Principles of Quantum Mechanics and Laws of Wave Optics from One Mathematical Formula

Abstract

Finding that in the formula of expansion of a function into a series of wave-like functions the coefficients are its Fourier transforms, if existed, we deduce mathematically all the principles and hypothesis that illustrated physicists utilized to build quantum mechanics a century ago, beginning with the duality particle-wave principle of Planck and including the Schr ö dinger equations. By the way, we find a simple Fourier transform relation between Dirac momentum and position bras and a useful permutation relation between operators in phase and Hilbert spaces. Moreover, from the found particle-wave duality formula we prove and obtain again essentially by mathematical analysis all the laws of wave optics concerning reflections, refractions, polarizations, diffractions by one or many identical 3D objects with various forms and dimensions.