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

Abstract

We have solved the two-dimensional time-dependent Schrödinger equation for a single particle in the presence of a non-uniform magnetic field for initial speed of 10-100 m/s, mass of the particle at 1-10 m p , where m p is the mass of a proton. Magnetic field at the origin of 5-10 T, charge of 1-4 e, where e is the charge of the particle and gradient scale length of 2.610 × 10 −5 -5.219 m. It was numerically found that the variance, or the uncertainty, in position can be expressed as dσ 2 r /dt = 4.1 v 0 /qB 0 L B , where m is the mass of the particle, q is the charge, v 0 is the initial speed of the corresponding classical particle, B 0 is the magnetic field at the origin and L B is the gradient scale length of the magnetic field. In this expression, we found out that mass, m does not affect our newly developed expression.

Quantum mechanical grad-B drift velocity operator in a weakly non-uniform magnetic field

“…The two-dimensional Schr€ odinger equation is solved numerically [29][30][31][32][33] and theoretically for a wavefunction w at position r and time t…”

“…The diffusion, however, is a quadratic function change, such as Dp 2 and Dr 2 , and is not properly accounted for in the existing classical and neoclassical theories. Corresponding to these facts, the authors conducted the quantum mechanical analyses [29][30][31][32][33] on a single charged particle in the presence of external electromagnetic fields, focusing especially on the time development of variance in position and momentum. For the plasma mentioned above, the deviation r r ðtÞ of the ions would reach the interparticle separation n À1=3 in a time interval of the order of 10 À4 s. After this time, the wavefunctions of neighboring particles would overlap, as a result the conventional classical analysis may lose its validity.…”

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.…”