Mathematical proof of electromagnetic wave nature of photons in hydrogen emission spectrum and theoretical derivation of Planck’s constant*

The photons emitted from electron transitions are electromagnetic waves, but there is no mathematical demonstration as to why this is so. According to the electromagnetic theory, electromagnetic radiation is emitted as magnetic dipole radiation when a charge revolving in an orbit of radius r₀ flows as an alternating current. This paper presents a mathematical proof for the emission of infrared-to-ultraviolet electromagnetic waves of the hydrogen emission spectrum when an alternating current is generated by the oscillation and revolution of an electron around a hydrogen-atom proton at a radius r_nor due to external energy. The mathematical results have demonstrated that the photon is not wave–particle duality of light but an electromagnetic wave of energy form, and the empirical Planck’s constant, h, is mathematically derived for the first time. Therefore, the electromagnetic spectrum above the infrared region can be explained with oscillating and revolving particles with a constant electric charge.