Path-integral approach to the Schrödinger current

A. Marchewka and Z. Schuss
Phys. Rev. A 61, 052107 – Published 10 April 2000
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Abstract

Discontinuous initial wave functions, or wave functions with a discontinuous derivative and a bounded support, arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schrödinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a unidirectional current at the boundary of the support. We use path integrals to define current and unidirectional current, and to provide a direct derivation of the expression for current from the path-integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short-time propagation of an initial wave function with compact support for cases of both a discontinuous derivative and a discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time Δt is O(Δt3/2), and the initial unidirectional current is O(Δt1/2). This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time Δt is O(Δt1/2), and the initial unidirectional current is O(Δt1/2). This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is O(Δt). This gives a decay law.

  • Received 10 November 1999

DOI:https://doi.org/10.1103/PhysRevA.61.052107

©2000 American Physical Society

Authors & Affiliations

A. Marchewka*

  • Department of Physics, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel

Z. Schuss

  • Department of Mathematics, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel

  • *Electronic address: ntmarche@wisemail.weizmann.ac.il
  • Electronic address: schuss@math.tau.ac.il

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Vol. 61, Iss. 5 — May 2000

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