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Il Nuovo Cimento B

Canonical transformation on an open system

Qiankai Yao, Bing Ma, Xue-Chao Feng, Jun-Lan Feng, Demin Li

Published Jan 1, 2004• 0 citations
Publisher

Abstract

A time-dependent Lagrangian is constructed. This time-dependent Lagrangian can help us to quantize the damped system in the frameworks of canonical quantization, and solve the non-autonomous quantum problem in different representations. We evaluate the measuring effect on the quantum system and find a linear canonical transformation between the original system and the measured one. Finally, the damped harmonic oscillator is discussed.

DOI
10.1393/ncb/i2002-10013-2

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A time-dependent Lagrangian is constructed. This time-dependent Lagrangian can help us to quantize the damped system in the frameworks of canonical quantization, and solve the non-autonomous quantum problem in different representations. We evaluate the measuring effect on the quantum system and find a linear canonical transformation between the original system and the measured one. Finally, the damped harmonic oscillator is discussed.
In this paper, an alternative approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.
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.
In this paper, a new approach for constructing Lagrangians for driven and undriven linearly damped systems is proposed, by introducing a redefined time coordinate and an associated coordinate transformation to ensure that the resulting Lagrangian satisfies the Helmholtz conditions. The approach is applied to canonically quantize the damped harmonic oscillator and although it predicts an energy spectrum that decays at the same rate to previous models, unlike those approaches it recovers the classical critical damping condition, which determines transitions between energy eigenstates, and is therefore consistent with the correspondence principle. It is also demonstrated how to apply the procedure to a driven damped harmonic oscillator.