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.

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.

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.

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 8 -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. Previously, we found out that the variance, or the uncertainty, in position can be expressed as dσwhere 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 research, it was numerically found that the variance, or the uncertainty, in total momentum can be expressed as dσ 2 P /dt = 0.57 qB 0 v 0 /L B . In this expression, we found out that mass, m does not affect both our newly developed expression for uncertainty in position and total momentum.

We have solved the Heisenberg equation of motion for the time evolution of the position and momentum operators for a non-relativistic spinless charged particle in the presence of a weakly non-uniform 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 non-uniform 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.