Merge branch 'master' into local

src/1_einstein_model.md

@@ -232,17 +232,17 @@ n(\omega,T)=\frac{1}{ {\rm e}^{\hbar\omega/k_{\rm B}T}-1}
 $$
 The Bose-Einstein distribution describes the occupation probability of a state at a given energy $\hbar \omega$. Using the Bose-Einstein distribution, we can calculate the expectation value of the energy stored in the oscillator  
 $$
-\bar{\varepsilon}=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1}
+\langle E \rangle=\frac{1}{2}\hbar\omega_0+\frac{\hbar\omega_0}{ {\rm e}^{\hbar\omega_0/k_{\rm B}T}-1}
 $$
 
 The left plot below shows the Bose-Einstein distribution vs energy. We see that low-energy states are more likely to be occupied than high-energy states. The right plot shows the increasing thermal energy in the oscillator for increasing temperature and highlights the zero-point energy $\hbar\omega_0/2$ that remains in the oscillator at $T=0$ because of the uncertainty principle. Moreover, we see that the energy in the oscillator only starts to increase significantly when $kT>\hbar \omega$. I.e., the heat capacity only becomes significant for $kT>\hbar \omega$ and goes to zero when $T\rightarrow0$ 
 
 ```python
 xline = [1, 1];
-yline = [0, 2];
+yline = [0, 1];
 fig, (ax, ax2) = pyplot.subplots(ncols=2, figsize=(10, 5))
 omega = np.linspace(0.1, 2)
-ax.plot(omega, 1/(np.exp(omega) - 1), '-', xline, yline, 'k--')
+ax.plot(omega, 1/(np.exp(omega) - 1), '-', xline, yline, 'r--')
 ax.set_ylim(0, top=3)
 ax.set_xlim(left=0)
 ax.set_xlabel('$\hbar \omega$')
@@ -251,20 +251,21 @@ ax.set_xticklabels(['$0$', '$k_B T$'])
 ax.set_ylabel('$n$')
 ax.set_yticks([1, 2])
 ax.set_yticklabels(['$1$', '$2$'])
-ax.text(1, 0.5, r'$\hbar \omega = kT$', ha='left', color='r');
+draw_classic_axes(ax, xlabeloffset=.2)
+ax.text(1, 1, r'$\hbar \omega = kT$', ha='left', color='r');
 temps = np.linspace(0.01, 2)
 ax2.plot(temps, 1/2 + 1/(np.exp(1/temps)-1))
 ax2.set_ylim(bottom=0)
 ax2.set_xlabel('$k_B T$')
 ax2.set_xticks([0, 1])
-ax2.set_xticklabels(['$0$', r'$\hbar \omega$'])
-ax2.set_ylabel(r"$\bar\varepsilon$")
+ax2.set_xticklabels(['$0$', r'$\hbar \omega_0$'])
+ax2.set_ylabel(r"$\langle E \rangle$")
 ax2.set_yticks([1/2])
-ax2.set_yticklabels([r'$\hbar\omega/2$'])
+ax2.set_yticklabels([r'$\hbar\omega_0/2$'])
 draw_classic_axes(ax2, xlabeloffset=.15)
 ```
 
-We now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\bar{\varepsilon}$ with respect to $T$.
+We now calculate the heat capacity per atom $C$ explicitly. To do so, we need to differentiate $\langle E \rangle$ with respect to $T$.
 $$
 \begin{multline}
 C = \frac{\partial\bar{\varepsilon}}{\partial T}