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Journal of Sound and Vibration

On a non-linear harmonic oscillator

M. Lakshmanan

Published Jun 1, 1979• 6 citations
Publisher
DOI
10.1016/0022-460x(79)90592-3

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Overdamping is a regime in which friction is sufficiently large that the motion either decays to its equilibrium position or it crosses the equilibrium position exactly once before returning monotonically towards the equilibrium position. The phenomena of overdamping has been studied classically and quantum mechanically only for the case of the linear damped harmonic oscillator. Here we study the classical and quantum dynamics of a family of over-damped non linear systems. The main objective of this paper is to find a Lagrangian and Hamiltonian framework to study over-damped non linear systems and to show that a quantum mechanical description can be developed in the momentum representation. Our results reduce to the well known solution of the linear damped harmonic oscillator when the non linear part is set to zero.
We introduce a quantum antibunching effect. The quantum antibunching effect of the K eigenstates of the K th powere ( K greater than or equal to 3) of the annihilation operator in the Q-deformed non-harmonic oscillator is investigated. The physical meaning of the K states are explored. The results show that there is the quantum antibunching effect in all of those states. All of them can be generated by a linear superposition of generalized coherent states produced by the time-dependent Q-deformation non-harmonic oscillator at different instants.