Density of states in 2 dimensional box

1. May 8, 2015

hideelo

I am trying to calculate the density of energy states in a two dimensional box. The way my professor did this is by first calculating the amount of states with their energy less than some energy e and taking its derivative with respect to e. In order to see how many energy states there are with energy less than e we are first calculating the amount of momentum states with a k vector less than some k and then translating it into the corresponding condition for energy.

There is one step in the derivation where my professor makes a jump which I do not understand, and I need some help with. I understand that for some given k vector in 2-dimensions, it has magnitude

k = √(kx2 + kx2 ) where kx = π nx/Lx and the same for y.

she then makes the jump that the amount of states with k vectors of magnitude less than some given k is

(1/4)(πk2)/(π/Lx + π/Ly)

I do not understand how she arrived at this, any help would be appreciated

2. May 9, 2015

Jilang

I think you may have written it down wrong. The expression needs to be dimensionless. If the + sign were a x sign that might work. The expression should represent the area of the positive quadrant of an ellipse in k space divided by the area taken up by each state.

3. May 9, 2015

Jilang

Edit. That should be a circle a k space and an ellipse in n space.

4. May 9, 2015

hideelo

Yep, you're right. And thanks