Consider a 1D simple harmonic oscillator with mass $m$ and spring constant $k$. The Hamiltonian is given in the usual way by:
$$
H = \frac{p^2}{2m}+\frac{k}{2}x^2.
$$
(a) Compute the classical partition function using the following expression:
$$
Z = \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty} dx e^{-\beta H(p,x)}.
$$
(b) Using the solution from (a), compute the expectation value for the energy.
(c) Now you are ready to compute the heat capacity. Check that you get the \textit{law of Dulong-Petit} but with a missing factor. Can you explain why?
(d) Which value for the heat capacity would you obtain if you considered a solid in 3D that consists of N atoms in harmonic wells?
### Exercice 2. Einstein model: quantum version
Consider a 1D quantum harmonic oscillator. The eigenstates of it's Hamiltonian are:
$$
E_n = \hbar\omega(n+\frac{1}{2}),
$$
(a) Compute the quantum partition function using the following expression:
$$
Z = \sum_j e^{-\beta E_j}.
$$
(b) Using the partition function, compute the expectation value for the energy.
(c) Now you are ready to compute the heat capacity. Check that for the high temperature limit we get the same as in Exercice 1(c).
### Exercice 3. Boson statistics
(a) What is the meaning of the occupation number? Give an expression for the occupation number for phonons.
### Exercise title
(b) Recall the harmonic oscillator energy levels. What does it mean, in the Einstein's picture, to be at the level $n=3$?
### Exercise title
### Exercise 4. Total heat capacity of a diatomic material
(a) Consider now a diatomic material with $2N$ atoms (N of each type). Which is the total heat capacity of this material. (?)
