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

[x, p] = i\hbar

We call it First Commutation Relation. Equivalently we conjecture the validity of another commutation relation

[H, t] = i\hbar

We call it Second Commutation Relation. Based on these two assumptions, we build up Quantum Mechanics as follows

From First Commutation Relation:

[p, x] = -i\hbar \rightarrow p = -i\hbar\frac{\partial{}}{\partial{x}}

From Second Commutation Relation:

[H, t] = i\hbar \rightarrow H = i\hbar\frac{\partial{}}{\partial{t}}

From Classical Mechanics, the definition of Hamiltonian for a particle is

H = \frac{P^2}{2m} + V

Substitute H and P into this equation

H=\frac{P^2}{2m} + V = -\frac{\hbar^2}{2m}\frac{\partial{}^2}{\partial{x^2}} + V = i\hbar\frac{\partial{}}{\partial{t}}

This produces Schrodinger Equation. Next we consider the case for photon. The Hamiltonian for photon is given by Einstein’s Special Theory of Relativity

H^2 = p^2c^2 + m^2c^4

Since photon has no mass

H^2 = p^2c^2

Substitute our formula for Momentum and Hamiltonian

H^2 = -\hbar^2\frac{\partial{}^2}{\partial{x^2}}c^2=-\hbar^2\frac{\partial{}^2}{\partial{t^2}}

We can easily see

\frac{\partial{}^2}{\partial{x^2}} = \frac{1}{c^2}\frac{\partial{}^2}{\partial{t^2}}

It reproduces Maxwell’s Light Equation. Given a standard EM wave equation to be

\psi = Ae^{i(kx - wt)}

We can find its energy by hamiltonian operator

H\psi = i\hbar\frac{\partial{}}{\partial{t}}Ae^{i(kx-\omega t)} = \hbar\omega\psi \rightarrow E=\hbar\omega

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.


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s