Numerical Analysis of Quantum Mechanical ∇B Drift

Abstract

We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a nonuniform magnetic field for initial speeds of 10-100 m/s. By linear extrapolation, it is shown that the variance, or the uncertainty, in position would reach the square of the interparticle separation n −2/3 with a number density of n = 10 20 m −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: Plasmas may behave more-or-less like extremely-low-density liquids, not gases, since the size of each particle is of the same order of the interparticle separation.

Numerical Analysis of Quantum Mechanical ∇<i>B </i>Drift II

“…This normalized grid size is sufficiently small to use as noted in Ref. [1]. Table 1 Normalized parameter for mass of the particle, charge, magnetic flux density, velocity, length and time.…”

“…In the previous paper [1], we have shown that the quantum mechanical variance in position may reach the square of the interparticle separation in a time interval of the order of 10 −4 sec for typical magnetically confined fusion plasmas with a number density of n ∼ 10 20 m −3 and a temperature of T ∼ 10 keV.…”

“…We use successive over relaxation (SOR) scheme for time integration in our numerical calculation. Calculation is done on a GPU (Nvidia GTX-580: 512 cores/3 GB @1.54 GHz) [1,2].…”

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].…”

“…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.…”