Solutions lecture 7

src/7_tight_binding_model_sol.md

@@ -32,6 +32,7 @@ Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'partic
 Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ with $E=\hbar\omega$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}} = a\sqrt{\frac{\kappa}{m}}\cos(\frac{ka}{2})$$ $$ g(\omega) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \bigg ] = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$
 
 ```python
+pyplot.figure()
 pyplot.subplot(1,2,1)
 k = np.linspace(-pi/2, pi/2, 300)
 pyplot.plot(k, np.cos(k))