Models of Damped Oscillators in Quantum Mechanics

TL;DR

This paper explores various models of quantum damped oscillators, deriving explicit Green functions, gauge transformations, and analyzing the time evolution of energy and position expectation values within a general quadratic Hamiltonian framework.

Contribution

It introduces explicit Green functions and gauge transformations for quantum damped oscillators, applying factorization techniques and analyzing expectation value dynamics.

Findings

Explicit Green functions in elementary functions

Time evolution of energy expectation values derived

Classical equations of motion obtained for expectation values

Abstract

We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.