Consider a 1D simple harmonic oscillator with mass $m$ and spring constant $k$. The Hamiltonian is given in the usual way by:
### Exercise 1: Heat capacity of a classical oscillator.
Let's refresh the connection of this topic to statistical physics.
You will need to look up the definition of partition function and how to use it to compute expectation values.
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:
1. Compute the classical partition function using the following expression:
$$
Z = \int_{-\infty}^{\infty}dp \int_{-\infty}^{\infty} dx e^{-\beta H(p,x)}.
$$
2. Using the solution of 1., compute the expectation value of the energy, and the expectation value of .
3. Compute the heat capacity. Check that you get the law of Dulong-Petit but with a different prefactor.
4. Explain the difference in the prefactor by considering the number of degrees of freedom.
(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:
### Exercise 2: Quantum harmonic oscillator
Consider a 1D quantum harmonic oscillator. Its eigenstates are:
$$
E_n = \hbar\omega(n+\frac{1}{2}),
$$
(a) Compute the quantum partition function using the following expression:
1. Sketch the wave function of this harmonic oscillator for $n=3$.
2. Compute the quantum partition function using the following expression:
$$
Z = \sum_j e^{-\beta E_j}.
$$
3. Using the partition function, compute the expectation value of the energy.
4. Compute the heat capacity. Check that in the high temperature limit you get the same result as in Exercise 1.1.
- What temperature can be considered high?
- What is the expectation value of $n$?
(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 4. Total heat capacity of a diatomic material
(b) Recall the harmonic oscillator energy levels. What does it mean, in the Einstein's picture, to be at the level $n=3$?
Naturally occurring lithium has [two stable isotopes](https://en.wikipedia.org/wiki/Isotopes_of_lithium): $^6$Li (7.5%) and $^7$Li (92.5%). Let us extend the Einstein model to take into account the different masses of different isotopes.
### 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. (?)
1. Assume that the strength of the returning force $k$ experienced by each atom is the same. What is the difference in the oscillation frequencies of different isotopes of lithium in the lithium crystal?
2. Write down the total energy of lithium assuming that all $^6$Li atoms are in $n=2$ vibrational state, and all $^7$Li atoms are in $n=4$ vibrational state.
3. Write down the total energy of lithium at a temperature $T$ by modifying the Einstein model.
