  Quantum diffusion is studied via dissipative Madelung hydrodynamics.
Initially the wave packet spreads ballistically, than passes for an instant
through normal diffusion and later tends asymptotically to a sub-diffusive law.
It is shown that the apparent quantum diffusion coefficient is not a universal
physical parameter since it depends on the initial wave packet preparation. The
overdamped quantum diffusion of an electron in the field of a periodic
potential is also investigated; in this case the wave packet spreads
logarithmically in time. Thermo-quantum diffusion of heavier particles as
hydrogen, deuterium and tritium atoms in periodic potentials is studied and a
simple estimate of the tunneling effect is obtained in the frames of a
quasi-equilibrium semiclassical approach. The effective thermo-quantum
temperature is also discussed in relation to the known temperature dependence
of muon diffusivity in solids.
