Quantum-Mechanical Description of a Brownian Particle

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Abstract
It is shown that the motion of a Brownian particle in the Smoluchowski approximation may be described by a Schrödinger‐like equation defining a complex probability amplitude whose norm is the same as the stochastic probability density. Furthermore, the quantum dynamical operators have a physical meaning which arises in a natural way, from the stochastic nature of the process. These operators satisfy the usual commutation relations and thus the uncertainty principle. Here the constant ℏ is replaced by a parameter depending on the characteristics of the system. In particular, the potential‐energy operator for a Brownian particle subject to no external forces can be interpreted as a Rayleigh dissipative function.

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