Can the de broglie formula still hold in a medium with reduced speed of light?

[Andeplane]

Hi, I was discussing a question with a Ph.D-student at my uni that neither him or the professor were able to answer.

The situation is the following;
Light enters a medium going from i.e. vacuum. We know that the energy of the photons in the vacuum is
$$E = h\nu$$
and we have the de broglie formula
$$p = \frac{h}{\lambda}$$

The relation [tex]c=\lambda\nu[\tex] tells us that the wavelength [tex]\lambda[\tex] decreases if the speed of light decreases (since frequency is constant). Does the de broglie formula still hold now? In that case, the momentum increases as the wavelength decreases, eventually reaches infinity when the speed of light goes to zero.

In my eyes, this cannot be true, so the de broglie formula cannot hold for reduced speed of light (also remember that when the light goes out of the medium again, the speed and hence the momentum is back to 'normal' again.

If we look at the particle interpretation of light, the photon will probably hit electrons and excite them and quickly send a new photon again. The reduced speed of light might be a consequence of many collisions.

The problem isn't quite clear, but the de broglie formula, does it hold in non-vacuum material?

 

[Matterwave]

The reduced speed of light is only an apparent effect of the collisions (absorptions and re-emissions). At any time the light is actually traveling, it's traveling at c, I believe.

 

[DrDu]

No, that's all ok. That the momentum changes when entering the medium is not astonishing and that it goes back to the value before entering the medium when leaving isn't either. Thats analogous to a comet approaching the sun where its momentum increases and decreases again when leaving the sun. The only astonishing point is that the momentum is inversely proportional to velocity, but that is more due to different conventions on the zero point of energy in relativistic and non-relativistic problems.

 

[jtbell]

The FAQ at the top of the General Physics forum has an entry that addresses photons and light in a medium.

 

[DrDu]

An even better analogy from solid state physics: Electrons have different "effective" masses in different semiconductors. This effective masses are due to the electrons carrying with them a polarization cloud of different size in the different media.
We now can think of an electron changing from one semiconductor to another without change of energy: p^2/2m will remain constant but m will change. That means that p1/p2=sqrt(m1/m2) but v1/v2=sqrt(m2/m1), so in one medium the electrons have a higher momentum but are slower.
The analogy is probably very close as both the optical polarization and the mass of the electrons are in both cases given as the omega dependent terms of the self energy.