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For a photon, $$E = pc$$ from the Einstein Energy Equation. Or $$E = hf$$ with $$p = hf/c$$

From the Heisenberg uncertainty principle $$(\Delta x)*(\Delta p) = h/2$$ The maximum value for $\Delta p$ for real values of momentum and also for imaginary cases (I think I didn't really find the maximum of the function using calculus) is $2hf/c$ as the photon can either be going straight up or straight down. Then the minimum value for $\Delta x$ is $c/(4f)$ ... so as frequency increases, one can determine the position to arbitrary large amounts, but at a fixed frequency there will be a uncertainty of $c/(4f)$.

I am not sure if this is true or can be determined by experiment, or if this relation breaks down at large frequencies. It would be interesting to see what happens in an experiment with extremely high frequency (Energy photons and seeing how small $\Delta x$ became, and saw if it tends to follow this relation)

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  • $\begingroup$ A part of your question is wrong, specifically the Heisenberg's uncertainty principle. (∆x) * (∆p) = h/4π , not h/2. $\endgroup$
    – Orion 73
    Commented Apr 7, 2020 at 11:03

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This is actually a bit easier to understand than for conventional particles.

First let's define two things:

  1. A photon's momentum is determined by its wavelength: $p=h \lambda$
  2. A photon's position is determined by the wavepacket

The wave packet is an idea you don't normally think of in introductory physics. In order to have a position, the photon can't just be a perfect sine wave, because that has no defined position (it goes on forever).

an example of a wave packet

So now you can say where the photon is, but in order to make this wave packet, you had to add some extra wavelengths in. If you just imagine taking the Fourier transform of the figure, it won't just be one wavelength.

Therefore, in order to define the position, you had to add some uncertainty in the momentum.

Another example is sound. In this case the tradeoff is frequency vs time. A pure sine wave goes on forever, but the same argument. But to make a shorter sound (that occurs at a well-defined time) you can't specify the wavelength/frequency as precisely. The extreme example is a snare drum hit or a clap, which happen at precise times, but don't really have a sensible frequency that describes them.

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