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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Operators of the classical field and semiclassical electrodynamics

Eugeni P. Gordov, Stanislav D. Tvorogov

Published Mar 15, 1974• 6 citations
Publisher

Abstract

The operators of the classical electromagnetic field, whose eigenvalues are the corresponding classical fields, are introduced. The use of eigenfunctions of classical field operators in describing the transition from the free quantum field to the classical one is considered. The scheme for the derivation of the semiclassical approximation from nonrelativistic quantum electrodynamics is improved. The limits of applicability of the semiclassical approximation are considered.

DOI
10.1103/physrevd.9.1711

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International Journal of Quantum Chemistry•Jan 1, 1999•0 citations
The operators of the classical electromagnetic field, whose eigenvalues are the corresponding classical fields, are introduced. The use of eigenfunctions of classical field operators in describing the transition from the free quantum field to the classical one is considered. The scheme for the derivation of the semiclassical approximation from nonrelativistic quantum electrodynamics is improved. The limits of applicability of the semiclassical approximation are considered.
A complete orthonormal set of eigenfunctions of the vector-potential operator is introduced. It is shown that the semiclassical approximation is naturally determined in the basis of the functions introduced. The extraction of the semiclassical approximation from nonrelativistic quantum electrodynamics allows the determination of the limits of applicability of the semiclassical approach. It is shown that the semiclassical approximation is inapplicable at sufficiently large photon densities. The advantages of using our set of eigenfunctions for the description of the interaction of a strong quantized field with a quantum system are discussed.
The electromagnetic field operators in the Heisenberg picture are found in both the minimal and multipolar forms of nonrelativistic quantum electrodynamics within the electric dipole approximation. These are made up of free- and source-field terms. The free electric- and magnetic-field operators in the two formalisms differ since the creation/annihilation operators are manifestly distinct. Their explicit forms in the Heisenberg picture are given. However, the source fields are formally the same when expressed in terms of dipole moment operators at positive retarded time. The cancellations that lead to this equivalence are presented. In many applications it is necessary to express the operators at retarded time in terms of creation and annihilation operators at the initial time. Explicit expressions for the fields up to second order in terms of the transition dipole moments are given. These enable the formulas for induced dipole moments to be derived within the context of quantum electrodynamics whereas previous derivations have been based on semiclassical electrodynamics.
This paper discusses the quantization of electromagnetic fields interacting with nonrelativistic atoms and molecules in the dipole approximation. Quantizing the field variables D and H and taking the interaction density −ε0−1P·D, we obtain a quantum theory corresponding to classical electrodynamics with no free currents and the induced dipole moments as the only sources of the radiation field. The resulting equations are found to agree with those Power and Zienau obtained from causality requirements. A comparison with the conventional approach is given.
Semiclassical models can be used, see e.g. Scully and Zubairy [Quantum Optics; Cambridge University Press: Cambridge, 1997], to simulate some distinguishing features of quantum optics. In this paper we show that one can also proceed successfully in the opposite direction. A quantum-like model for the classical random electromagnetic field is proposed. Averages with respect to classical random fields can be approximated by quantum-like averages given by the von Neumann trace-formula: . The operators ρ and can be easily determined as classical quantities. Here ρ is obtained by normalization of the covariance operator of the random field. It has all the features of the von Neumann density operator. Additionally an analogue of the quantum observable is given by the second derivative of a functional (‘classical variable’) f (E, B) of the classical electromagnetic field ψ = (E, B).
The interaction of nonrelativistic matter with the quantized electromagnetic field is investigated in the classical limit of large photon numbers. Quantization of both matter, say an atom, and the field results in a time-independent Schr\"odinger equation (TISE). However, for very strong fields (quantum mechanically, large photon numbers) this is impractical to solve. The standard approach then is simply to replace the quantized field by a classical field to give a time-dependent Schr\"odinger equation (TDSE) for the atom alone. Here we show how this TDSE can be derived from the TISE for atom plus field, illustrating at each stage the approximations that are necessary to treat the field classically. An important difficulty at the semiclassical stage is a breakdown of the approximation at classical turning points. We show how the use of coherent field states can circumvent this problem. In the limit that the field can be treated classically, time emerges from the Maxwell equations and a TDSE for the atom alone results.
A form of classical electrodynamic field exists which gives exact agreement with the operator field of quantum electrodynamics (QED) for the Lamb shift of a harmonically bound point electron. Here it is pointed out that this form of classical theory, with its physically acceptable interpretation, is the result of an unconventional resolution of a mathematically ambiguous term in classical field theory. Finally, a quantum classical correspondence principle is shown to exist in the sense that the classical field and expectation value of the QED operator field are identical, if retardation is neglected in the latter. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 655–658, 1999