Solutions lecture 7

src/7_tight_binding_model_sol.md

@@ -55,7 +55,7 @@ pyplot.tight_layout();
 
 ### Subquestion 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 $\frac{d\omega}{dk}$. Draw a coordinate system **next** to the dispersion with *$g(\omega)$ on the y-axis* in which you graph $\big(\frac{d\omega}{dk}\big)^{-1} \approx \frac{1}{\frac{d\omega}{dk}}$.
+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 $\frac{d\omega}{dk}$. Draw a coordinate system **next** to the dispersion with *$g(\omega)$ on the y-axis* in which you graph $\big(\frac{d\omega}{dk}\big)^{-1}$.
 
 ??? hint "Plots"
 
@@ -81,7 +81,7 @@ The Schrödinger equation is given as: $|\Psi\rangle = \sum_n \phi_n |n\rangle$
 
 Solving the Schrödinger equation yields dispersion: $$E(k) = E_0 -t\cos(ka) -t'\cos(2ka)$$
 
-### Subquestiion 3
+### Subquestion 3
 
 $$m_{eff} = \frac{\hbar^2}{2a^2}\frac{1}{t\cos(ka)+4t'\cos(2ka)}$$
 
@@ -117,8 +117,8 @@ pyplot.plot(k1, m(k1,2),'b');
 pyplot.plot(k2, m(k2,2),'b');
 pyplot.plot(k3, m(k3,2),'b',label='t=2t\'');
 pyplot.xlabel('$k$'); pyplot.ylabel('$m_{eff}(k)$');
-pyplot.xticks([-pi,0,pi],[r'$-\pi/a$',0,r'$\pi/a$']);
-pyplot.yticks([-10,10],[]);
+pyplot.xticks([-1.6,0,1.6],[r'$-\pi/a$',0,r'$\pi/a$']);
+pyplot.yticks([0],[]);
 pyplot.tight_layout();
 
 k1 = np.linspace(-1.58, -0.81, 300);
@@ -137,7 +137,7 @@ pyplot.plot(k1, m(k1,10),'k');
 pyplot.plot(k2, m(k2,10),'k');
 pyplot.plot(k3, m(k3,10),'k',label='t=10t\'');
 
-pyplot.legend()
+pyplot.legend();
 ```