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ON THE QUANTIZATION OF DAMPED HARMONIC OSCILLATOR

Subrata K. Ghosh, Amitava Choudhuri, B. Talukdar

Published Jan 1, 2009• 7 citations
OpenAlex

Abstract

We derive results for two constants of the motion of a one-dimensionaldamped harmonic oscillator with position-dependent frictional coefficientand use them to obtain two alternative Lagrangian representations, whichare not connected by a gauge term. The Hamiltonians corresponding tothese Lagrangianslead tocanonicallyinequivalent phase-spacedescriptions.We could, however, make use of a perturbation theoretic approach to quan-tize the classical motion using both Hamiltonians and thus demonstratethat the corresponding quantum systems are entirely different.PACS numbers: 03.20.+i, 03.30.+p, 03.65.–w

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We have analysed the quantization schemes presented in references [1, 2] for a damped harmonic oscillator and found out that there are some defects in volved in these schemes. In order to remedy these defects, we assume the coordinate system fluctuating with a velocity g(t) in addition to damping force f(t) proportional to the velocity and suggest a method of quantization for this system based on the full quantum theory of lasers. It is proved that the usual Heisenberg relation xp-px=ih can be maintained, but the quantum term in statistical average value of physical variable does not damp as shown in reference [1].

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We derive results for two constants of the motion of a one-dimensionaldamped harmonic oscillator with position-dependent frictional coefficientand use them to obtain two alternative Lagrangian representations, whichare not connected by a gauge term. The Hamiltonians corresponding tothese Lagrangianslead tocanonicallyinequivalent phase-spacedescriptions.We could, however, make use of a perturbation theoretic approach to quan-tize the classical motion using both Hamiltonians and thus demonstratethat the corresponding quantum systems are entirely different.PACS numbers: 03.20.+i, 03.30.+p, 03.65.–w
An explicitly time independent Lagrangian functional of a three-dimensional damped harmonic oscillator has been proposed. I derive results for the motion of the three-dimensional damped harmonic oscillator with a pure imaginary three dimensional vector and oscillator’s position-dependent friction coefficient. The Hamiltonians corresponding to the Lagrangian is also explicitly time independent. The choice of functional form of the friction coefficient on the oscillator position determines and plays a vital role in the form of the equation of motion classically and quantum mechanically. One choice of the form of the friction coefficient I made lead to breaking the symmetry of the isotropy of oscillations in the three dimensional space.
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.
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 have analysed the quantization schemes presented in references [1, 2] for a damped harmonic oscillator and found out that there are some defects in volved in these schemes. In order to remedy these defects, we assume the coordinate system fluctuating with a velocity g(t) in addition to damping force f(t) proportional to the velocity and suggest a method of quantization for this system based on the full quantum theory of lasers. It is proved that the usual Heisenberg relation xp-px=ih can be maintained, but the quantum term in statistical average value of physical variable does not damp as shown in reference [1].