dorsal/arxiv
View SchemaTime-of-arrival probabilities and quantum measurements
| Authors | Charis Anastopoulos, Ntina Savvidou |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0509020 |
| URL | https://arxiv.org/abs/quant-ph/0509020 |
| DOI | 10.1063/1.2399085 |
| Journal | J. Math. Phys. 47, 122106 (2006) |
Abstract
We study the construction of probability densities for time-of-arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time-of-arrival appear naturally when one considers histories. The definition of time-of-arrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of Schr\"odinger's equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in history-based schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit--there are however problems related to the quantum Zeno effect and decoherence. We finally employ the histories-based description to construct Positive-Operator-Valued-Measures (POVMs) for the time-of-arrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.
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"abstract": "We study the construction of probability densities for time-of-arrival in\nquantum mechanics. Our treatment is based upon the facts that (i) time appears\nin quantum theory as an external parameter to the system, and (ii) propositions\nabout the time-of-arrival appear naturally when one considers histories. The\ndefinition of time-of-arrival probabilities is straightforward in stochastic\nprocesses. The difficulties that arise in quantum theory are due to the fact\nthat the time parameter of Schr\\\"odinger\u0027s equation does not naturally define a\nprobability density at the continuum limit, but also because the procedure one\nfollows is sensitive on the interpretation of the reduction procedure. We\nconsider the issue in Copenhagen quantum mechanics and in history-based schemes\nlike consistent histories. The benefit of the latter is that it allows a proper\npassage to the continuous limit--there are however problems related to the\nquantum Zeno effect and decoherence. We finally employ the histories-based\ndescription to construct Positive-Operator-Valued-Measures (POVMs) for the\ntime-of-arrival, which are valid for a general Hamiltonian. These POVMs\ntypically depend on the resolution of the measurement device; for a free\nparticle, however, this dependence cancels in the physically relevant regime\nand the POVM coincides with that of Kijowski.",
"arxiv_id": "quant-ph/0509020",
"authors": [
"Charis Anastopoulos",
"Ntina Savvidou"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.2399085",
"journal_ref": "J. Math. Phys. 47, 122106 (2006)",
"title": "Time-of-arrival probabilities and quantum measurements",
"url": "https://arxiv.org/abs/quant-ph/0509020"
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