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 Schrodinger 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.
Highlights
From the find that a function f (r ) may be expanded into a series of functions ( ) eikr with coefficients equal to (2π)3 2 multiplies the Fourier transform f k of f (r ) we arrive to obtain that a particle moving with celerity v0, momentum p0 creates a wave, confirming the wave-particle duality principle conceived by Planck and Einstein in 1900-1905
Someone has said that “Physics is the studies of Nature, how matter and radiation behave, move and interact thorough space and time
Mathematics, on the other hand, is logical deductive reasoning based on initial assumption
Summary
From the find that a function f (r ) may be expanded into a series of functions ( ) eikr with coefficients equal to (2π) multiplies the Fourier transform f k of f (r ) we arrive to obtain that a particle moving with celerity v0 , momentum p0 creates a wave, confirming the wave-particle duality principle conceived by Planck and Einstein in 1900-1905. By the same formula giving quantum mechanics’ principles we realize that the product of a wave eik0r and an object described by a function f (r ) is a sum ( ) over eikr with coefficients equal to (2π) f k − k0. This opens a simple way to calculate the amplitude of diffraction of a wave by a 3D object such as a semi-space which leads to the Descartes, Snell’s laws, Fresnel equations, by a set of identical objects having different geometric forms such as plane which leads to the Braag’s formula, pyramid, sphere, etc. Details of the finds are explained successively in the following paragraphs
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