Deriving the density of states for waves in a blackbody

[daselocution]

Homework Statement

Hey all,

I am having trouble following some of the notes that my professor posted with regards to waves inside a blackbody; here is what he posted: (the part in bold is what I am just not understanding)

"Inside the blackbody box, we need for the position of the walls of the box to be a node for each dimension. For a one-dimensional cavity, e.g., we would choose x = 0 at one wall and then require the other end to be 2a/λ = n (where a is the length of of a side of the cube, which he defined as the shape of this particular blackbody). How many such waves can we fit inside a box of dimensions a^3? We can figure out how to calculate this geometrically by looking at a cube of points in a 3-dimensional space where each point represents a coordinate defined by (nx, ny, nz) and nx = 1, 2, 3, ... and the same for ny and nz. This is a graph in which only positive integer values for nx, ny, and nz are allowed and we want to know the size of the cube (i.e. the number of points defined by the coordinates we just defined) that fit inside of the 1/8 of a sphere whose radius is r = (2a/λ) with shell thickness givenby dr = (2a/λ^2)dλ. The net result of the density of states calculation is

N(λ)dλ= (2πa^3)/(cλ^2) dλ

I have absolutely no idea how he made the jump to that result, and he didn't indicate any other calculations in the note page.

The Attempt at a Solution

Since there was a dλ in the density of states he arrived at, I thought he might have taken the dV/dr of a sphere such that dv=4πr^2 dr

As he noted in the paragraph above, we should only be looking in 1/8 of this region, so I multiplied this dv*1/8 (which is the same effect as just multiplying the original volume by 1/8...) and got

dv=(πr^2)/2 dr

I then substituted dr=(2a/λ^2) as indicated in the above paragraph and got:

dV=aπr²a/λ²
I then substituted in r²=4a²/λ²...

What I did is clearly wrong. I have no idea how he arrived at having a 2 in the numerator rather than a 4, or how he got a c in the denominator. Any help any of you could give me would be HUGELY appreciated.

Thanks

 

[TSny]

It looks to me that your derivation is essentially correct! The instructor's result is clearly incorrect as N(λ)dλ should be dimensionless. You still need to multiply your result by 2 to take into account the 2 independent polarizations for each mode.

More often, the derivation of the density of states is carried out in terms of frequency ($\nu$) or wavenumber (k) instead of wavelength. But you can then easily express these results in terms of wavelength. For example, you can compare your result with that given http://www.phys-astro.sonoma.edu/people/faculty/tenn/p314/BlackbodyRadiation.pdf .

 

[daselocution]

Ah thank you very much. I am glad to see that I am not quite as lost as I thought I was, your input is much appreciated!

 

[daselocution]

You still need to multiply your result by 2 to take into account the 2 independent polarizations for each mode.

Could you please explain to me what you mean by the two independent polarizations for each mode? Does this mean that the standing wave could be sin or -sin? I guess I never really understood this when my professor mentioned it...

 

[TSny]

For a standing wave on a vibrating string, the string could oscillate in a vertical plane. Then you would have vertical polarization. Or, the string could oscillate in a horizontal plane. Other directions of polarization could be obtained by superposition of the vertical and horizontal modes. So, you can take the horizontal and vertical modes as 2 independent polarizations from which all other polarizations can be constructed by superposition. Likewise for electromagnetic waves where the polarization is determined by the direction of the electric field.