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Physical Review A

Green’s functions via path integrals for systems with position-dependent masses

L. Chétouani, Liès Dekar, Théophile F. Hammann

Published Jul 1, 1995• 72 citations
Publisher

Abstract

The position-dependent mass Green's function is connected to the constant-mass Green's function by means of a time and space rescaling. Application is made for step and rectangular-barrier potentials and masses. Reflection and transmission coefficients are deduced from the continuity conditions.
DOI
10.1103/physreva.52.82

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The path integral for the propagator is expanded into a perturbation series, which can be exactly summed in the case of \ensuremath{\delta}-function perturbations giving a closed expression for the (energy-dependent) Green function. Making the strength of the \ensuremath{\delta}-function perturbation infinite repulsive produces a totally reflecting boundary, hence giving a path integral solution in half spaces in terms of the corresponding Green function. The example of the Wood-Saxon potential serves by an appropriate limiting procedure to obtain the Green function for the step potential and the finite potential well in the half space, respectively.
The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.
The use of Green's functions is valuable when solving problems in electrodynamics, solid-state physics, and many-body physics. However, its role in quantum mechanics is often limited to the context of scattering by a central force. This work shows how Green's functions can be used in other examples in quantum mechanics courses. In particular, we introduce time-independent Green's functions and the Dyson equation to solve problems with an external potential. We calculate the reflection and transmission coefficients of scattering by a Dirac delta barrier and the energy levels and local density of states of the infinite square well potential.