# Structure and properties of Hughston's stochastic extension of the Schroedinger equation

## Abstract

Hughston has recently proposed a stochastic extension of the Schroedinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics. (c) 2000 American Institute of Physics.

- Authors:

- Institute for Advanced Study, Princeton, New Jersey 08540 (United States)

- Publication Date:

- OSTI Identifier:
- 20216302

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 41; Journal Issue: 5; Other Information: PBD: May 2000; Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; SCHROEDINGER EQUATION; STOCHASTIC PROCESSES; HILBERT SPACE; EIGENSTATES; DENSITY MATRIX; QUANTUM MECHANICS; THEORETICAL DATA

### Citation Formats

```
Adler, Stephen L., and Horwitz, Lawrence P.
```*Structure and properties of Hughston's stochastic extension of the Schroedinger equation*. United States: N. p., 2000.
Web. doi:10.1063/1.533255.

```
Adler, Stephen L., & Horwitz, Lawrence P.
```*Structure and properties of Hughston's stochastic extension of the Schroedinger equation*. United States. doi:10.1063/1.533255.

```
Adler, Stephen L., and Horwitz, Lawrence P. Mon .
"Structure and properties of Hughston's stochastic extension of the Schroedinger equation". United States. doi:10.1063/1.533255.
```

```
@article{osti_20216302,
```

title = {Structure and properties of Hughston's stochastic extension of the Schroedinger equation},

author = {Adler, Stephen L. and Horwitz, Lawrence P.},

abstractNote = {Hughston has recently proposed a stochastic extension of the Schroedinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics. (c) 2000 American Institute of Physics.},

doi = {10.1063/1.533255},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 5,

volume = 41,

place = {United States},

year = {2000},

month = {5}

}