Waves, Energy, Blackbodies and Modes

[Quelsita]

I am going over some notes and am trying to fit some pieces together. For some reason I keep confusing myself as to what exactly a "mode" is. Is a mode a wave? or a frequency?

Also, how does a mode relate to the degrees of freedom for a particle in a system?


Thanks!

 

[Andy Resnick]

In the context of blackbody (cavity) radiation, a mode is an eigenstate of the radiation field within the cavity. That is, it's a configuration of the electromagnetic field that can exist stably within the cavity: for simple cavitygeometries, analytic expressions for the modes can be written down fairly easily (sines and cosines, bessel functions, hermite polynomials ,etc). Modes will vary with frequency, and can also have different spatial profiles.

Each mode can also be considered a degree of freedom of the electromagnetic field- an arbitrary field can be decomposed into a mode distribution.

Does that help?

 

[Quelsita]

a mode is an eigenstate of the radiation field within the cavity. That is, it's a configuration of the electromagnetic field that can exist stably within the cavity.

That makes sense but "eigenstate" threw me off a bit. So, if an eigenvalue is a scalar, then that just means an eigenstate for, say a photon, would be every possible path it can take?
This is why a mode is a degree of freedom, because it describes the possible configurations?

Modes will vary with frequency, and can also have different spatial profiles.

So, if you consider a blackbody radiating energy like a standing wave, it has a certain amount of modes possible which increase with frequency.
I know that modes have equal energy and as frequency increases, the energy becomes infinitely large but how does a degree of freedom have energy?

Am I making sense, or just confusing things further?

 

[Andy Resnick]

Forget about the photon picture for now- we are discussing the field *within* a cavity

http://en.wikipedia.org/wiki/Transverse_mode (for example pictures.)
http://en.wikipedia.org/wiki/Longitudinal_mode (for wavelength/frequency info)

Each of the transverse modes are the same wavelength and frequency. If you like, they are standing wave patterns- just like the longitudinal modes are standing wave patterns. For a cubic cavity, the mode structure is very simple- sines and cosines in all three dimensions, with an additional factor of 2 for polarizations.

The missing piece (so far) from this discussion is thermal equilibrium- what is the mode density for a radiation field at thermal equilibrium? As you point out, with increasing frequency there is increasing energy, so we have to take into account the energy dependence of the modes.

http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/rayj.html#c2
http://hyperphysics.phy-astr.gsu.edu/Hbase/mod6.html