add an exercise about phonon numbers

src/2_debye_model.md

@@ -131,10 +131,37 @@ draw_classic_axes(ax, xlabeloffset=0.3)
 
 ## Exercises
 
+### Exercise 1: What is a phonon
+
+The probability to find an atom of a 1D solid that originally had a position $x$ at a displacement $\delta x$ is shown on this plot:
+
+```python
+def psi_squared(delta_x, x):
+    return delta_x**2 * np.exp(-delta_x**2) * np.sin(4*np.pi*x)**2
+
+x = np.linspace(0, 1, 200)
+delta_x = np.linspace(-2, 2, 200)
+
+pyplot.imshow(psi_squared(delta_x.reshape((-1, 1)), x.reshape((1, -1))), cmap='gist_heat_r', extent=(0, 3, -1, 1))
+pyplot.ylabel(r'$\delta x$')
+pyplot.xlabel(r'$x$')
+pyplot.xticks((0, 3), ('$0$', '$L$'))
+pyplot.yticks((), ())
+cbar = pyplot.colorbar()
+cbar.set_ticks(())
+cbar.set_label(r'$|\psi^2|$')
+```
+
+Describe how many phonons in which $k$-state this solid has.
+
+??? hint
+
+    There are $n=2$ phonons in the state with $k=4\pi/L$ and $n=2$ phonons in a state with $k=-4\pi/L$.
+
 ### Exercise 1: Debye model: concepts
 1. Describe the concepts of k-space and density of states.
 2. Calculate the density of state $g(\omega)$ and $g(k)$  for a 3D, 2D and 1D systems with linear dispersion $\omega=vk$.
-3. _draft: sketch the state of a standing wave. Here I'm not sure what you exactly want to see, Anton, and how to write it down in an easy way._
+3. Discuss what it means to have $n=3$ phonons occupying a state with $k=(0, 0, 2\pi/L)$. Draw the amplitudes of the atomic displacements in a state with $
 
 ###  Exercise 2: Debye model in 2D
 1. State the assumptions of the Debye theory.