Solutions lecture 7

src/8_many_atoms_sol.md

@@ -69,7 +69,7 @@ pyplot.xlabel('$ka$'); pyplot.ylabel(r'$E-\epsilon$')
 pyplot.xlim([-2*pi,2*pi])
 pyplot.ylim([-2*(t1-t2),2*(t1-t2)])
 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.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], -2*(t1-t2)*1.1,2*(t1-t2)*1.1, linestyles='dashed');
 pyplot.hlines([-np.abs(t1-t2), np.abs(t1-t2)], -2*pi, 2*pi, linestyles='dashed');
 pyplot.fill_between([-3*pi,3*pi], -np.abs(t1-t2), np.abs(t1-t2), color='red',alpha=0.2);
@@ -89,8 +89,7 @@ Both $v_g$ and $m_{eff}$ can be calculated using the respective formulae given i
 We know $g(E) = \frac{dN}{dk} \frac{dk}{dE} = \frac{L}{2\pi} \frac{1}{v_g}$ with $v_g$ from the previous subquestion. Graphically the density of states looks accordingly:
 
 ```python
-pyplot.subplot(1,2,1)
-pyplot.subplot(1,2,1)
+pyplot.subplot(1,3,1)
 k = np.linspace(-2*pi, 2*pi, 400)
 t1 = 1;
 t2 = 1.5;
@@ -105,7 +104,7 @@ pyplot.xlim([-2*pi,2*pi])
 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.subplot(1,2,2)
+pyplot.subplot(1,3,2)
 w = np.sqrt(t1**2 + t2**2+2*t1*t2*np.cos(k))
 pyplot.hist(w,30, orientation='horizontal',ec='black',color='b');
 pyplot.hist(-w,30, orientation='horizontal',ec='black',color='b');
@@ -114,6 +113,15 @@ pyplot.ylabel(r'$E-\epsilon$')
 pyplot.yticks([],[])
 pyplot.xticks([],[])
 pyplot.tight_layout();
+
+pyplot.subplot(1,3,3)
+w = -(t1+t2)*np.cos(k/2)
+pyplot.hist(w,60, orientation='horizontal',ec='black',color='r');
+pyplot.xlabel(r'$g(E)$')
+pyplot.ylabel(r'$E-\epsilon$')
+pyplot.yticks([],[])
+pyplot.xticks([],[])
+pyplot.tight_layout();
 ```