Density of states for a free electron

[kottur]

Homework Statement

1. Find the density of orbitals (often called 'density of states') for a free electron gas in
one dimension, in a box of length L.

2. Find the density of orbitals for a free electron gas in two dimensions, in a box with
area A. Compare with the three dimensional case, eqn. (19) on page 187.

eqn. (19): $D(\epsilon)=\frac{V}{2\pi^{2}}\left(\frac{2m}{h^{2}}\right)^{\frac{3}{2}}\epsilon^{\frac{1}{2}}$

Homework Equations

The one above for the second problem because I can't use it for a one dimensional box because it's got V (Volume in it).

The Attempt at a Solution

I'm not sure what equation I should be using, also I'm confused of Fermi and Bose. My professor told me there are two ways to find density of states, one with Fermi and one with Bose (if I understood correctly).

Any help is much appreciated! I really want to understand this! :)

 

[vela]

There is no single equation you should be using. Your text probably derived the density of states in three dimensions for you. The problem is asking you to do the analogous derivations for the one-dimensional and two-dimensional cases. The idea is to make you think through what they did and apply those same techniques so that you'll understand where the final formula came from. So the place to start is to look up the derivation in your book.

The difference between fermions and bosons are how they fill up the available states, which will in turn affect the density of states. How do fermions and bosons behave differently?

 

[kottur]

Fermions have "half" spins and bosons have "whole" spins. I think I need to use fermions because my textbook uses it to derive the formula for 3D DOS.

 

[vela]

You're right about the spins, but what does that have to do with how they fill up states?