There is a concept in Schrodinger equation that has never been taught in any of my Quantum Mechanics lectures.
We assume the validity of the following commutation relation
We call it First Commutation Relation. Equivalently we conjecture the validity of another commutation relation
We call it Second Commutation Relation. Based on these two assumptions, we build up Quantum Mechanics as follows
From First Commutation Relation:
From Second Commutation Relation:
From Classical Mechanics, the definition of Hamiltonian for a particle is
Substitute H and P into this equation
This produces Schrodinger Equation. Next we consider the case for photon. The Hamiltonian for photon is given by Einstein’s Special Theory of Relativity
Since photon has no mass
Substitute our formula for Momentum and Hamiltonian
We can easily see
It reproduces Maxwell’s Light Equation. Given a standard EM wave equation to be
We can find its energy by hamiltonian operator
It predicts correct energy of each photon as predicted by Einstein and Planck.
As a concluding remark, the assumption of the validity of two commutation relation turns out to be generally true even in relativistic case. Interestingly, one can also show that the two commutations are essentially equivalent. Dropping the First Commutation Relation, our Quantum Mechanics builds itself upon the incompatibility of energy and time, which it’s absolutely mysterious, utterly intriguing and extremely suggestive that an internal structure of space-time must remain undiscovered that governs this incompatibility.