M. Marsili, ICTP

This lesson started the study of magnetic systems and, in particular, the study of **Paramagnetism**. The basic element of these systems are particles, that can be thought of as atoms, with a magnetic moment or 'spin'. These particles interact with an external magnetic field. If there is dissipation in the system, meaning that it can transfer rotational energy of the atom to some other degrees of freedom, then the system of spins can minimize it's energy. The energy that each spin contributes to the total Hamiltonian is: E_{I} = -? H cos ?_{i}.

The total energy the microstate is the sum over all spins and from there one can calculate the **partition function **by summing over all possible microstates.

Thermodynamical quantities can be computed as it is customary. It is interesting first of all to calculate the average magnetization. It is found that it decreases with temperature. This is to be expected on

physical grounds, the larger the temperature, the less relevant the contribution of the interaction between the magnetic moment and the magnetic field to the total energy.

The picture changes in the quantum system of spins. There, the **angular momentum**, and hence the magnetic moment, is quantized. The total magnetic moment and the magnetic moment along the direction of the magnetic field were calculated and the differences with the classical case were stressed

as well as how the classic limit is obtained.

As a complementary tool you can also see some lessons on Statistical Mechanics given in the <span class="yt-user-name author">Stanford</span> University.

Lecture 9 | Modern Physics: Statistical Mechanics May 25, 2009 - Leonard Susskind picks up on magnets, phase transitions, and mean field transitions. He goes on to explain chemical potential.