Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.

The quantum Brownian motion of a particle in an external potential V (x) is treated using the master equation for the Wigner distribution function W (x, p, t) in phase space (x, p). A heuristic method of determination of diffusion coefficients in the master equation is proposed. The time evolution equation so obtained contains explicit quantum correction terms up to o(h 4 ) and in the classical limit, h → 0, reduces to the Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation for W (x, p, t) coinciding with that of Agarwal (1971 Phys. Rev. A 4 739). In the non-inertial regime, by applying the Brinkman expansion of the momentum distribution in Weber functions (Brinkman 1956 Physica 22 29), the corresponding semiclassical Smoluchowski equation is derived.

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The quantum Brownian motion of a particle in a cosine periodic potential V͑x͒ =−V 0 cos͑x / x 0 ͒ is treated using the master equation for the time evolution of the Wigner distribution function W͑x , p , t͒ in phase space ͑x , p͒. The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.

The translational Brownian motion of a particle in a tilted washboard potential is considered. The dynamic structure factor and longest relaxation time are evaluated from the solution of the governing Langevin equation by using the matrix continued fraction method. The longest relaxation time is compared with the Kramers theory of the escape rate of a Brownian particle from a potential well as extended to the Kramers turnover region by Mel'nikov [Physics Reports 209, 1 (1991)]. It is shown that in the low temperature limit, the universal Mel'nikov expression for the escape rate provides a good estimate of the longest relaxation time for all values of dissipation including the very low damping (VLD), very high damping (VHD), and turnover regimes. For low barriers (where the Mel'nikov method is not applicable) and zero tilt, analytic equations for the relaxation times in the VLD and VHD limits are derived.