Action-angle variables and fluctuation-dissipation relations for a driven quantum oscillator
S. Krinsky · Physical Review A · 1985
For a classical oscillator subject to a time-dependent perturbation, the average change in the action variable 〈\ensuremath{\Delta}J〉=〈J-${J}_{0}$〉 is related to its fluctuation 〈(\ensuremath{\Delta}J${)}^{2}$〉 by 〈\ensuremath{\Delta}J〉=(1/2)\ensuremath{\partial}〈(\ensuremath{\Delta}J${)}^{2}$〉/\ensuremath{\partial}${J}_{0}$, where ${J}_{0}$ is the initial value of J and the average is over the initial value ${\ensuremath{\theta}}_{0}$ of the angle variable. In this paper, the quantum-mechanical generalization of this relation is derived, and we discuss the correspondence between our work and the usual treatment of the fluctuation-dissipation theorem for a system which initially is described by a canonical distribution.