- #1
TIF141
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- Homework Statement
- I'm trying to work through the derivation of the density of states function in a Fermi sphere. I understand the derivation except for one part when n(k)dk changes to n(E)dE with the only explanation being "by definition".
- Relevant Equations
- Volume of k-space per electron energy state = ##\frac{L^3}{4 \pi^3}##
Hence energy states per unit volume is ##\frac{4 \pi^3}{L^3}##
Volume of k-space between spheres of radius ##k## and ##k + dk## is ##4 \pi k^2 dk##.
Number of states in that volume of k-space, ##n(k)dk## is: $$n(k)dk = (\frac{L^3}{4 \pi^3}) \cdot 4 \pi k^2 dk = \frac{L^3}{\pi^2}dk$$.
Then the notes state that by defintion, ##n(k)dk = n(E)dE##, and hence $$n(E)d(E) = \frac{L^3}{\pi^2}dk$$.
I don't quite see why this is true - isn't it the case that each coordinate in k-space has 2 energy states associated with it? Then how is the number of energy states in that volume that same as the number of k states? If someone can clear up my confusion that would be great, thanks!
Then the notes state that by defintion, ##n(k)dk = n(E)dE##, and hence $$n(E)d(E) = \frac{L^3}{\pi^2}dk$$.
I don't quite see why this is true - isn't it the case that each coordinate in k-space has 2 energy states associated with it? Then how is the number of energy states in that volume that same as the number of k states? If someone can clear up my confusion that would be great, thanks!
