[Paper Review] Slightly generalized Maxwell classical electrodynamics can be applied to inneratomic phenomena

В. М. Симулик, I.Yu. Krivsky|ArXiv.org|Jan 21, 2002

Quantum Mechanics and Applications4 references8 citations

TL;DR

This paper proposes a slightly generalized classical Maxwell electrodynamics that describes intraatomic phenomena—such as the hydrogen atom's fine structure—without invoking quantum mechanics or the Dirac equation. By introducing medium-dependent electric and magnetic permeabilities (ε, μ) derived from the Coulomb potential and a mass parameter, the model reproduces the Sommerfeld-Dirac formula for energy levels, derives the Bohr postulates and Lamb shift as classical consequences, and establishes unitary equivalence between Maxwell and Dirac equations, extending classical theory's reach into the microworld.

ABSTRACT

In order to extend the limits of classical theory application in the microworld some weak generalization of Maxwell electrodynamics is suggested. It is shown that slightly generalized classical Maxwell electrodynamics can describe the intraatomic phenomena with the same success as relativistic quantum mechanics can do. Group-theoretical grounds for the description of fermionic states by bosonic system are presented briefly. The advantages of generalized electrodynamics in intraatomic region in comparison with standard Maxwell electrodynamics are demonstrated on testing example of hydrogen atom. We are able to obtain some results which are impossible in the framework of standard Maxwell electrodynamics. The Sommerfeld - Dirac formula for the fine structure of the hydrogen atom spectrum is obtained on the basis of such Maxwell equations without appealing to the Dirac equation. The Bohr postulates and the Lamb shift are proved to be the consequences of the equations under consideration. The relationship of the new model with the Dirac theory is investigated. Possible directions of unification of such electrodynamics with gravity are mentioned.

Motivation & Objective

• To extend classical electrodynamics into the microworld by weakly generalizing Maxwell's equations.
• To provide a non-quantum-mechanical, classical explanation for atomic phenomena traditionally requiring quantum theory.
• To demonstrate that the Sommerfeld-Dirac formula, Bohr postulates, and Lamb shift emerge naturally from generalized Maxwell equations.
• To establish a unitary relationship between Maxwell and Dirac equations, suggesting a bosonic origin for fermions.
• To lay foundations for unifying classical electrodynamics with gravity through a medium-based formulation.

Proposed method

• Introduce a medium with position-dependent electric and magnetic permeabilities ε(x) = 1 - (Φ(x) + m₀)/ω and μ(x) = 1 - (Φ(x) - m₀)/ω, where Φ(x) = -Ze²/r.
• Use the generalized Maxwell equations with sources: curl H - ∂₀(εE) = jₑ, curl E + ∂₀(μH) = jₘₐg, div(εE) = ρₑ, div(μH) = ρₘₐg.
• Apply the formalism to the hydrogen atom by modeling the electron as a stationary electromagnetic-scalar wave in a medium with specific ε and μ.
• Establish unitary equivalence between the generalized Maxwell equations and the Dirac equation via linear relations between field amplitudes.
• Derive energy levels by solving the stationary form of the generalized equations, showing discreteness arises from medium properties, not quantization.
• Reproduce the Lamb shift by adding a quasipotential to Φ(x), interpreting it as a classical polarization effect of the medium, not vacuum polarization.

Experimental results

Research questions

• RQ1 Can classical Maxwell electrodynamics be extended to describe intraatomic phenomena without invoking quantum mechanics?
• RQ2 Does the generalized Maxwell formalism reproduce the fine structure of the hydrogen spectrum as accurately as the Dirac equation?
• RQ3 Can the Bohr postulates and the Lamb shift emerge as classical consequences of the generalized equations rather than as postulates or quantum corrections?
• RQ4 Is there a unitary relationship between the generalized Maxwell equations and the Dirac equation, suggesting a bosonic structure for fermions?
• RQ5 Can this classical framework be extended to include gravity, suggesting a unified description of electromagnetic and gravitational phenomena?

Key findings

• The Sommerfeld-Dirac formula for the fine structure of the hydrogen spectrum is derived directly from the generalized Maxwell equations without using the Dirac equation.
• The Bohr postulates are shown to be consequences of the generalized Maxwell equations, emerging from the boundary conditions and medium properties.
• The Lamb shift is reproduced as a classical electrodynamical effect due to medium polarization, not vacuum polarization, by adding a quasipotential to the Coulomb potential.
• The model establishes a unitary equivalence between the stationary Dirac equation and the stationary generalized Maxwell equations with gradient-like sources.
• The electron is interpreted as a classical stationary electromagnetic-scalar wave, suggesting a bosonic structure for fermions without requiring quantum field theory.
• The model provides a classical, non-probabilistic interpretation of the Dirac wave function Ψ as a combination of electromagnetic and scalar field strengths, eliminating the need for the Copenhagen interpretation.