How Do You Calculate the Density of States for Massless Particles in a 3D Cube?

[jammydav93]

Homework Statement

Calculate the single particle density of states for massless particles with dispersion E=h_bar ck for a 3D cube of volume V

Homework Equations

E=pc, p=E/c,
dp=dE/c, d^3p = 4pi*p^2 dp
k=sqrt(k_x^2+k_y^2+k_z^2)
k_j = 2pi/L l_j (j=x,y,z)

The Attempt at a Solution

I have tried calculating the density of states in the exact same way as I do for a massive particle but using different energy relations.

Sum(all K)
= sum(all kx,ky,kz)
= int(dl_x dl_y dl_z)
= int ((2pi/Lh_bar)^3 d^3p)
= int ((2pi/Lh_bar)^3 4pi*p^2 dp)
= int ((2pi/Lh_bar)^3 4pi*E^2/c^3 dE)

D(E) = (2pi/Lh_bar)^3 4pi*E^2/c^3

The powers are correct for E and C however I seem to have a dependce on the volume (1/L^3 = 1/V) which I should not be getting - does anyone know why I am getting this?

Thanks,
James

 

[TSny]

k_j = 2pi/L l_j (j=x,y,z)
-----------------------
int(dl_x dl_y dl_z)
= int ((2pi/Lh_bar)^3 d^3p)

From $k_j = \frac{2\pi}{L}l_j$, what do you get for $dl_j$ in terms of $dk_j$?

 

[jammydav93]

I get:

dl_j = L/2pi dk_j = L/(2pi*h_bar) dp_j

Sorry that must have been a typo, I still seem to have an L^3 term which won't go away though.

I now get:
= int(dl_x dl_y dl_z)
= int ((L/2pi*h_bar)^3 d^3p)
= int ((L/2pi*h_bar)^3 4pi*p^2 dp)
= int ((L/2pi*h_bar)^3 4pi*E^2/c^3 dE)

D(E) = (L/2pi*h_bar)^3 4pi*E^2/c^3

 

[TSny]

OK, I think that's corect. But the density of states is often defined on a "per unit volume" basis. If that's what you need, how would you fix your result so that D(E)dE represents the number of states per unit volume with energy between E and E+dE?

 

[paranormal]

Hello James,

Thank you for sharing your solution attempt. It seems like you have the right idea, but there is a small mistake in your calculation. The correct expression for the density of states for massless particles in a 3D cube of volume V is:

D(E) = (V/h_bar^3) * (2pi)^3 * E^2 / c^3

The mistake in your calculation is that you have used the volume of the cube in terms of its side length (V = L^3) instead of using the actual volume V. This is why you have a dependence on the volume in your final expression. By using the correct volume V, you should get the correct result without any dependence on V.

I hope this helps clarify your solution and allows you to arrive at the correct expression for the density of states for massless particles in a 3D cube.

Best,