A Concise Study about Fourier Transform and Principles of Quantum Mechanics

Based on the hypothesis that the position-representation of a physical state <x| > is the Fourier transform of its momentum-representation <P| > and that the time-representation <t| > is the inverse Fourier transform of its energy-representation <E| >, we are able to find the Planck relation E = hv, the de Broglie relation P = . Afterward, utilizing the Dirac delta function (x) and the property FTxf(x) = -i , pFTxf(x) = -i , pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [ ] = -i , the Schrödinger equations, the Heisenberg uncertainty principle in quantum mechanics, the annihilation & creation of a photon from excitation & de-excitation of an atom according to Bohr.