Solutions lecture 7

src/8_many_atoms_sol.md

@@ -36,8 +36,8 @@ A unit cell consits of exactly one $t_1$ and exactly $t_2$ hopping.
 ### Subquestion 2
 
 Using the hint we find:
-$$ E \phi_0 = \epsilon \phi_0 + t_1 \psi_0 + t_2 \psi_{n-1} $$
-$$ E \psi_0 = \epsilon \psi_0 + t_2 \phi_{n+1} + t_2 \phi_0 $$
+$$ E \phi_0 = \epsilon \phi_0 + t_1 \psi_0 + t_2 \psi_0 e^{-ika} $$
+$$ E \psi_0 = t_1 \phi_0 e^{ika} + t_2 \phi_0 + \epsilon \psi_0 $$
 
 Notice that the hopping, in this case, is without the '-'-sign!
 
@@ -67,7 +67,7 @@ pyplot.plot(k, -np.sqrt(t1**2 + t2**2+2*t1*t2*np.cos(k)),'b')
 
 pyplot.xlabel('$ka$'); pyplot.ylabel(r'$E-\epsilon$')
 pyplot.xlim([-2*pi,2*pi])
-pyplot.xticks([-2*pi, -pi, 0, pi,2*pi], [r'$-2ka$',r'$-ka$', 0, r'$ka$',r'$2ka$'])
+pyplot.xticks([-2*pi, -pi, 0, pi,2*pi], [r'$-2\pi$',r'$-\pi$', 0, r'$\pi$',r'$2\pi$'])
 pyplot.yticks([-t1-t2, -np.abs(t1-t2), 0, np.abs(t1-t2), t1+t2], [r'$-t_1-t_2$',r'$-|t_1-t_2|$', '0', r'$|t_1-t_2|$', r'$t_1+t_2$']);
 pyplot.vlines([-pi, pi], -3, 3, linestyles='dashed');
 pyplot.hlines([-np.abs(t1-t2), np.abs(t1-t2)], -2*pi, 2*pi, linestyles='dashed');
@@ -79,3 +79,9 @@ pyplot.legend(loc='lower center');
 
 Notice that the red shaded area is not a part of the *Band structure* anymore!
 
+### Subquestion 5
+
+Both $v_g$ and $m_{eff}$ can be calculated using the respective 
+
+
+