Numerical Analysis of Schrödinger Equation for a Magnetized Particle in the Presence of a Field Particle

Abstract

We have solved the two-dimensional time-dependent Schödinger equation for a magnetized proton in the presence of a fixed field particle with an electric charge of 2×10−5 e, where e is the elementary electric charge, and of a uniform megnetic field of B = 10 T. In the relatively high-speed case of v 0 = 100 m/s, behaviors are similar to those of classical ones. However, in the low-speed case of v 0 = 30 m/s, the magnitudes both in momentum mv = |mu|, where m is the mass and u is the velocity of the particle, and position r = |r| are appreciably decreasing with time. However, the kinetic energy K = m u 2 /2 and the potential energy U = qV , where q is the electric charge of the particle and V is the scalar potential, do not show appreciable changes. This is because of the increasing variances, i.e. uncertainty, both in momentum and position. The increment in variance of momentum corresponds to the decrement in the magnitude of momentum: Part of energy is transfered from the directional (the kinetic) energy to the uncertainty (the zero-point) energy.

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.…”