Quantum mechanical expansion of variance of a particle in a weakly nonuniform 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 nonrelativistic spinless charged particle in the presence of a weakly nonuniform 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 nonuniform 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 0608628 (Japan)
 Faculty of Engineering, Hokkaido University, Sapporo 0608628 (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, Shunichi. Quantum mechanical expansion of variance of a particle in a weakly nonuniform electric and magnetic field. United States: N. p., 2016.
Web. doi:10.1063/1.4960834.
Chan, Poh Kam, Kosaka, Wataru, & Oikawa, Shunichi. Quantum mechanical expansion of variance of a particle in a weakly nonuniform electric and magnetic field. United States. doi:10.1063/1.4960834.
Chan, Poh Kam, Kosaka, Wataru, and Oikawa, Shunichi. 2016.
"Quantum mechanical expansion of variance of a particle in a weakly nonuniform 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 nonuniform electric and magnetic field},
author = {Chan, Poh Kam and Kosaka, Wataru and Oikawa, Shunichi},
abstractNote = {We have solved the Heisenberg equation of motion for the time evolution of the position and momentum operators for a nonrelativistic spinless charged particle in the presence of a weakly nonuniform 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 nonuniform 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 = 2016,
month = 8
}

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