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.

Quantum mechanical plasma diffusion is studied using a semi-classical model with two different characteristic lengths; one is the average interparticle separation, and the other is the magnetic length. The diffusion coefficients D derived in this study show a dependence on several plasma parameters, such as temperature T , mass m, density n, and magnetic field B, similar to that observed experimentally. The numerical values of the diffusion coefficient D in this study are as large as that for neoclassical diffusion. We have pointed out in this study that (i) for distant encounters in typical fusion plasmas of T = 10 keV and n = 10 20 m −3 , the average potential energy U ∼ 30 meV is as small as the uncertainty in energy ΔE ∼ 40 meV, and (ii) for a magnetic field B = 3 T, the spatial size of the wavefunction in the plane perpendicular to the magnetic field is as large as B ∼ 1.4 × 10 −8 m, which is much larger than the typical electron wavelength λ e ∼ 10 −11 m.

We have solved the two-dimensional time-dependent Schödinger equation for a particle with and without the interparticle potential in a fusion plasma. It was shown that spatial extent of a free particle grows monotonically in time. Such expansion leads to a spatial extent or size of a proton of the order of the average interparticle separation Δ ≡ n −1/3 ∼ 2 × 10 −7 m in a time interval of 10 6 × Δ /v th ∼ 10 −7 sec for a plasma with a density n ∼ 10 20 m −3 and a temperature T = mv 2 th /2 ∼ 10 keV. It was also shown that, under a Coulomb potential, the wavefunction of a charged particle first shrink and expand in time. In the expansion phase, at times t ≥ 10 −10 sec, the size of particle in the presence of a Coulomb potential is much larger than that in the absence of it.

We have constructed a semiclassical collisional diffusion model. In this model, a field particle is treated as either a point charge or a spatially distributed charge. The test particle is treated as a distributed point charge with Gaussian distribution. It was shown that the collisional changes in velocity in our model is of the same order as the classical theory for a typical proton in a fusion plasma of T = 10 keV and n = 10 20 m −3 . It was also shown that the spatial extent of the distribution, or the quantum-mechanical uncertainty in position, for the test particle obtained in our model grows in time, and becomes of the order of the average interparticle separation Δ ≡ n −1/3 during a time interval τ r ∼ × 10 7 Δ /g th , where g th = √ 2T/m is the thermal speed, with m being the mass of the particle under consideration. The time interval is 3-4 order of magnitudes smaller than the collision time. This suggests that particle transport cannot be understood in the framework of classical mechanics, and that the quantum-mechanical distribution of individual particles in plasmas may cause the anomalous diffusion.