The vacuum electromagnetic fields and the Schrodinger picture

TL;DR

This paper investigates how to incorporate vacuum electromagnetic fluctuations and radiation reaction forces into the Schrödinger picture for a nonrelativistic charged harmonic oscillator, highlighting issues of double counting and proposing a solution.

Contribution

It demonstrates how to correctly include electromagnetic vacuum fluctuations and radiation reaction in the Schrödinger picture without double counting, extending previous Heisenberg picture analyses.

Findings

Zero-point electromagnetic fluctuations relate to the momentum operator in Schrödinger picture.

Including both vacuum fluctuations and radiation reaction causes double counting.

Proper inclusion of only radiation reaction and thermal fields avoids double counting.

Abstract

Several authors have used the Heisenberg picture to show that the atomic transitions, the stability of the ground state and the position-momentum commutation relation [x,p]=ih, can only be explained by introducing radiation reaction and vacuum electromagnetic fluctuation forces. Here we consider the simple case of a nonrelativistic charged harmonic oscillator, in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrodinger picture. We consider the effects of both classical zero-point and thermal electromagnetic vacuum fields. We show that the zero-point electromagnetic fluctuations are dynamically related to the momentum operator p=-ih d/dx used in the Schrodinger picture. Consequently, the introduction of the zero-point electromagnetic fields in the vector potential A_x(t) used in the Schrodinger equation, generates ``double counting'', as was shown recently by A.J. Faria et al. (Physics Letters A 305 (2002) 322). We explain, in details, how to avoid the ``double counting'' by introducing only the radiation reaction and the thermal electromagnetic fields into the Schrodinger equation.