dorsal/arxiv
View SchemaTime-of-arrival in quantum mechanics
| Authors | Norbert Grot, Carlo Rovelli, Ranjeet S. Tate |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9603021 |
| URL | https://arxiv.org/abs/quant-ph/9603021 |
| DOI | 10.1103/PhysRevA.54.4676 |
| Journal | Phys.Rev. A54 (1996) 4679 |
Abstract
We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\"odinger current. We derive a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time-representation'' of the quantum mechanics of a free particle, in which the time-of-arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics.
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"abstract": "We study the problem of computing the probability for the time-of-arrival of\na quantum particle at a given spatial position. We consider a solution to this\nproblem based on the spectral decomposition of the particle\u0027s (Heisenberg)\nstate into the eigenstates of a suitable operator, which we denote as the\n``time-of-arrival\u0027\u0027 operator. We discuss the general properties of this\noperator. We construct the operator explicitly in the simple case of a free\nnonrelativistic particle, and compare the probabilities it yields with the ones\nestimated indirectly in terms of the flux of the Schr\\\"odinger current. We\nderive a well defined uncertainty relation between time-of-arrival and energy;\nthis result shows that the well known arguments against the existence of such a\nrelation can be circumvented. Finally, we define a ``time-representation\u0027\u0027 of\nthe quantum mechanics of a free particle, in which the time-of-arrival is\ndiagonal. Our results suggest that, contrary to what is commonly assumed,\nquantum mechanics exhibits a hidden equivalence between independent (time) and\ndependent (position) variables, analogous to the one revealed by the\nparametrized formalism in classical mechanics.",
"arxiv_id": "quant-ph/9603021",
"authors": [
"Norbert Grot",
"Carlo Rovelli",
"Ranjeet S. Tate"
],
"categories": [
"quant-ph",
"gr-qc"
],
"doi": "10.1103/PhysRevA.54.4676",
"journal_ref": "Phys.Rev. A54 (1996) 4679",
"title": "Time-of-arrival in quantum mechanics",
"url": "https://arxiv.org/abs/quant-ph/9603021"
},
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