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Physical Review A

Heisenberg’s quantization of dissipative systems

Merced Montesinos

Published Jul 7, 2003• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.68.014101

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Comment on “Quantization of the damped harmonic oscillator” [J. Math. Phys. 59, 082105 (2018)]

Francisco Marcelo Fernández

Serhan et al. [J. Math. Phys. 59, 082105 (2018)] derived a quantum-mechanical Hamiltonian operator from the classical equations of motion for the damped harmonic oscillator and obtained its eigenvalues and eigenfunctions by canonical quantization. They claim that the eigenvalues are real, but here we show that they are in fact complex.

Journal of Mathematical Physics•Sep 1, 2019•2 citations
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.
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.
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.
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.
In this study, we developed a Hamiltonian framework for the anisotropic harmonic oscillator and applied the Hamilton-Jacobi equations, which are particularly well-suited for this approach. We were able to find the action function by including boundary conditions in the study of a dissipative anisotropic harmonic oscillator. To quantize the system, we employed three distinct methods: the WKB approximation, canonical quantization, and the creation and annihilation operator technique. The quantization of the Heisenberg equations was achieved by reformulating them using Poisson bracket quantum variables. Through the application of creation and annihilation operators, we derived the energy levels and quantum states of the system. The results demonstrated complete consistency across the different methods. This model exhibits strong agreement among the solutions derived from various approaches, highlighting its importance in advancing the understanding of dissipative quantum systems.
Phenomenological Equations of Motion for Dissipative Systems Lagrangian Hamiltonian and Hamilton-Jacobi Formulation of the Classical Dissipative Systems Noether's Theorem and Non-Noether Conservation Laws Dissipative Forces Derived from Many-Body Problems A Particle Coupled to a Field and the Damped Motion of a Central Particle Coupled to a Heat Bath Quantization of Dissipative Systems in General and of Explicitly Time-Dependent Hamiltonians in Particular Density Matrix and the Wigner Distribution Function for Damped Systems Path Integral Formulation of a Damped Harmonic Oscillator Quantization of the Motion of an Infinite Chain Heisenberg's Equations of Motion for a Particle Coupled to a Heat Bath Quantum Mechanical Models of Dissipative Systems and the Concept of Optical Potential.