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

TL;DR

This paper presents a concise phase space method for deriving the propagator of Hamiltonians with spatially-dependent kinetic energy, including position-dependent mass and curved space scenarios, simplifying complex quantum mechanical problems.

Contribution

It introduces a unified phase space approach to derive propagators for Hamiltonians with position-dependent kinetic energy, including arbitrary ordering and curved space cases.

Findings

Derived propagator for Hamiltonians with spatially-dependent kinetic energy.

Unified scheme for constant-mass and curved space cases.

Applicable to arbitrary discretization choices.

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.