On the Quantum-Mechanical Fokker—Planck Equation

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Abstract
Taking the Wigner function as the basic N‐body distribution function, quantum‐mechanical modifications to the Fokker—Planck equation are studied using the two different approaches introduced, respectively, by Kirkwood and Prigogine. In Sec. II the weak coupling limit is studied using the time‐smoothing technique first introduced by Kirkwood; the results of this section can perhaps form the basis of an approximate treatment of the transport phenomena of pure liquids. In Sec. III we treat the problem of the Brownian motion of a heavy particle employing the perturbation techniques developed by Prigogine and co‐workers; in this section it is found that the form of the quantum‐mechanical Fokker—Planck equation is identical to the classical equation, but that the friction coefficient contains quantum‐mechanical corrections. Finally, it is noted that the weak coupling result obtained by the time‐smoothing technique is identical to the approximation of small momentum transfer (Sec. III), if in the latter case the scattering cross section is represented in the Born approximation.