It is known that there is a close analogy, exhibited by the use of five-dimensional tensor notation, between the electromagnetic equations in free space and the first- and second-order equations of the quantum theory. In the present paper an analogy is traced between two tensors, one of which is related to the magnetic and electric moments of a doublet while the other represents the electric and magnetic field-strengths.
It is known that there is a close analogy, exhibited by the use of five-dimensional tensor notation, between the electromagnetic equations in free space and the first- and second-order equations of the quantum theory. In the present paper an analogy is traced between two tensors, one of which is related to the magnetic and electric moments of a doublet while the other represents the electric and magnetic field-strengths.
Abstract The form of the wave equation for a non-rotating electron suggests that it enters into the theory very much in the same way as the wave equation associated with electromagnetic theory. It would be expected to be derivable from equations of the first order corresponding to Maxwell's equations. It has been suggested that the function Ψ might enter by means of a relation such as s = grad Ψ (1) where s replaces the current four vector of the electromagnetic theory. The difficulty in connection with this procedure is to account for the phenomena associated with electronic rotation. Dirac has shown how to overcome this difficulty and has derived first order equations which can be derived from generalisations of Maxwell's equations. There are certain difficulties with regard to the form of Dirac's results which have been much discussed and some of them have been removed. There are two unsatisfactory points in the treatment of this question. One is the introduction of an operator (h/2πi ∂/∂xα - eϕα) into the equations when it is desired to pass from a non-electromagnetic problem to one in which an electromagnetic field is present. The second difficulty lies in the occurrence of a term in mc. Darwin has pointed out this difficulty and considers that it is due to our inability to calculate electromagnetic mass in the quantum theory.
Due to the advent of Quantum Mechanics' 100th anniversary in 2025, we wrote this review paper in order to present a discussion that addresses the foundations of this theory. And since the creation of this Mechanics and other quantum theories was guided, for instance, by correspondence principles that needed to be identified between them and other well-established physical theories, this paper will be devoted to discussing the correspondence between these quantum theories and Maxwell's theory of electromagnetism. More precisely, what we will do throughout this paper is discuss how the Maxwell's electromagnetic theory in free space and the stronger formulation of the correspondence principle already pointed, together, to the basis of a Quantum Mechanics that was only be formulated half a century later. And, in order to make this very clear, we will show that the quantum-mechanical description of a photon can already be identified, simply, by manipulating Maxwell's equations in free space with mathematical resources that, for instance, were already well known before the advent of Quantum Mechanics.
Abstract 1. The procedure usually adopted in the Quantum Theory for the estimation of the intensities of a spectrum line includes two stages : (1) the electric moment, p, of the distribution responsible for the spectrum line is calculated by means of Schrodinger’s integrals or some equivalent method ; (2) the rate of loss of energy due to radiation is calculated by the classical formula due to H. Hertz, viz., 2p2/3c3, just as if the distribution were concentrated in a dipole of precisely the same moment p. The first stage involves no difficulties beyond those met with in the evaluation of the integrals or matrices with which we are concerned, the only principles needed being those of the particular quantum theory adopted. But the second stage is open to grave objections and can only be justified on the ground of necessity owing to the absence of any completely satisfactory quantum theory of the electromagnetic field. Sometimes an appeal is made to Bohr’s Correspondence Principle, but this itself involves a somewhat hazardous extrapolation from infinitely large down to small quantum numbers and cannot take the place of a rigorous calculation. A logically consistent method is possible, based on Maxwell’s theory as represented by his electromagnetic equations and their solutions in terms of the classical retarded potential integrals. It may be objected that we do not know how far Maxwell’s equations can be applied to the interiors of atoms, in particular to distributions like those of Schrodinger, but precisely the same objection applies to the customary use of the classical expression for the radiation from an oscillating dipole. Moreover, the attempts made by Heisenberg and Pauli, Dirac and others to formulate a quantum theory of the electromagnetic field have all been based on the assumption that such a theory must be consistent with Maxwell’s equations. The direct application of the methods of classical electromagnetic theory to Schrodinger’s distributions can be justified at least as a rigorous attempt to determine their electromagnetic field and the radiation from them, and the validity of the procedure will have to be judged by the agreement, or otherwise, of the results obtained with experience. In this way we avoid the assumption that a spatially extended distribution can be treated as if its electric moment were concentrated in a point; in fact, the results obtained in this paper for the more important spectrum line series show definitely that the assumption is false, because it leads to results in many cases quite inconsistent with a rigorous application of Maxwell’s theory.
It is pointed out that the electromagnetic interaction $\ensuremath{-}\stackrel{^}{\ensuremath{\mu}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{E}$ between an atom and a field, in which the atom is modelled as a point dipole of moment $\stackrel{^}{\ensuremath{\mu}}$ interacting with the electric field $\stackrel{^}{E}$, is not strictly consistent with both quantum mechanics and Maxwell's equations. Quantum mechanics allows magnetic point dipoles but not electric point dipoles.
Maxwell equations (Faraday and Ampere-Maxwell laws) can be presented as a three component equation in a way similar to the two component neutrino equation. However, in this case, the electric and magnetic Gauss's laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have also shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable and well understood quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum theories.
The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.