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Density of states in 2 dimensional box

  1. May 8, 2015 #1
    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. jcsd
  3. May 9, 2015 #2
    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.
  4. May 9, 2015 #3
    Edit. That should be a circle a k space and an ellipse in n space.
  5. May 9, 2015 #4
    Yep, you're right. And thanks
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