The results for deriving the density of states in different dimensions is as follows:
- 3D: $g(k)dk = 1/(2\pi)^3 4 \pi k^2 dk$
- 2D: $g(k)dk = 1/(2\pi)^2 2 \pi k dk$
- 1D: $g(k)dk = 1/(2\pi) 2 dk$
I get for the 3d one the $4\pi k^2 dk$ is the volume of a sphere between $k$ and $k + dk$. Similarly for 2D we have $2\pi kdk$ for the area of a sphere between $k$ and $k + dk$. However I am unsure why for 1D it is $2dk$ as opposed to $2 \pi dk$. So could someone explain to me why the factor is $2dk$?