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

Physical Review A

Abraham-Lorentz equation in quantum mechanics

Masao Ozaki, Shigeru Sasabe

Published Aug 17, 2009• 4 citations
Publisher

Abstract

Following Crisp's method based on neoclassical theory for a radiating electron, we have derived the nonrelativistic Lorentz-Dirac equation from quantum mechanics by means of Ehrenfest's theorem. It is found that the dipole approximation is essential to obtain the correct form of the equation of motion.
DOI
10.1103/physreva.80.024102

Related papers

Lorentz Atom Revisited by Solving the Abraham–Lorentz Equation of Motion

Jürgen Bosse

By solving the non-relativistic Abraham–Lorentz (AL) equation, I demonstrate that the AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing the appropriate parameters Ω and Γ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarisability, which many authors “derived” from the AL equation in recent years, is shown to violate Kramers–Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as the light frequency ω is neither too close to nor too far from the resonance frequency Ω. The polarisability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and are shown to agree with the quantum mechanical calculations of the same quantities. In particular, oscillator parameters Ω and Γ deduced by treating the atom as a quantum oscillator are found to be equivalent to those derived from the classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz’s model that was suggested long before the advent of quantum theory.

•Jan 1, 2017•2 citations

Lorentz Atom Revisited by Solving the Abraham–Lorentz Equation of Motion

Jürgen Bosse

Abstract By solving the non-relativistic Abraham–Lorentz (AL) equation, I demonstrate that the AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing the appropriate parameters Ω and Γ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarisability, which many authors “derived” from the AL equation in recent years, is shown to violate Kramers–Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as the light frequencyωis neither too close to nor too far from the resonance frequency Ω. The polarisability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and are shown to agree with the quantum mechanical calculations of the same quantities. In particular, oscillator parameters Ω and Γ deduced by treating the atom as a quantum oscillator are found to be equivalent to those derived from the classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz’s model that was suggested long before the advent of quantum theory.

arXiv:1604.05688•Zeitschrift für Naturforschung A•Jul 15, 2017•2 citations

The Dirac Equation

Kenneth G. Dyall, Knut Fægri

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.

Oxford University Press eBooks•Jul 5, 2007•0 citations

Classical Tunneling from the Lorentz-Dirac Equation

Frederik Denef B, Joris Raeymaekers, Urban M. Studer +1 more

The classical equation of motion of a charged point particle, including its radiation reaction, is described by the Lorentz-Dirac equation. We found a new class of solutions that describe tunneling (in a completely classical context!). For nonrelativistic electrons and a square barrier, the solution is elementary and explicit. We show the persistance of the solution for smoother potentials. For a large range of initial velocities, initial conditions may leave a (discrete) ambiguity on the resulting motion.

arXiv:hep-th/9602066•Feb 13, 1996•0 citations

Quantum mechanics of radiation damping

P.M.V.B. Barone, A. O. Caldeira

We develop a nonrelativistic formulation for the quantum dynamics of an electron coupled to its own radiation field. For this purpose, we have applied the Feynman-Vernon approach to the composite system in order to obtain the reduced density operator of the electron. In the classical limit, some well-known results, such as the Abraham-Lorentz equation of motion, are reproduced. We have applied the resulting formalism to the problem of interference in order to investigate the possible effects of the incoherent modes of the electromagnetic radiation on the interference fringes. The results allow us to conclude that the coupling to the radiation field is not enough for one to observe a strong influence of those modes on the interference phenomenon.

Physical Review A•Jan 1, 1991•87 citations

Quantum Mechanics Derived from Boltzmann's Equation for the Planck Aether

Friedwardt Winterberg

Abstract In the Planck aether hypothesis it is assumed that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying a nonrelativistic law of motion. If described by a quantum mechanical two-component nonrelativistic nonlinear operator field equation, this model has a spectrum of particles greatly resembling the particles of the standard model, with Lorentz invariance as a derived dynamical symmetry valid in the limit of energies small com­ pared to the Planck energy. Here we show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of the Boltzmann equation for the positive and negative Planck masses, with departures from quantum mechanics suppressed by the Planck length.

Zeitschrift für Naturforschung A•Jun 1, 1995•0 citations
By solving the non-relativistic Abraham–Lorentz (AL) equation, I demonstrate that the AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing the appropriate parameters Ω and Γ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarisability, which many authors “derived” from the AL equation in recent years, is shown to violate Kramers–Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as the light frequency ω is neither too close to nor too far from the resonance frequency Ω. The polarisability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and are shown to agree with the quantum mechanical calculations of the same quantities. In particular, oscillator parameters Ω and Γ deduced by treating the atom as a quantum oscillator are found to be equivalent to those derived from the classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz’s model that was suggested long before the advent of quantum theory.
Abstract By solving the non-relativistic Abraham–Lorentz (AL) equation, I demonstrate that the AL equation of motion is not suited for treating the Lorentz atom, because a steady-state solution does not exist. The AL equation serves as a tool, however, for deducing the appropriate parameters Ω and Γ to be used with the equation of forced oscillations in modelling the Lorentz atom. The electric polarisability, which many authors “derived” from the AL equation in recent years, is shown to violate Kramers–Kronig relations rendering obsolete the extracted photon-absorption rate, for example. Fortunately, errors turn out to be small quantitatively, as long as the light frequencyωis neither too close to nor too far from the resonance frequency Ω. The polarisability and absorption cross section are derived for the Lorentz atom by purely classical reasoning and are shown to agree with the quantum mechanical calculations of the same quantities. In particular, oscillator parameters Ω and Γ deduced by treating the atom as a quantum oscillator are found to be equivalent to those derived from the classical AL equation. The instructive comparison provides a deep insight into understanding the great success of Lorentz’s model that was suggested long before the advent of quantum theory.
The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.
The classical equation of motion of a charged point particle, including its radiation reaction, is described by the Lorentz-Dirac equation. We found a new class of solutions that describe tunneling (in a completely classical context!). For nonrelativistic electrons and a square barrier, the solution is elementary and explicit. We show the persistance of the solution for smoother potentials. For a large range of initial velocities, initial conditions may leave a (discrete) ambiguity on the resulting motion.
We develop a nonrelativistic formulation for the quantum dynamics of an electron coupled to its own radiation field. For this purpose, we have applied the Feynman-Vernon approach to the composite system in order to obtain the reduced density operator of the electron. In the classical limit, some well-known results, such as the Abraham-Lorentz equation of motion, are reproduced. We have applied the resulting formalism to the problem of interference in order to investigate the possible effects of the incoherent modes of the electromagnetic radiation on the interference fringes. The results allow us to conclude that the coupling to the radiation field is not enough for one to observe a strong influence of those modes on the interference phenomenon.
Abstract In the Planck aether hypothesis it is assumed that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying a nonrelativistic law of motion. If described by a quantum mechanical two-component nonrelativistic nonlinear operator field equation, this model has a spectrum of particles greatly resembling the particles of the standard model, with Lorentz invariance as a derived dynamical symmetry valid in the limit of energies small com­ pared to the Planck energy. Here we show that quantum mechanics itself can be derived from the Newtonian mechanics of the Planck aether as an approximate solution of the Boltzmann equation for the positive and negative Planck masses, with departures from quantum mechanics suppressed by the Planck length.