It seems that they use \frac{2}{\pi} as the states per unit length, why?
We get many questions about solutions, but it is much more useful to try to derive the DOS starting from the question and see if/where you get stuck. If you get stuck you can post your derivation here and I am very happy to discuss it.
I don’t even know where to start. The lecture notes are very clear but it does not seem to be correct. g(E)=\sum \frac{dn}{dk}\frac{dk}{dE}, using k=\frac{1}{a}\sqrt{\frac{E-E_g}{t_{cb}}} . We arrive at g(E)=\frac{1}{2a\sqrt{t_{cb}(E-E_g)}}\sum \frac{dn}{dk}. I don’t have a formula for n, but in the lecture notes (e.g. tight binding model) it says that in 1D: g(E)=\frac{L}{2\pi}\sum|\frac{dk}{dE}|, which seems to suggest that \frac{dn}{dk}=\frac{L}{2\pi}. But this is incorrect!!
Is there a formula for n(k) that i’m missing?
but you are completely right: dn/dk = L/2\pi in 1D, although you still have to multiply by a factor 2 if you want to account for negative k.
To understand how to derive a density of states, I suggest to start by writing the number of states in a certain energy range as a sum over k-values: n_{states} = \sum_k = \frac{L}{2\pi}\int_{-k_0}^{k_0} dk = 2\frac{L}{2\pi}\int_{0}^{k_0} dk . Then, using the dispersion, you can write this as \int g(E) dE. Note that another factor 2 will enter for the spin degeneracy