Quantum friction in the c-number picture: The damped harmonic oscillator

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Abstract
Considering the Lagrangian proposed by Havas, that describes the classical damped motion of a particle, new momentum and position are defined in order to write a Hamiltonian that is subsequently quantized and expressed in terms of non‐Hermitian operators. Using the c‐number formalism proposed by Lax and Yuen, we associate to the quantum Liouville equation a Fokker–Planck one in terms of c‐numbers. From the properties of this equation we obtain the mean values of the position, momentum, and energy of a brownian particle and we also verify the uncertainty principle. We observe that when the system is considered under the Markov hypothesis, the stochastic force is intimately related to the uncertainty principle and to the zero point energy.