Some Arguments for the Wave Equation in Quantum Theory

TL;DR

This paper explores the connection between the wave equation and quantum theory by analyzing electromagnetic solutions, non-radiation conditions, and demonstrating energy quantization consistent with atomic spectral lines.

Contribution

It establishes that certain relations satisfying wave equations imply electromagnetic energy quantization and non-radiation conditions in all inertial frames.

Findings

Energy in electromagnetic fields is quantized.

Non-radiation condition is equivalent to wave equations for charge and current.

Results align with properties of the Balmer series.

Abstract

We clarify some arguments concerning Jefimenko's equations, as a way of constructing solutions to Maxwell's equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.