Analogue of the S.E. for photons?

[TriKri]

Hello! What wave equation describes the motion of light? And how do you show that light will necessarily get different speeds for different frequencies in diffractive materials? This would be the analogue of the Schrödinger equation for photons.

 

[DrDu]

Maybe the Maxwell equations?

 

[TriKri]

I mean, there is the electromagnetic wave equation, but according to that all frequencies travel with the same speed, which is clearly not the case for all medias.

 

[tom.stoer]

You can include matter using the tensors for permittivity and permeability ε = ε(E,B) and μ = μ(E,B) and writing
D = εE for the displacement and
H = B/μ for the magnetization

Formally the Maxwell equations look the same using D and H, but you should keep in mind that in matter ε and μ become field-strength dependent tensors.

 

[TriKri]

How can that explain that light doesn't always travel with the same speed in all medias and for all frequencies?

 

[tom.stoer]

Because the Maxwell equations in media are non-linear in E and B due to ε = ε(E,B) and μ = μ(E,B). That means that the (originally) linear differential equations become non-linear which prevents us from solving them with a simple ansatz in x-ct and x+ct as is the case for the equations in vacuum

 

[DrDu]

But the different speed of light already happens for systems with a linear dependence of D on E. So it's not primarily a question of non-linearity.

 

[TriKri]

You say that you can write ε and μ as functions, ε(E,B) and μ(E,B); what do they look like in reality? Or maybe even better, how can you write ε and μ as functions of the frequency or the wavelength? Then you can derive ε and μ as functions of E and B from those.

 

[DrDu]

I wouldn't write permeability and permittivity as functions of E and B but rather D(E). In the optical region, one usually choses H=B, so mu=1. The dependence of
Then, at least in homogeneous media, one can write for the Fourier transformed fields:
$$D(\omega,k)=\epsilon(\omega,k) E(\omega,k)+$$ terms quadratic (and higher orders) in E.
Note that epsilon is a tensor. The terms of higher order are responsible for the effects of non-linear optics like frequency doubling etc.
Approximate analytical expressions for epsilon exist e.g. for a homogeneous gas of electrons. Look for Lindhard dielectric function.