Brownian motion of a quantum harmonic oscillator

Julian Schwinger

Fabián H. Gaioli, Edgardo T. Garcia-Alvarez

Takeyasu Kotera, Morikazu Toda

The Brownian motion of a harmonic oscillator is treated quantum-mechanically and the same method is applied to the relaxation of a spin. The method is quite general and applicable to a system displaced from the equilibrium state by an arbitrary amount initially. The oscillator or the spin is interacting with a thermostat composed of a great number of harmonic oscillators. The equation of motion is solved in operator form taking average as to the thermostat to the second order of the interaction parameter. The Langevin equation, the Kramers-Chandra-sekhar equation for a harmonic oscillator and the Bloch equation for a static field are derived.

Xi-Zhen Wu, Zhuxia Li, Joachim Alexander Maruhn +2 more

Based on a simplified Hamiltonian model the transition from quantum to classical diffusion behaviours for a system has been shown by studying the dependence of the quantum Brownian motion in a harmonic oscillator potential on the viscosity and temperature of the system. The approximation of a locally harmonic oscillator and reduced density operator propagator in the harmonic oscillator are generalised to calculate the escape rate over the barrier for the system in a fission potential consisting of a ground-state well and barrier. Compared with the results of the Fokker-Planck equation the model in this paper is completely quantum mechanical.

Eugen Merzbacher

Stefan I. Tsonchev, Philip Pechukas

A binary collision model for phenomena of quantum dissipation is developed. Unlike the harmonic oscillator model, widely used for over thirty years, this model assumes nonlinear coupling between system and environment, and is applicable to both bosonic and fermionic baths. The system interacts with an ideal bath through binary collisions only. Solutions for the classical and quantum-mechanical problems in the case of free Brownian motion are presented, and the quantum-classical correspondence for nonequilibrium processes is established. It is shown that in the Brownian motion limit the two models lead to identical dynamical behavior, provided the coupling coefficients in the harmonic oscillator Hamiltonian are temperature dependent. For cases of bath particles of finite mass and number the two models lead to different results. Linear response theory for the model is developed, and the results are compared with those for the harmonic oscillator model. At the end, possible applications of the model are suggested.