Quantum-Mechanical, Microscopic Brownian Motion
J. McKenna, H. L. Frisch
Recently, the Fokker-Planck (F-P) equation, satisfied by the phase-space distribution function of a Brownian (B) particle, has been derived from the Liouville equation, satisfied by the phase-space distribution function of the composite system, B particle plus fluid. We have started from a strictly quantum-mechanical formulation of the problem, using a density-matrix description of the composite system, and have shown that in an appropriate classical limit the F-P equation results. To lowest order in $\ensuremath{\hbar}$, our technique gives the same expression for the friction coefficient obtained earlier from the Liouville equation. In principle it yields corrections to the F-P equation to all orders in $\ensuremath{\hbar}$. We have shown that for a uniform fluid, the quantum corrections to the F-P equation to first order in $\ensuremath{\hbar}$ appear only as a shift in the friction coefficient. If the classical friction coefficient ${\ensuremath{\zeta}}_{0}$ is replaced by ${\ensuremath{\zeta}}_{0}+\ensuremath{\hbar}{\ensuremath{\zeta}}_{1}$, with an appropriate ${\ensuremath{\zeta}}_{1}$, then the F-P equation is given correctly to order $\ensuremath{\hbar}$. There is a difference in the mass dependence of ${\ensuremath{\zeta}}_{0}$ and ${\ensuremath{\zeta}}_{1}$ which suggests the existence of an isotope effect in the mobility of a particle moving through a quantum fluid. The technique employed in the derivation is to consider the expectation value of the density matrix of the system in a state in which the position and momentum of the particles are known with maximum accuracy (Kennard packet). This expectation value $\ensuremath{\mu}$ is an appropriate, positive, semiclassical phase-space distribution function. An appropriate equation is derived for $\ensuremath{\mu}$, which to lowest order in $\ensuremath{\hbar}$ is the Liouville equation. The remainder of the derivation is similar to the derivation in the classical case (a suitable projection technique is used).
Physical Review•May 6, 1966•38 citations