The damped linear and the forced harmonic oscillator are used as standard examples for a dynamical system with a time-dependent Hamilton function to investigate the problem of constructing a Hilbert state space and evolution operators in this space.
Harmonic oscillators are important in physical applications and in demonstrating the working of the postulates of quantum formalism. A classical harmonic oscillator has a state space, i.e., its phase space, coordinated by the oscillator’s position x and momentum p . Observables correspond to real-valued functions on the state space. The evolution of the state is determined by Hamilton’s equations, which reduce to Eq. (27.12) . The time dependence of the state automatically determines the time dependence of observables.
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.
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.