In this paper, we review a new treatment of classical radiation damping, which resolves a well-known contradiction in the Abraham-Lorentz equation that has long been a concern. This radiation damping problem has already been solved in quantum mechanics by the method introduced by Friedrichs. Based on Friedrichs’ treatment, we solved this long-standing problem by classicizing quantum mechanics by replacing the canonical commutation relation in quantum mechanics with the Poisson bracket relation in classical mechanics. As an application of our new approach, we will analyze the anomalous damping experienced by electrons undergoing cyclotron motion inside a waveguide near the cut-off frequency. This anomalous damping cannot be described using the existing Abraham-Lorentz equation.
In this paper, we review a new treatment of classical radiation damping, which resolves a well-known contradiction in the Abraham-Lorentz equation that has long been a concern. This radiation damping problem has already been solved in quantum mechanics by the method introduced by Friedrichs. Based on Friedrichs’ treatment, we solved this long-standing problem by classicizing quantum mechanics by replacing the canonical commutation relation in quantum mechanics with the Poisson bracket relation in classical mechanics. As an application of our new approach, we will analyze the anomalous damping experienced by electrons undergoing cyclotron motion inside a waveguide near the cut-off frequency. This anomalous damping cannot be described using the existing Abraham-Lorentz equation.
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.
Recent development of spectral analysis of the Liouville-von Neumann equation has revealed the fact that irreversibility is a rigorous dynamical property of Poincaré non-integrable systems with an infinite degrees of freedom interacting among each other through resonance coupling. In the present work we discuss this role of resonance in some examples of matter-field coupling systems for both classical and quantum mechanics: the one is a classical motion of a charged particle in electromagnetic waveguide, and the other is the decoherence problem of quantum matter-field interacting systems. In the first part of this dissertation, we study an accelerated motion of a charged classical dipole molecule with frequency ω1 inside the rectangular waveguide. If the particle is in free space, it is well known that its accelerated motion will eventually stop by radiating the field through the resonance interaction. This result is the so-called radiation damping. For the case in the waveguide, there are two possible situations, due to the existence of the cut-off frequency ωc of the waveguide. Under the cut-off frequency electromagnetic wave cannot propagate inside the waveguide. The stability of the dipole depends on the relation between ω1 and ωc. For ω1 < ωc, the dipole cannot resonate with the field. This corresponds to the Poincaré integrable system. For this case the dipole keeps its accelerated motion without emitting the radiating field. Therefore the radiation damping of the dipole molecules is suppressed inside the waveguide under the absence of resonance interaction. The motion of this steady state somewhat resembles a quantum ground state. We show that this steady state is dressed by electromagnetic field. The overlap of the dressing field leads to a force analogous to van der Waals force in quantum mechanics. The critical frequency determined by ω1 = ωc gives a critical size of the waveguide. For heavy molecules, such as HCl, this is of order 10−5m. We show that the size of the dressing field is the same order of the size of the waveguide. Hence we have a macroscopic size of the dressing in the waveguide. For ω1 > ωc, the dipole can resonate with the field, and the system becomes non-integrable in the sense of Poincaré. As a result, the accelerated motion eventually stop by emitting the resonance field. This corresponds to the problem of classical radiation damping. We show that there is non-negligible deviation of exponential decay in a short time scale of the order t ∼ 1/ω1. This corresponds to the quantum Zeno effect, well known in quantum unstable systems. After this period, the dipole decays exponentially in time by emitting the resonance field. We found by choosing ωc very close to ω1, we can increase the decay rate 105 times faster than the case where the dipole is in the free space, at the same time the emitted field travel 10−4 times slower than the speed of light. This is again a consequence of the existence of the cut-off frequency in the waveguide. Indeed, the cut-off frequency leads to a non-linear dispersion relation for the electromagnetic field. To some extent, the electromagnetic field is sticky inside the waveguide. Due to the large decay rate and slower speed of light, the size of the wavepacket emitted by the dipole is significantly small (about 10cm for HCl). This is even smaller than the quantum case in free space, where the wavepacket of the field emitted by the decay of electron in hydrogen atom is about 1m. In the second part of this dissertation, we study a quantum matter-field coupled system. We focus our attention on the problem of quantum decoherence in a system of a particle coupled with a field, the Hamiltonian of which has a similar structure to the problem of classical radiation damping mentioned above. We apply the complex spectral representation of the Liouville-von Neumann operator that gives a rigorous approach to the irreversible processes. We focus our attention on the time evolution of the field, which is commonly neglected in phenomenological approaches to the decoherence problem. We found a signature of decoherence in the field which has a characteristic time dependence proportional to t that comes from the secular effect between the particle and the field through the resonance interaction that breaks time-symmetry.
Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.
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.