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According to refractive index's formula, we say that speed of light is other mediums is smaller than that of speed of light in air/vacuum. What quantum mechanical effects govern the increase in speed of light when it moves from another medium back into air/vacuum?

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Light is a classical emergent phenomenon from the underlying quantum mechanical framework of photons. The wavefunctions of the photons are solutions of a quantized Maxwell's equation, and thus in its complex form has the electric and magnetic fields that in coherence add up to the E and B amplitudes of the classical electromagnetic waves.

To get an intuition of how photons that just have energy (i.e. frequency*h) and spin build up the classical wave see how the polarization of classical waves arises from the spin of the photon this image:

enter image description here

With this background:

What quantum mechanical effects govern the increase in speed of light when it moves from another medium back into air/vacuum ?

The photons always move with velocity c. Within a medium, due to photon-lattice or photon-molecule interactions travel longer lengths within the medium so that apparent speed of the light front being built up by the photons in the medium is slower, even though the photons always move with velocity c from interaction to interaction. Getting out of the transparent medium they build up the light front with no interactions and velocity c.

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  • $\begingroup$ Have there been experiments to confirm that photons always travel at c? Seems like that would be a testable hypothesis . $\endgroup$ – Lambda Apr 5 '17 at 14:31
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    $\begingroup$ @Lambda All of particle physics data is based on Lorenz transformation calculations, and those are completely dependent on the speed of photons being c. see also this physics.aps.org/synopsis-for/10.1103/PhysRevLett.106.243602 $\endgroup$ – anna v Apr 5 '17 at 14:55
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When light travels through space, it always moves at c. Why do we say it moves through water slower than in air? (Or in general, slower in a denser medium than in a less dense medium?)

Imagine light is a attendant that gives parking tickets. The attendant always walks at speed c. If there are no cars on the block, the attendant travels that distance at an apparent speed still equal to c.

Now, say there is a car with an expired meter on the block. The attendant walks at speed c, stops for some time to write a ticket at the car, then continues walking the rest of the block at speed c. As you might guess, the attendant represents a photon of light and cars represent atoms in the matter the light is traveling through. Not all atoms are at the right energy level to absorb and re-emit the photon, so cars requiring a ticket represent the atoms which can actually absorb and re-emit the photon. So when light passes through a region with one atom that will absorb & re-emit it, when a parking attending walks a block with one expired meter, what is the apparent speed through the block, $\frac{distance traveled}{time taken}$?

Well, the attendant traveled the distance of the block d. For the time t, let's break it into three segments: time spent walking the distance $d_1$ to the car at speed c ($t_1=d_1/c$), spending time $t'$ writing the ticket, then time $t_2=d_2/c$ walking the remaining distance $d_2$. Note that the two distances make up the entire distance of the block: $d=d_1+d_2$. So the apparent speed is:

$$c_{app}=\frac{distance}{time} \\ =\frac{d}{t_1+t'+t_2} \\ =\frac{d}{d_1c^{-1}+t'+d_2c^{-1}} \\ = \frac{cd}{d_1+ct'+d_2} \\ = \frac{cd}{d+ct'}\\ =\frac{c}{1+ct'/d}\\$$

$$ c_{app}=\Big(\frac{1}{1+ct'd^{-1}}\Big)c$$

This is the apparent speed of light traveling a distance of $d$ through a medium, with $t'$ being the total time spent "inside" atoms (and bonds) between absorptions and re-emissions. Note that according to this expression $c_{app} \leq c$, with $c_{app}=c$ only when $t'=0$ (when the photon spends no time being absorbed "in" the atom). If the parking attendant stops at a car, indeed any number of cars, his apparent speed is less than c. If he doesn't stop at any cars, his speed is exactly c.

Denser matter (like water) contains more atoms in a given amount of space than less denser matter (such as air). So on a dense block, our parking attendant will be stopping at more cars before reaching the end of the block, lowering his apparent speed.

Thus, light appears to move slower through denser mediums because it spends time being "inside" atoms. But whenever a photon is literally traveling, it travels at c. The space between atoms is a vacuum.

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