View article

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 (ℏ 4) and in the classical limit, ℏ→ 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.