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Journal of Mathematical Physics

Quantum mechanics of the damped pulsating oscillator

M. Sebawe Abdalla, Ursula A.T. Ramjit

Published Jan 1, 1989• 19 citations
Publisher

Abstract

A harmonic oscillator subject to the combined effects of damping and pulsating is represented by a Kanai–Caldirola Hamiltonian. The equations of motion are solved in the Heisenberg picture in the case of weak pulsation. The rotating-wave approximation (RWA) is used to obtain the motion in the neighborhood of the principal resonance. The RWA Schrödinger equation is solved exactly and pseudostationary and quasicoherent states are described. The transition probability between quasicoherent and coherent states is obtained and the gain in energy is discussed.

DOI
10.1063/1.528590

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•Jan 1, 2011•0 citations
A harmonic oscillator subject to the combined effects of damping and pulsating is represented by a Kanai–Caldirola Hamiltonian. The equations of motion are solved in the Heisenberg picture in the case of weak pulsation. The rotating-wave approximation (RWA) is used to obtain the motion in the neighborhood of the principal resonance. The RWA Schrödinger equation is solved exactly and pseudostationary and quasicoherent states are described. The transition probability between quasicoherent and coherent states is obtained and the gain in energy is discussed.
A canonical description of a harmonic oscillator with energy dissipation is sought which combines the advantages of the Kanai-Caldirola Hamiltonian and a simple model of strangulation previously considered by the authors. The treatment is in the Heisenberg picture of quantum mechanics or alternatively in classical mechanics. A strangulation \ensuremath{\beta} is superimposed on the damping (or growth) \ensuremath{\gamma}, represented by an oscillator mass m(t)=m0{m}_{0}m0​exp(2\ensuremath{\gamma}t). In general, the expected dissipation of energy occurs only if \ensuremath{\beta}>\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert}. If \ensuremath{\beta}<\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert} the attempted strangulation induces a long-term growth in energy unless a special initial motion is chosen. A slight deviation from this initial motion results in a temporary decay in energy followed by growth. \ensuremath{\beta}<0 always induces a growth in energy.
A quantum-mechanical model of a damped harmonic oscillator (both with time-independent and time-dependent parameters) is studied in the framework of the linear Schr\"odinger equation with a Hermitian nonstationary Hamiltonian. Integrals of the motion of this equation and their eigenstates, including coherent states, are constructed. The influence of an external harmonic force to the time evolution of various average values calculated over coherent states is considered, including the resonance case. The specific symmetry of the Hamiltonian leading to the new concept of loss-energy states is discussed.
We introduce a modified model for the problem of a damped harmonic oscillator in presence of a driving force. The model is treated from the point of view of quantum mechanics. The wave function in the Schr¨ odinger picture is obtained. The connection between the quasicoherent state and pseudostationary state is discussed. The constants of motion are also introduced and the eigenfunctions and the corresponding eigenvalues for pair of quadratic invariants are obtained. The phenomena of squeezing and the Poissonian distribution are considered. It is shown that the system is sensitive to the variation of the damping factor, ∞, as well as of the modulation factor, „.