Fourier Transform, Dirac Commutator, Energy Conservation, and Correspondence Principle for Electrical Engineers

Abstract

The canonical commutation relation is a fundamental postulate of the quantum theory regarding the operators needed in quantum description of physical observables. It is shown that the Fourier transform is derivable from this seemingly simple postulate along with the basic properties of the position and momentum operators. Further discussions on the canonical commutation relation reveal its connection to a more fundamental notion that energy must be conserved. This discussion also unveils the mathematical homomorphism between the classical and quantum theories for systems represented by sum separable Hamiltonians. Another link between the classical and quantum theories is established by the correspondence principle which states that the classical theory emerges from quantum theory in the limit of vanishingly small Planck constant. Finally, the quantum Maxwell's equations, which have been derived in our previous works, are presented and briefly discussed, and the 3-D mode transform is derived that can be interpreted as a generalization of the Fourier transform. We present both the details and meanings of the 3-D mode transform which will serve as a foundation for a full 3-D quantum finite-difference time-domain method.