diff --git a/src/2_debye_model.md b/src/2_debye_model.md index 744e7ee230f75af41aa1aff4d9fcaa82b54e498e..0017ad86a03d0d14eb3172b7d14ad3df7f16cc39 100644 --- a/src/2_debye_model.md +++ b/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.