- #1
roam
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Homework Statement
I need some help with the following problem:
Homework Equations
##\rho(k) dk = \frac{L}{\pi} dk##
##L=Na##
##\omega^2= \omega_m^2 \ sin^2 (qa/2)##
The Attempt at a Solution
The density of states is given by:
##g(\omega)= \rho (k) / \frac{dw}{dk}##
Where
##\frac{d\omega}{dk} = \frac{a}{2} \ cos \frac{qa}{2}##
##g(\omega) = \frac{L}{\pi} \frac{2}{a} \frac{1}{\omega_m \ cos (qa/2)}##
Using the identity
##sin^2 x + cos^2 x =1 \implies cos x = \sqrt{1-sin^2 x} , \ cos(qa/2)=\sqrt{1-sin^2(qa/2)}##
We get
##g(\omega)=\frac{L}{\pi} \frac{2}{a \omega_m \sqrt{1-sin^2(qa/2)}} = \frac{2Na}{a \pi \omega_m \sqrt{1-sin^2(qa/2)}}##
##\therefore g(\omega)= \frac{N}{\pi} \frac{2}{\sqrt{\omega_m^2 -\omega^2}}##
##\therefore g(\omega)= \frac{N}{\pi} \frac{2}{\sqrt{\omega_m^2 -\omega^2}}##
But this is not the correct answer. Why is there a "2" on the numerator, and how can we get rid of this factor of 2?
Did I make a mistake, or is this a typo in the question?
Any help is greatly appreciated.
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