# Applying the Heisenberg uncertainty principle to photons [duplicate]

The speed of light is a universal constant, so we definitely know the speed of the photons. If we know the speed, then we should not have any information about their location, because of Heisenberg's uncertainty principle. But I'm one hundred percent sure when light goes through my window.

Why is this so?

• Possible duplicate of Application of Heisenberg's uncertainty principle on photons
– Luke
Sep 4, 2018 at 13:46
• This has a very misleading title, and might be a duplicate of physics.stackexchange.com/q/223573
– Luke
Sep 4, 2018 at 13:46
• (The picture in your post doesn't show the light rays itself, it shows light reflected off the dust in the air. Not sure whether that has anything to do with your question) Sep 4, 2018 at 15:31
• The important thing is that light comes through the window. Interaction with the particles is what makes us see it the way it is and it has nothing to do with my question. Sep 4, 2018 at 15:34
• @Bergi I think that probably is relevant. I think the asker believes that their eyes are detecting light that is travelling along the lines shown in the photograph, rather than light that's reflected off dust into their eyes. Sep 4, 2018 at 16:57

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)

Other people have commented on the distinction between momentum and speed, but there is also a distinction between speed and velocity. Simply knowing an object's speed does not tell you its velocity. If we represent the momentum with a vector, there can be uncertainty in both its length and direction. Thus, even if you know to high precision the frequency of a photon, there will still be uncertainty as to its momentum. Since, given a fixed uncertainty in direction, the uncertainty in momentum is proportional to frequency, the position of high frequency photons can, all else being equal, be located with greater precision that the position of low frequency photons. And given a fixed uncertainty in position, high frequency photons have less uncertainty in direction. When you try to send a beam of light in a particular direction, there is some minimum amount of spreading, and that minimum is inversely proportional to frequency (and thus directly proportional to wavelength) and inversely proportional to the size of the emitter. This is related to the fact that waves with higher wavelength diffract more: because there is more uncertainty in the direction of a long-wavelength photon compared to that of a short-wavelength one, the long-wavelength photon can bend around corners more easily. If you put a high-frequency photon through the double-slit experiment, the diffraction pattern will be more tightly spaced than if you use a low-frequency photon.

This is a less technical answer, by a non-physicist.

The uncertainty principle primarily applies when you examine individual particles. In layman's terms, it says that you can't know both the momentum and position of a particle perfectly accurately: the more accurately you determine one, the less accurate the other is.

However, quantum mechanics allows us to determine probabilities. And when you aggregate enormous numbers of particles, the total measurement comes very close to the expected probabilities. As an analogy, you can't predict whether an individual coin flip will be heads or tails, but if you perform millions of flips, you can be pretty sure that the number of heads will be about half of them.

There isn't anything special about light in your question. I'm also 100% certain about the location and momentum of my chair. Theoretically, the atoms could all suddenly relocate to different places in my room (or anywhere else, for that matter), but the probability is infinitessimal. A few particles might occasionally flit somewhere, but they're negligible when dealing with the chair as a whole.

And the same is true for the light going through your window. Many photons will deviate from the expected path, but when we deal with the enormous number in a ray of sunlight, the average is what we see.

• "I'm also 100% certain about the location and momentum of my chair" - that's a reasonable statement if you're rounding up from a tiny fraction below 100%, but if you mean there's genuinely zero uncertainty, I think your confidence is misplaced. Sep 5, 2018 at 4:10
• @Chappo I mean it in the same way that the OP said he's 100% sure that light is going through the window. Sep 5, 2018 at 16:59
• I'm sitting on the chair, and I have absolutely no concern that it might disappear out from under me (of course, it could break). Even physicists who know all the details of quantum mechanics know that these details can be ignored when dealing with macroscopic effects. Any uncertainty I might have about the location of the chair is far less than the precision of a measuring device I have access to (e.g. a household tape measure). Sep 5, 2018 at 17:03