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In quantum mechanics, I know that there is an uncertainty in the speed of the non-relativistic particles (i.e. Heisenberg's uncertainty principle). Is there also an uncertainty in speed of light in quantum physics?

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Actually I think you'll find that the uncertainty principle talks about position and momentum, not speed. The distinction is subtle, but matters a lot for light -- light moves at a fixed speed, but its momentum depends upon its wavelength.

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I know that there is an uncertainty in the speed of the non-relativistic particles (i.e. Heisenberg's uncertainty principle).

It is not the speed of particles, it is the simultaneous measurement ofposition and momentum that are constrained by the HUP, and the HUP holds for relativistic quantum mechanics too.

Lets keep it simple, the link given by PM2Ring gives a theoretical view.

Special relativity holds in the quantum level. Light is composed of photons, which have mass zero and their four vector

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by necessity connects their energy with the momentum , $E=pc$.

The photoelectric effect and innumerable other experiments have shown that the energy of a photon, describes it and is equal to $E=hν$ where ν is the frequency of light built by a large number of photons.

That is how the conundrum arises: if to identify a photon (or any zero mass particle) the Energy has to be known exactly, the momentum has to be known exactly, and by Heisenberg's principle the position is undefined. The quantum mechanical solution describing a photon is a plane wave, and plane waves are all over space time.

As experiments show that photons are localized, see for example this single photon at a time double slit experiment, where photons leave a footprint on the screen,(the accumulation shows the classical interference of light,) one uses the wave packet solutions to understand free photons.

wavepack

This means that the uncertainty goes to the exact value of the frequency/wavelength of the free photon measured/seen (as in the link ), if we want to locate it finely.

Fortunately to calculate distributions to compare to experimental results this is not necessary, as Quantum Field Theory and the Feynman diagram representations of the integrations work without this complication.

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