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I am trying to find an expression for the density of states of free two-dimensional electrons, as a function of energy, and I am really struggling.

I get that what I am looking for is the number of states per unit area of k-space per unit energy, and in general (3D), this is expressed as

density of states [tex]g(E) = \frac{1}{V}\frac{dN}{dE}[/tex]

However, since this is in 2D, the V is actually an area. In k-space, I think a unit of area is [tex]A=\frac{\pi}{L}\frac{\pi}{L} = \frac{\pi^2}{L^2}[/tex] since[tex] k=\frac{\pi}{L}[/tex] for the smallest allowed length in k-space.

So, what I need is some expression for the number of states, N(E), but presumably have to find it in terms of N(k) first.

So, what I said was that the free electron has energy [tex]E = \frac{\hbar^2}{2m_e}(k_x^2 + k_y^2)[/tex] so that when I know N in terms of k, I can easily convert it to N in terms of E.

So I think I eventually need to get [tex] g(E) = \frac{L^2}{\pi^2}\frac{dN(E)}{dE} [/tex]

However, this is where I have ran out of steam. I haven't been able to come up with an expression for the number of states as a function of wave number, N(k).

Can anyone give me a hand here? It would make a lovely Christmas present

Cheers.

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# The density of states in 2d?

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