  Einstein's Brownian motion of a quantum particle in a classical environment
is studied via virial and equipartition theorems. The effect of continuous
measurement in a strongly dissipative environment is accounted for and a
quantum generalization of the classical Einstein law of Brownian motion is
obtained. A thermo-quantum Smoluchowski diffusion equation is derived via a
generalization of the Madelung quantum hydrodynamics. The latter is applied for
description of the quantum tunneling at equilibrium and stationary states as
well as of the motion of an electron in metals, i.e. the Smoluchowski-Poisson
problem.
