# The von Neumann model of measurement in quantum mechanics

## Abstract

We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called 'Wigner's formula' emerges in the strong-coupling limit, while 'Kirkwood's quasi-probability distribution' is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.

- Authors:

- Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México, D. F. (Mexico)

- Publication Date:

- OSTI Identifier:
- 22264080

- Resource Type:
- Journal Article

- Journal Name:
- AIP Conference Proceedings

- Additional Journal Information:
- Journal Volume: 1575; Journal Issue: 1; Conference: Latin-American school of physics Marcos Moshinsky Elaf: Nonlinear dynamics in Hamiltonian systems, Mexico City (Mexico), 22 Jul - 2 Aug 2013; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0094-243X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; CORRELATIONS; COUPLING; DENSITY; INTERACTIONS; PROBABILITY; PROBES; STRONG-COUPLING MODEL

### Citation Formats

```
Mello, Pier A.
```*The von Neumann model of measurement in quantum mechanics*. United States: N. p., 2014.
Web. doi:10.1063/1.4861702.

```
Mello, Pier A.
```*The von Neumann model of measurement in quantum mechanics*. United States. doi:10.1063/1.4861702.

```
Mello, Pier A. Wed .
"The von Neumann model of measurement in quantum mechanics". United States. doi:10.1063/1.4861702.
```

```
@article{osti_22264080,
```

title = {The von Neumann model of measurement in quantum mechanics},

author = {Mello, Pier A.},

abstractNote = {We describe how to obtain information on a quantum-mechanical system by coupling it to a probe and detecting some property of the latter, using a model introduced by von Neumann, which describes the interaction of the system proper with the probe in a dynamical way. We first discuss single measurements, where the system proper is coupled to one probe with arbitrary coupling strength. The goal is to obtain information on the system detecting the probe position. We find the reduced density operator of the system, and show how Lüders rule emerges as the limiting case of strong coupling. The von Neumann model is then generalized to two probes that interact successively with the system proper. Now we find information on the system by detecting the position-position and momentum-position correlations of the two probes. The so-called 'Wigner's formula' emerges in the strong-coupling limit, while 'Kirkwood's quasi-probability distribution' is found as the weak-coupling limit of the above formalism. We show that successive measurements can be used to develop a state-reconstruction scheme. Finally, we find a generalized transform of the state and the observables based on the notion of successive measurements.},

doi = {10.1063/1.4861702},

journal = {AIP Conference Proceedings},

issn = {0094-243X},

number = 1,

volume = 1575,

place = {United States},

year = {2014},

month = {1}

}