Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Semi-canonical quantisation of dissipative equations

J. Geicke

Published Apr 21, 1989• 8 citations
Publisher

Abstract

In order to avoid the known problems of canonical quantisation of the dissipative equation mx+ alpha x n + delta V/ delta x=0, a modified approach is proposed. It is based on the classically conserved quantity E=mx 2 /2+V(x)+ alpha integral x n dx rather than on a Hamilton function. Linear, quadratic and cubic damping (n=1, 2 and 3) are discussed explicitly. For n=1 and n=3 normalisable (one-particle) states cannot be stable. However, for quadratic friction, which is classically a reversible process, the author shows, using the example of the harmonic oscillator, that stable normalisable eigenstates can exist.

DOI
10.1088/0305-4470/22/8/017

Related papers

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 A•Jul 7, 2003•10 citations

Quantisation of velocity-dependent forces αx2and αx4

J. Geicke

The eigenvalue spectrum E l (V,n,α) of normalisable states of the equation (αℏ/im) n ϕ (n-1) (x)-ℏ 2 /2m) ϕ"(x)+(V(x)-E)ϕ(x) = 0, recently proposed as a quantisation of classical forces -αx n - δV/δx, is discussed for even n. For n = 4 and a rectangular potential well an analytic argument shows that bound states for 0 < α < , → 0, can exist only if E l (V,4,α) → E l (0) (V) as α → 0 where E l (0) (V) are the eigenvalues of the corresponding conservative Schrodinger equation (α = 0). Numerical results indicate E l (V,4,α) - E l (0) (V) = c l (V) α 2 + O(α 3 ) as α → 0. For n = 2, E l (V,2,α) - E l (0) (V)= 1/2 α 2 ℏ 2 m -3 is exact and independent of V(x), and at a potential step (to the left since α > 0) a continuous band of normalisable states exists instead of the scattering states for α =0.

Journal of Physics A Mathematical and General•Sep 7, 1989•1 citation

Classical and Quantum Dissipative Systems

Mohsen Razavy

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.

WORLD SCIENTIFIC eBooks•Dec 13, 2016•85 citations

Quantization of the linearly damped harmonic oscillator

H. Dekker

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.

Physical Review A•Nov 1, 1977•167 citations

Canonical Quantisation

K. Kong Wan

Q27(1) Show that Hamilton’s equations of motion in Eq. (27.8) for a classical harmonic oscillator whose Hamiltonian is given by Eq. (27.11) are equivalent to Newton’s equation of motion.

•Aug 27, 2020•0 citations

CANONICAL QUANTIZATION OF DISSIPATIVE SYSTEMS

Ola A. Jarab’ah, Khaled I. Nawafleh, H. B. Ghassib

The canonical method is invoked to quantize dissipative systems using the WKB approximation. The wave function is constructed such that its phase factor is simply Hamilton’s principal function. The energy eigenvalue is found to be in exact agreement with the classical case. To demonstrate our approach, the three examples considered in our previous work (ESJ 9(30), 70-81, 2013) are quantized in detail: the damped harmonic oscillator, a system with a variable mass, and a charged particle in a magnetic field.

European Scientific Journal ESJ•Dec 31, 2013•5 citations
In order to avoid the known problems of canonical quantisation of the dissipative equation mx+ alpha x n + delta V/ delta x=0, a modified approach is proposed. It is based on the classically conserved quantity E=mx 2 /2+V(x)+ alpha integral x n dx rather than on a Hamilton function. Linear, quadratic and cubic damping (n=1, 2 and 3) are discussed explicitly. For n=1 and n=3 normalisable (one-particle) states cannot be stable. However, for quadratic friction, which is classically a reversible process, the author shows, using the example of the harmonic oscillator, that stable normalisable eigenstates can exist.
The eigenvalue spectrum E l (V,n,α) of normalisable states of the equation (αℏ/im) n ϕ (n-1) (x)-ℏ 2 /2m) ϕ"(x)+(V(x)-E)ϕ(x) = 0, recently proposed as a quantisation of classical forces -αx n - δV/δx, is discussed for even n. For n = 4 and a rectangular potential well an analytic argument shows that bound states for 0 < α < , → 0, can exist only if E l (V,4,α) → E l (0) (V) as α → 0 where E l (0) (V) are the eigenvalues of the corresponding conservative Schrodinger equation (α = 0). Numerical results indicate E l (V,4,α) - E l (0) (V) = c l (V) α 2 + O(α 3 ) as α → 0. For n = 2, E l (V,2,α) - E l (0) (V)= 1/2 α 2 ℏ 2 m -3 is exact and independent of V(x), and at a potential step (to the left since α > 0) a continuous band of normalisable states exists instead of the scattering states for α =0.
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.
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.
The canonical method is invoked to quantize dissipative systems using the WKB approximation. The wave function is constructed such that its phase factor is simply Hamilton’s principal function. The energy eigenvalue is found to be in exact agreement with the classical case. To demonstrate our approach, the three examples considered in our previous work (ESJ 9(30), 70-81, 2013) are quantized in detail: the damped harmonic oscillator, a system with a variable mass, and a charged particle in a magnetic field.