Density of states in 2 dimensional box

[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 = √(k_x² + k_x² ) where k_x = π n_x/L_x 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/L_x + π/L_y)

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

 

[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.

 

[Jilang]

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

 

[hideelo]

Yep, you're right. And thanks