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.

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