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Physical Review A

Propagator for two dispersing δ-function potentials

G. Scheitler, M. Kleber

Published Jul 1, 1990• 16 citations
Publisher

Abstract

A light particle is assumed to move in the field of two heavy particles that separate with constant velocity. The resulting field of the two-heavy fragments is approximated by two moving, one-dimensional zero-range potentials. The propagator for this problem is derived in closed form for arbitrary potential strengths. The resulting integral representation is suitable for numerical evaluation.

DOI
10.1103/physreva.42.55

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A light particle is assumed to move in the field of two heavy particles that separate with constant velocity. The resulting field of the two-heavy fragments is approximated by two moving, one-dimensional zero-range potentials. The propagator for this problem is derived in closed form for arbitrary potential strengths. The resulting integral representation is suitable for numerical evaluation.
The one-dimensional equation of motion for the canonical or Bloch density matrix is solved by the method of separation of variables for independent particles moving in a potential V(x). The solution is exact when third and higher derivatives of V(x) are zero, embracing thereby the examples of (a) harmonic potentials and (b) a constant electric field of arbitrary strength. The form of the Slater sum obtained involves then two functions of temperature only, to be determined. One obvious relation between these two functions is via the partition function. A further relation comes from the orthonormality of the wave functions appearing in the Slater sum: the analog of idempotency for the ground-state Dirac density matrix. These two conditions again are shown to lead back to the exact result for harmonic potentials. The results presented generalize the one-dimensional Thomas-Fermi theory to apply to a temperature range that is significantly wider than for the original method.