Density of orbitals in one dimension

[mooeypoo]

Hey guys,

 

I'm studying for my thermo exam, and I have this homework problem:

 

(a) Show that the density of orbitals of a free electron in one dimension is

[math]D_1(\epsilon)=(\frac{L}{\pi})(\frac{2m}{\hbar^2\epsilon})^{1/2}[/math]

 

Okay, so here's what I did:

 

Electrons have 2 possible spins, so my usual "n" would be doubled in occupancy, hence N=2n. And:

 

[math]\epsilon=\frac{\hbar^2\pi^2 n^2}{2m L^2}=\frac{\hbar^2\pi^2 N^2}{8m L^2}[/math]

 

And solving for N, gives me

 

[math]N=\frac{2 \sqrt{2m \epsilon}L}{\pi \hbar}[/math]

 

The energy of the highest filled orbital in the ground state of a free particle gas of fermions of spin 1/2 is

[math]\epsilon_F=\frac{\hbar^2}{2M}\left( \frac{3\pi^2N}{V} \right)^{2/3}[/math]

 

And the density of orbitals at [math]\epsilon_F[/math] is [math]D(\epsilon_F)=\frac{3N}{2\epsilon_F}[/math]

 

So what I was thinking, is putting in both equations into the D(e) equation, something like:

 

[math]D(\epsilon_F)=\frac{3}{2}\left( \frac{2\sqrt{2m\epsilon}L}{\hbar \pi} \right) \left( \frac{\hbar^2}{2M}\left( \frac{V}{3\pi^2 N} \right)^{2/3} \right)[/math]

 

And then insert "N" into the above in the V/N part.

Thing is, this is getting ridiculously challenging to simplify at some point, and further, I'm stuck with "V" in there, which isn't in the form that I'm supposed to present.

 

So after wracking my brain for a while, I went to see what the book's solution attempted. They took a whole different route, though, and I'm not entirely sure why mine doesn't work.

 

After setting up N and epsilon as I did, they stated simply that since

[math]\frac{d\epsilon}{\epsilon}=\frac{dN}{N}[/math]

and epsilon is epsilon_F, we plug in:

[math]D_1(\epsilon)=\frac{dN}{d\epsilon}=N/2\epsilon=\left( \frac{L}{\pi\hbar} \right) \left( \frac{2m}{\epsilon} \right)^{1/2}[/math]

 

Where the heck did the "dN/de" came from? If D is the density of orbitals at epsilon, how is dN/de related to it.. I don't get it. Also, if that IS true, then where did the [math]D=\frac{3N}{2\epsilon_F}[/math] come from?

 

I remember my professor said something about the difference being the dimensions (one dimensional vs 2- or 3-dimensional) but I don't quite understand what's going on with these. I'd have never thought of dN/de in this question, and yet it seems to have made it so simple it's solvable in 3 steps.

 

Thanks in advance,

 

~mooey