What is the density of states in k-space?

AI Thread Summary
The discussion centers on understanding the density of states in k-space, specifically proving the relationship D(k)d3k = (V/4π^3)4πk^2dk. The user grapples with the concept of density being related to volume, expressing confusion about the integration element d3k. They recognize that the term 4πk^2dk corresponds to the volume element in spherical coordinates. The user concludes that the discrepancy in their results compared to their notes is due to spin degeneracy, indicating a deeper understanding of the topic. Overall, the conversation highlights the complexities involved in deriving the density of states in k-space.
RadonX
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Homework Statement



I'm having a dumb moment in proving why the following is true:

D(k)d3k=\frac{V}{4\pi^{3}}4\pi k^{2}dk

Homework Equations



NA

The Attempt at a Solution



I realize that the second part 4\pi k^{2}dk is an integration element d3k. But the density of states I can't figure out.
To me it seems that density should be something ON volume. But here it's the other way around.

I'm sure I've done this before and it was fine. Just having a dumb moment that's wasting quite a bit of my time.
 
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I THINK I have it.
For each axis we have a state of a different k value at every;
\frac{2 \pi}{a}n
So rearranging for n and summing up (integrating) over each axis we end up with;
(\frac{a}{2 \pi})^3 \int d^{3} k
The factor of 2 difference between this answer and the one quoted in my notes is due to spin degeneracy.

Does all that sound right?
 
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