# Time-of-arrival probabilities and quantum measurements

## Abstract

In this paper 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 the Schroedinger'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 withmore »

- Authors:

- Department of Physics, University of Patras, 26500 Patras (Greece)
- (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 20861552

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 12; Other Information: DOI: 10.1063/1.2399085; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; HAMILTONIANS; PROBABILITY; QUANTUM MECHANICS; SCHROEDINGER EQUATION; STOCHASTIC PROCESSES

### Citation Formats

```
Anastopoulos, Charis, Savvidou, Ntina, and Theoretical Physics Group, Imperial College, SW7 2BZ, London.
```*Time-of-arrival probabilities and quantum measurements*. United States: N. p., 2006.
Web. doi:10.1063/1.2399085.

```
Anastopoulos, Charis, Savvidou, Ntina, & Theoretical Physics Group, Imperial College, SW7 2BZ, London.
```*Time-of-arrival probabilities and quantum measurements*. United States. doi:10.1063/1.2399085.

```
Anastopoulos, Charis, Savvidou, Ntina, and Theoretical Physics Group, Imperial College, SW7 2BZ, London. Fri .
"Time-of-arrival probabilities and quantum measurements". United States.
doi:10.1063/1.2399085.
```

```
@article{osti_20861552,
```

title = {Time-of-arrival probabilities and quantum measurements},

author = {Anastopoulos, Charis and Savvidou, Ntina and Theoretical Physics Group, Imperial College, SW7 2BZ, London},

abstractNote = {In this paper 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 the Schroedinger'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.},

doi = {10.1063/1.2399085},

journal = {Journal of Mathematical Physics},

number = 12,

volume = 47,

place = {United States},

year = {Fri Dec 15 00:00:00 EST 2006},

month = {Fri Dec 15 00:00:00 EST 2006}

}