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Foundations of Physics

The Vacuum Electromagnetic Fields and the Schrödinger Equation

A. J. de Faria, H. M. França, G. G. Gomes, R. C. Sponchiado

Published Jun 11, 2007• 3 citations
Publisher
DOI
10.1007/s10701-007-9152-6

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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.

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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.
This paper extends the preceding Schrödinger-based analysis to the relativistic Dirac framework and examines how gauge covariance, spin magnetism, and quantum-vacuum effects deepen the connection between quantum theory and Maxwell electrodynamics. Local phase invariance of the Dirac equation yields the electromagnetic field tensor, while the conserved Dirac four-current provides a covariant source for Maxwell’s equations. The Pauli limit recovers g=2g=2 and the Bohr magneton, and the Gordon decomposition separates convection and spin-magnetization currents. The Foldy–Wouthuysen expansion further connects relativistic quantum dynamics to spin–orbit and Darwin terms involving the electromagnetic fields. Hydrogen fine-structure and Darwin-shift benchmarks are evaluated quantitatively, while the Dirac velocity operator and Zitterbewegung scale are compared with the magnetic dipole length appearing in the SQC/MSC M1 formulation. The analysis is then extended to the quantum vacuum through the Schwinger critical fields and Euler–Heisenberg effective theory, including vacuum birefringence. The paper distinguishes structural consequences of relativistic quantum mechanics from electromagnetic field dynamics that must still be specified independently.
A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.