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arXiv:hep-th/9404163 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Detection of particles under a potential barrier

Alexander Vilenkin, Serge Winitzki

Published Oct 15, 1994• 3 citations
arXivHTMLPDF

Abstract

We introduce a model detector which registers the passage of a particle through the detector location, without substantially perturbing the particle wave function. (The exact time of passage is not determined in such measurements.) We then show that our detector can operate in a classically forbidden reigon and register particles passing through a certain point under a potential barrier. We show that it should be possible to observe the particle's track under the barrier.

DOI
10.1103/physrevd.50.5409

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We introduce a model detector which registers the passage of a particle through the detector location, without substantially perturbing the particle wave function. (The exact time of passage is not determined in such measurements.) We then show that our detector can operate in a classically forbidden reigon and register particles passing through a certain point under a potential barrier. We show that it should be possible to observe the particle's track under the barrier.
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We investigate the short-time dynamics of a delta-function potential barrier on an initially confined wave packet. There are mainly two conclusions: (A) At short times the probability density of the first particles that passed through the barrier is unaffected by it. (B) When the barrier is absorptive (i.e., its potential is imaginary) it affects the transmitted wave function at shorter times than a real potential barrier. Therefore, it is possible to distinguish between an imaginary and a real potential barrier by measuring its effect at short times only on the transmitting wave function.
We consider a particular model of a Source of independent particles and a macroscopic Detector that are both tuned to the same resonance frequency ν0≡1/P. Particles are emitted by the Source at exact multiples of the resonance period P, and the Detector absorbs them with a certain probability at any one of its points. The Detector may also announce the detection of the absorbed particle. Any particle that is not absorbed at a certain point passes through to a deeper layer in the interior of the Detector. Eventually, all particles will be absorbed, i.e., detected. We calculate the probability of detection for two particle time series generated by the same Source reaching the Detector with a time delay of δt and show that it manifests the illusion of collective (wave-like) interference with particle number conservation. We conclude that wave phenomena may reflect the nature of detectors rather than fundamental laws of physics.
Introduction In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons. Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [ a, b ]. Three kinds of time have been defined in this context. One, called the transmission time τ T ( a, b ), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τ R ( a, b ) is the average time spent within the barrier region by the particles that are eventually reflected.
A model for a one-dimensional quantum particle interacting with a point detector is developed and analyzed. The analysis leads to any simple, intuitively pleasing expression for the final density operator of the particle in terms of the freely propagating particle density operator, a projection operator representing the detector volume, and a probability density for the time at which the detector fires. This probability density is found explicitly to be the presence-time density given by ${\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}∕\ensuremath{\int}{\ensuremath{\mid}{\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)\ensuremath{\mid}}^{2}dt$ where ${\ensuremath{\psi}}_{f}\phantom{\rule{0.1em}{0ex}}(0,t)$ is the free particle wave function evaluated at the detector position (x=0)(x=0)(x=0). Moreover, it is shown that the resultant density operator transformation is a nonlinear transformation, not describable as a positive operator valued measure.