A Concise Study about Fourier Transform and Principles of Quantum Mechanics

Abstract

Based on the hypothesis that the  position-representation of a physical state <x|\(\alpha\)> is the Fourier transform of  its momentum-representation <P|\(\alpha\)> and that the time-representation <t|\(\alpha\)> is the inverse Fourier transform of its energy-representation <E|\(\alpha\)>, we are able to find the  Planck relation E = hv, the de Broglie relation P = \(\frac{h}{\lambda}\). Afterward, utilizing the Dirac delta function \(\delta\)(x) and the property FTxf(x) = -i\(\hbar\)\(\partial\), _pFTxf(x) = -i\(\hbar\)\(\partial\), _pF(p) we obtain the links between operators in ordinary and Hilbert spaces leading to the Dirac fundamental commutation relation [\(\hat{p}, \hat{x}\)] = -i\(\hbar\)\(\hat{I}\), 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.