# Applying the Heisenberg uncertainty principle to photons

The Heisenberg Uncertainty Principle does not involve speed. It involves momentum, and this is one of the places where that distinction is very important. Photons all travel at the same speed, yes, but their momentum can take on any value. As such, the uncertainty in its position and the uncertainty in its momentum are still linked in the same way they would be for ordinary matter.

In fact, due to the fact that photons are traveling at the speed of light, their energy and momentum are related by $E=pc$. Applying the usual Planck-Einstein relation $E=hf$, we can see that an uncertainty in the photon's momentum is also directly proportional to the uncertainty in the photon's frequency, which might be easier to picture (i.e. in the case of photons, the Heisenberg Uncertainty Principle links the uncertainty in their position and the uncertainty in their frequency).

If we know the speed, then I should not have any information about their location because of Heisenberg's uncertainty principle.

The uncertainty principle in its common forms says:

$$\Delta p\Delta x \ge \frac \hbar 2$$

Now if you know the speed of a photon you know absolutely nothing about the momentum of a photon because its momentum is $\frac{hf}{c}$ and you don't know the frequency $f$, and hence you do not know $\Delta p$ either.

Likewise, knowing it came through the window doesn't really say a lot about $\Delta x$ either.

Any measurement of $f$ will have an uncertainty to it and hence there will be an uncertainty to $x$.

I would like to mention some interpretation that made me understand these concept easier for myself.

Imagine that your window shrank into approximately wavelength size. You would certainly observe diffraction phenomenon under this condition.

You can interpret it as an outcome of uncertainty principle in a case when X axis is paralel to the wall where you installed your tiny window.

(After you decreased the uncertainty of photon position [on X axis], the photon increased its uncertainty of momentum [on X axis too] for the uncertainty principle to be conserved)