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arXiv:1002.1116

Damping Effect of Electromagnetic Radiation and Time-Dependent Schrodinger Equation

Ji Luo

Published Feb 5, 2010• 0 citations
arXivHTMLPDF

Abstract

The inexactness of the time-dependent Schrödinger equation of a charged particle in an external electromagnetic field is discussed in terms of the damping effect of the radiation. A possible improvement is to add a nonlinear term representing this effect to the linear Schrödinger equation. Conditions for the nonlinear term are investigated and it is demonstrated that the obtained nonlinear Schrödinger equation may present state evolutions similar to the wave-function reduction and transitions between stationary states.

DOI
10.48550/arxiv.1002.1116

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The inexactness of the time-dependent Schrödinger equation of a charged particle in an external electromagnetic field is discussed in terms of the damping effect of the radiation. A possible improvement is to add a nonlinear term representing this effect to the linear Schrödinger equation. Conditions for the nonlinear term are investigated and it is demonstrated that the obtained nonlinear Schrödinger equation may present state evolutions similar to the wave-function reduction and transitions between stationary states.
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Electromagnetic interactions of a single charged particle in the presence of damping are investigated theoretically in this work. Classically, by considering the explicit example of a particle in a constant magnetic field, it is shown that the minimal coupling prescription is inadequate for the infinitely large number of Hamiltonians from which the equations of motion in the presence of damping are derivable by Hamilton's equations. The use of minimal coupling procedure to switch on electromagnetic interactions in phenomenological quantum equations of damped motion is shown to lead to incorrect equations in the classical limit. It is necessary to consider the microscopic models of damping to allow for electromagnetic interactions in these systems. Two microscopic models for damping involving a linear coupling with a heat bath of oscillators are considered in detail and it is shown that correct Heisenberg equations of motion follow upon elimination of bath degrees of motion so that damping and Lorentz force do not affect each other. However, the field couples to the mechanical momentum as mv-(e/c)A rather than to the canonical momentum. The linearly coupled quantal case is analogous to the classical situation. The Schr\"odinger-Langevin equation contains an extra term Fr{\mathcal{F}}^{\mathrm{r}}Fr A\ensuremath{\cdot}dr'; however, the gauge invariance character and the Aharonov-Bohm effect are unaffected. Two nonlinear models of damping are studied by considering the exchange of bosons in the different states of the central particle. In this situation the Lorentz force is not recovered in the classical limit.
The states of the atom, determined by the motion of electrons, are stationary in the presence of all internal atomic forces and of interactions with constant external fields. When electrons interact with the field associated with electromagnetic radiation, the atomic states are no longer stationary, and the time-dependent perturbation treatment has to be applied to calculate the effect produced by the radiation, which is a function of time. Consequently, as shown in Section 9.3 , the time evolution of the atom will be described in terms of transitions between the stationary states of the unperturbed atomic system.