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Heisenberg’s quantization of dissipative systems

Merced Montesinos

The quantization of the damped one-dimensional harmonic oscillator is carried out. This is implemented by first writing the classical equations of motion that describe the system in a Hamiltonian form, the Hamiltonian being time independent explicitly. Canonical quantization is then implemented by using Heisenberg's picture. When friction is turned off the results agree with those of the one-dimensional harmonic oscillator.

Physical Review AJul 7, 200310 citations

QUANTIZATION OF THE DAMPED HARMONIC OSCILLATOR*

Herman Feshbach, Yoel Tikochinsky

Transactions of the New York Academy of SciencesNov 1, 1977116 citations

Quantization of the damped harmonic oscillator revisited

Mario Cesar Baldiotti, R. Fresneda, Dmitri M Gitman

arXiv:1005.4096Physics Letters AMar 7, 201142 citations

Quantum dynamics of the damped harmonic oscillator

T G Philbin

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.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.

arXiv:1205.2545New Journal of PhysicsAug 31, 201232 citations

Quantum dynamics of the damped harmonic oscillator

T. G. Philbin

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.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.

Aug 17, 201618 citations

Quantization of Simple Harmonic Oscillator

Hiu Yung Wong

Springer eBooksJan 1, 20250 citations

Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

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