dorsal/arxiv
View SchemaA Path Intergal Approach to Current
| Authors | A. Marchewka, Z. Schuss |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9903076 |
| URL | https://arxiv.org/abs/quant-ph/9903076 |
| DOI | 10.1103/PhysRevA.61.052107 |
| Journal | Phys.Rev.A61:052107,2000 |
Abstract
Discontinuous initial wave functions or wave functions with discontintuous derivative and with 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 Schr\"odinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a uni-directional current at the boundary of the support. We use path integrals to define current and uni-directional current and give 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 initial wave function with compact support for both the cases of discontinuous derivative and discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\Delta t$ is $O(\Delta t^{3/2})$ and the initial uni-directional current is $O(\Delta t^{1/2})$. 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 $\Delta t$ is $O(\Delta t^{1/2})$ and the initial uni-directional current is $O(\Delta t^{-1/2})$. 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 $O(\Delta t)$. This gives a decay law.
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"abstract": "Discontinuous initial wave functions or wave functions with discontintuous\nderivative and with bounded support arise in a natural way in various\nsituations in physics, in particular in measurement theory. The propagation of\nsuch initial wave functions is not well described by the Schr\\\"odinger current\nwhich vanishes on the boundary of the support of the wave function. This\npropagation gives rise to a uni-directional current at the boundary of the\nsupport. We use path integrals to define current and uni-directional current\nand give a direct derivation of the expression for current from the path\nintegral formulation for both diffusion and quantum mechanics. Furthermore, we\ngive an explicit asymptotic expression for the short time propagation of\ninitial wave function with compact support for both the cases of discontinuous\nderivative and discontinuous wave function. We show that in the former case the\nprobability propagated across the boundary of the support in time $\\Delta t$ is\n$O(\\Delta t^{3/2})$ and the initial uni-directional current is $O(\\Delta\nt^{1/2})$. This recovers the Zeno effect for continuous detection of a particle\nin a given domain. For the latter case the probability propagated across the\nboundary of the support in time $\\Delta t$ is $O(\\Delta t^{1/2})$ and the\ninitial uni-directional current is $O(\\Delta t^{-1/2})$. This is an anti-Zeno\neffect. However, the probability propagated across a point located at a finite\ndistance from the boundary of the support is $O(\\Delta t)$. This gives a decay\nlaw.",
"arxiv_id": "quant-ph/9903076",
"authors": [
"A. Marchewka",
"Z. Schuss"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.61.052107",
"journal_ref": "Phys.Rev.A61:052107,2000",
"title": "A Path Intergal Approach to Current",
"url": "https://arxiv.org/abs/quant-ph/9903076"
},
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