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Im trying to calculate the impulse of the particles of a light ray in water assuming its freely traveling through and not bumping into any molecules.

This i what i tried:

We know that:

$$E=h \cdot f$$ $$p=\frac{h}{2\cdot \pi}\cdot k$$

Photons travel with the speed of light. In this case it should be the speed of light in water.

By Einstein: $$E=\gamma \cdot m \cdot c_{vacuum}^2 = h \cdot f$$ Also: $$p=\gamma \cdot m \cdot c_{water}$$

It follows: $$p=\frac{E \cdot c_{water}}{c_{vacuum}^2}=\frac{h \cdot f \cdot c_{water}}{c_{vacuum}^2}$$

But according to wikipedia: the dispersion relation is just $\omega = k \cdot c_{water}$ Then: $$p=\frac{h}{2 \cdot \pi} \cdot k = \frac{h \cdot f}{c_{water}}$$

So what is correct? And where did I go wrong? Can someone help me? Im so stupid i cant solve this at all :(

user2276094
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  • "In this case it should be the speed of light" photons always travel at c. It is the emergent light from multitudes of that energy photons that changes velocity in medium. See this answer of mine to get an intuition https://physics.stackexchange.com/questions/677393/why-only-the-wavelength-and-speed-of-refracted-light-traveling-inside-a-transpar/677398#677398 – anna v Nov 27 '21 at 14:51
  • So the reason the whole wave travels a bit slower is due to dispersion and the photons have longer travel paths? They still travel at c_vaccum though? – user2276094 Nov 27 '21 at 15:06
  • Yes, thats right. It is mathematically complicated and needs quantum field theory to calculate, but in effect, photons are always at vacuum traveling with c, in longer paths. If they do not move through vaccum, they will interact quantum mechanically with the particle or the field that occupies the space, and get out of the classical beam. – anna v Nov 27 '21 at 15:50

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