A mathematical modeling concept of the time dependent Schrödinger's equation using Maxwell's equations and special relativity.
Abstract
This theoretical study aims to derive the time dependent Schrödinger's equation using the electric field wave equation derived from Maxwell's equations and Einstein's relativistic energy-momentum relation. First, the electric field wave equation is derived from Maxwell's equation. Where the solution of the wave equation is a function of the frequency and wave function, and in order to associate the solution of the wave equation with the properties of particles, the frequency is expressed in terms of photon energy and the wave function is expressed in terms of photon momentum and it was done using special relativity, and then it became clear that it is possible to express energy and momentum of the photon with partial differential forms called operators. . By symmetry, it is very reasonable to assume that a particle can have the wave-particle duality. Therefore, last, it was established a wave equation for a moving particle by replacing the vector of the electric field with the wave function where the two terms are related to the numerical density of the photon, and the expression of energy and momentum of particle with their operators.
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