Solutions lecture 7

src/7_tight_binding_model_sol.md

@@ -16,4 +16,4 @@ Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'partic
 
 2. 
 
-Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$ with $E=\hbar\omega$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}} = a\sqrt{\frac{\kappa}{m}}\cos(\frac{ka}{2})$$ $$ g(k) = \frac{L}{2\pi}\frac{d}{d\omega} \big (\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \big ) = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$
+Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$ with $E=\hbar\omega$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}} = a\sqrt{\frac{\kappa}{m}}\cos(\frac{ka}{2})$$ $$ g(k) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \bigg ] = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$