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Journal of Physics A Mathematical and General

Inconsistency of a proposed constrained dynamics for the damped harmonic oscillator

Nivaldo Agostinho Lemos

Published Sep 7, 1990• 0 citations
Publisher

Abstract

It is argued that the method introduced recently by Wang (1987) to quantise the damped harmonic oscillator is untenable because it does not reproduce the standard results for the quantum oscillator in the limit of arbitrarily small damping. The method is also shown to be inconsistent if the friction coefficient is allowed to take any positive value. Furthermore, it is established that no moving wavepacket can be constructed within Wang's formalism.

DOI
10.1088/0305-4470/23/17/022

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It is argued that the method introduced recently by Wang (1987) to quantise the damped harmonic oscillator is untenable because it does not reproduce the standard results for the quantum oscillator in the limit of arbitrarily small damping. The method is also shown to be inconsistent if the friction coefficient is allowed to take any positive value. Furthermore, it is established that no moving wavepacket can be constructed within Wang's formalism.
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.
The damped harmonic oscillator is used to represent the damping processes in nature. Therefore it is necessary to find the Hamiltonian for the damped oscillator. The Hamiltonian is developed for the harmonic oscillator with frictional force proportional to velocity. The solution of the Schr\"odinger equation gives nonstationary states. The quantum friction coefficient is derived to be zero for massless particles. It is thought that velocity-dependent forces exist near the nucleus and therefore, the scattering amplitude for the deep inelastic process is derived.