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Stochastic Schrödinger equation for a quantum oscillator with dissipation. (English. Russian original) Zbl 1108.82025

Math. Notes 78, No. 6, 867-873 (2005); translation from Mat. Zametki 78, No. 6, 934-940 (2005).
Summary: We construct an exact solution of the stochastic Schrödinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the \(Q\)-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrödinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.

MSC:

82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
81S25 Quantum stochastic calculus
35R60 PDEs with randomness, stochastic partial differential equations
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics

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References:

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