From Maxwell’s equations to Quantum Mechanics: an introduction

In this elementary but original introduction to the close relationship between the quantum mechanics in complex tempered distribution space and Maxwell’s electromagnetism, we start from an elementary (but representative) case. We prove that, in the chosen particular case, the Maxwell’s equations in empty space can be reformulated as a unique equation inside a subspace of solenoidal smooth vector fields. Moreover, we reinterpret the curl operator as the magnitude of the wave-vector operator. The unique complex equation condensing the two curl Maxwell’s equations reveals to be a faithful representation of the quantization of the Einstein’s energy relation for photons, or, if preferred, the relativistic SchrÅNodinger equation for a massless particle.