Free Electron Density of States question

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SUMMARY

The discussion centers on the relationship between free electron waves and their wavevector k, specifically addressing the factor of 2π in the expression < k | k' > = (2.pi)^3 . d( k - k'). The participant questions the motivation behind this factor and its connection to the density of states, which is stated to be 1 / (2.pi)^3. The resolution lies in recognizing that the integral of exp(ikx) over all x yields 2π δ(k), indicating a connection to Fourier Transforms.

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Master J
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I have seen thee following relation in regards to free electron waves with wavevector k :

< k | k' > = (2.pi)^3 . d( k - k') where d() is a Dirac delta function.

Why the 2.pi factor?? I can't seem to motivate it. Also, from this, it is stated that the density of states is then 1 / (2.pi)^3 ? I don't see how this relates to that factor at all?? how does one deduce that from the above?
 
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This follows from the fact that the integral of exp(ikx) over all x is 2π δ(k).
 
Ah I see. That integral looks like a Fourier Transform?
 

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