Numerical Analysis of Quantum-Mechanical Non-Uniform <i>E × B </i>Drift

Abstract

We have numerically solved the two-dimensional time-dependent Schödinger equation for a magnetized proton in the presence of a uniform electric field and a nonuniform magnetic field with a gradient scale length of L B . It is shown that the particle mass and the electric field do not affect the time rate of variance change at which variance increases with time, and their characteristic times are of the order of L B /v 0 sec with v 0 being the initial particle speed.

Numerical analysis of quantum-mechanical non-uniform E×B drift: Non-uniform electric field

“…(1) with an appropriate initial condition in x-y plane, using the finite difference method (FDM) in space with the Crank-Nicolson scheme [1][2][3][4][5].…”

“…Here, {ψ n } stands for the discretized wavefunction, the superscript n represents the time-label, I and H are the unit matrix and the numerical Hamiltonian matrix [1][2][3][4][5]. Assuming the Coulomb gauge ∇ · A = 0, the numerical Hamiltonian matrix H ≡ { H i, j } is written as follows,…”

“…The charged particles drift in the presence of a magnetic field B, the drifts include ∇B drift, curvature drift and E × B drift if there exist an electric field E. The two-dimensional time-dependent Schrödinger equation have been already solved for a charged particle in the presence of a non-uniform magnetic field and a uniform electric field, in which it was shown that the variance, or the uncertainty, in position σ 2 r (t) grows with time [1][2][3][4][5]. For the typical fusion plasma with a temperature T ∼ 10 keV and a number density of n ∼ 10 20 m −3 , the standard deviation σ r (t) would reach the interparticle separation n −1/3 in a time interval of the order of 10 −4 sec.…”

“…For the typical fusion plasma with a temperature T ∼ 10 keV and a number density of n ∼ 10 20 m −3 , the standard deviation σ r (t) would reach the interparticle separation n −1/3 in a time interval of the order of 10 −4 sec. After this time the wavefunctions of neighboring particles would overlap, as a result the conventional classical analysis may lose its validity [1]. In Ref.…”

“…In Ref. [1] mentioned above, the uniform electric field have been assumed. In this paper, quantum mechanical effects of a nonuniform electric field and a non-uniform magnetic field will be studied.…”