# Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field

## Abstract

We have solved the Heisenberg equation of motion for the time evolution of the position and momentum operators for a non-relativistic spinless charged particle in the presence of a weakly non-uniform electric and magnetic field. It is shown that the drift velocity operator obtained in this study agrees with the classical counterpart, and that, using the time dependent operators, the variances in position and momentum grow with time. The expansion rate of variance in position and momentum are dependent on the magnetic gradient scale length, however, independent of the electric gradient scale length. In the presence of a weakly non-uniform electric and magnetic field, the theoretical expansion rates of variance expansion are in good agreement with the numerical analysis. It is analytically shown that the variance in position reaches the square of the interparticle separation, which is the characteristic time much shorter than the proton collision time of plasma fusion. After this time, the wavefunctions of the neighboring particles would overlap, as a result, the conventional classical analysis may lose its validity. The broad distribution of individual particle in space means that their Coulomb interactions with other particles become weaker than that expected in classical mechanics.

- Authors:

- Graduate School of Engineering, Hokkaido University, Sapporo 060-8628 (Japan)
- Faculty of Engineering, Hokkaido University, Sapporo 060-8628 (Japan)

- Publication Date:

- OSTI Identifier:
- 22599908

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Plasmas; Journal Volume: 23; Journal Issue: 8; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; CHARGED PARTICLES; CLASSICAL MECHANICS; COLLISIONS; DISTRIBUTION; EQUATIONS OF MOTION; EXPANSION; LENGTH; MAGNETIC FIELDS; NUMERICAL ANALYSIS; PARTICLES; PLASMA; PROTONS; QUANTUM MECHANICS; RELATIVISTIC RANGE; TIME DEPENDENCE; WAVE FUNCTIONS

### Citation Formats

```
Chan, Poh Kam, Kosaka, Wataru, and Oikawa, Shun-ichi.
```*Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field*. United States: N. p., 2016.
Web. doi:10.1063/1.4960834.

```
Chan, Poh Kam, Kosaka, Wataru, & Oikawa, Shun-ichi.
```*Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field*. United States. doi:10.1063/1.4960834.

```
Chan, Poh Kam, Kosaka, Wataru, and Oikawa, Shun-ichi. Mon .
"Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field". United States.
doi:10.1063/1.4960834.
```

```
@article{osti_22599908,
```

title = {Quantum mechanical expansion of variance of a particle in a weakly non-uniform electric and magnetic field},

author = {Chan, Poh Kam and Kosaka, Wataru and Oikawa, Shun-ichi},

abstractNote = {We have solved the Heisenberg equation of motion for the time evolution of the position and momentum operators for a non-relativistic spinless charged particle in the presence of a weakly non-uniform electric and magnetic field. It is shown that the drift velocity operator obtained in this study agrees with the classical counterpart, and that, using the time dependent operators, the variances in position and momentum grow with time. The expansion rate of variance in position and momentum are dependent on the magnetic gradient scale length, however, independent of the electric gradient scale length. In the presence of a weakly non-uniform electric and magnetic field, the theoretical expansion rates of variance expansion are in good agreement with the numerical analysis. It is analytically shown that the variance in position reaches the square of the interparticle separation, which is the characteristic time much shorter than the proton collision time of plasma fusion. After this time, the wavefunctions of the neighboring particles would overlap, as a result, the conventional classical analysis may lose its validity. The broad distribution of individual particle in space means that their Coulomb interactions with other particles become weaker than that expected in classical mechanics.},

doi = {10.1063/1.4960834},

journal = {Physics of Plasmas},

number = 8,

volume = 23,

place = {United States},

year = {Mon Aug 15 00:00:00 EDT 2016},

month = {Mon Aug 15 00:00:00 EDT 2016}

}