Solutions lecture 7

src/7_tight_binding_model_sol.md

@@ -29,31 +29,31 @@ Hint: What kind of particles obey Bose-Einstein statistics? What kind of 'partic
 
 2.
 
-Group velocity is given as $v=\hbar^{-1}\frac{\partial E}{\partial k}$ with $E=\hbar\omega$ and $g(\omega) = \frac{dN}{d\omega} = \frac{dN}{dk}\frac{dk}{d\omega}$. So we find: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}} = a\sqrt{\frac{\kappa}{m}}\cos(\frac{ka}{2})$$ $$ g(\omega) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \bigg ] = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$
+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: $$ v(k) = \frac{a}{2}\sqrt{\frac{2\kappa}{m}}\frac{\sin(ka)}{\sqrt{1-\cos(ka)}}$$ $$ g(\omega) = \frac{L}{2\pi}\frac{d}{d\omega} \bigg [\frac{2}{a}\sin^{-1}(\sqrt{\frac{m}{\kappa}}\frac{\omega}{2}) \bigg ] = \frac{L}{2\pi a} \sqrt{\frac{m}{\kappa}} \frac{1}{\sqrt{1-\frac{m\omega^2}{4\kappa}}}$$
 
 3.
 
 ```python
-pyplot.figure()
 pyplot.subplot(1,2,1)
-k = np.linspace(-pi/2, pi/2, 300)
-pyplot.plot(k, np.cos(k))
-pyplot.xlabel('$k$'); pyplot.ylabel('$v(k)$')
-pyplot.xticks([-pi/2, 0, pi/2], [r'$-\pi/2$', 0, r'$\pi/2$'])
-pyplot.yticks([0, 1], [0, r'$2\sqrt{\frac{\kappa}{m}}$'])
-pyplot.hlines([1], -pi/2, 0, linestyles='dashed')
+k = np.linspace(-pi+0.01, pi-0.01, 300)
+pyplot.plot(k, np.sin(k)/(np.sqrt(1-np.cos(k))));
+pyplot.xlabel('$k$'); pyplot.ylabel('$v(k)$');
+pyplot.xticks([-pi, 0, pi], [r'$-\pi/2$', 0, r'$\pi/2$']);
+pyplot.yticks([-np.sqrt(2), 0, np.sqrt(2)], [r'$-2\sqrt{\frac{\kappa}{m}}$', 0, r'$2\sqrt{\frac{\kappa}{m}}$']);
+pyplot.tight_layout();
 
 pyplot.subplot(1,2,2)
-w = np.linspace(-0.95, 0.95, 300)
-g = 1/np.sqrt(1-w**2)
-pyplot.plot(w, g)
-pyplot.xlabel('$k$'); pyplot.ylabel('$v(k)$')
-pyplot.xticks([-pi/2, 0, pi/2], [r'$-\pi/2$', 0, r'$\pi/2$'])
-pyplot.yticks([0, 1], [0, r'$2\sqrt{\frac{\kappa}{m}}$'])
-pyplot.hlines([1], -pi/2, 0, linestyles='dashed')
+w = np.linspace(-0.95, 0.95, 300);
+g = 1/np.sqrt(1-w**2);
+pyplot.plot(w, g);
+pyplot.xlabel(r'$\omega$'); pyplot.ylabel('$g(w)$');
+pyplot.xticks([-1, 0, 1], [r'$-2\sqrt{\frac{k}{m}}$', 0, r'$2\sqrt{\frac{k}{m}}$']);
+pyplot.yticks([0.5, 1], [0, r'$\frac{L}{2\pi a}\sqrt{\frac{\kappa}{m}}$']);
+pyplot.tight_layout();
+pyplot.show()
 ```
 
 4.
 
-
+Hint: The group velocity is given as $v = \frac{d\omega}{dk}$, draw a coordinate system **under** or **above** the dispersion graph with $k$ on the x-axis in which you draw $\frac{d\omega}{dk}$.