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American Journal of Physics

Quantizing the damped harmonic oscillator

Bernard Yurke

Published Dec 1, 1986• 22 citations
Publisher

Abstract

A damped harmonic oscillator can be modeled as a manifestly conservative system by replacing the dissipative element with a string or transmission line of infinite extent. This conservative system can be quantized in a straightforward manner using the standard techniques of canonical quantization. The system may be used to illustrate various aspects of the quantum mechanics of a particle interacting with a field.

DOI
10.1119/1.14730

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A damped harmonic oscillator can be modeled as a manifestly conservative system by replacing the dissipative element with a string or transmission line of infinite extent. This conservative system can be quantized in a straightforward manner using the standard techniques of canonical quantization. The system may be used to illustrate various aspects of the quantum mechanics of a particle interacting with a field.
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys the Kramers–Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems.
Abstract. The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys Kramers-Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems. PACS numbers: 03.65.-w, 03.65.Yz, 03.70.+k 1.
In this work, a suitable Hamiltonian that describes the damped harmonic oscillator is constructed. Besides, the Hamilton-Jacobi equation for this dissipative system is written and the action function is obtained. Then, the system is quantized using two methods: the WKB approximation and the canonical quantization. In addition, evolution of the probability density is plotted at different times, and it is observed that the width of these densities decreases with time. The average energy of the system is also calculated and found to decrease with time.
Quantizing dissipative systems like damped harmonic oscillators remains an ongoing challenge in theoretical physics due to their non-conservative nature.This study addresses this issue by employing the method of doubling the degrees of freedom, transforming an open, dissipative system into a closed, conservative one through the introduction of a time-reversed mirror system.The aim is to derive a quantum model consistent with classical mechanics using the Hamilton-Jacobi formalism and to formulate an equivalent Schrödinger equation.Through canonical transformations and analytical solutions, we demonstrate that damping effectively vanishes in the quantum model.These results offer new insights into the quantization of non-conservative systems and contribute to the advancement of quantum open-system theory.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.