-parameter classes. As one application, we solve the puzzling case of Yang and Yee KEO. We also find, under certain conditions, some kind of duality between von Roos ordering and one of the new classes.
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.
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.
In earlier work, a Hamiltonian describing the classical motion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated.Here we derive the quantum propagator for the Hamiltonian in different representations, one corresponding to momentum space, the other to position, and the third to a natural choice of "velocity" variables.We call attention to the following noteworthy fact: the Hamiltonian contains three parameters which do not in any way influence the motion of the position of the particle.However, at the quantum level, the propagator, even in the position representation, depends in an intricate way on these classically irrelevant parameters.This creates considerable doubt as to the validity of such a quantization procedure, as the physical results predicted differ for various Hamiltonians, all of which describe the dissipative dynamics equally well.