Introduction to Mathematical Physics/N body problems and statistical equilibrium/Exercises

Exercice:

Paramagnetism. Consider a system constituted by $N$ atoms located at nodes of a lattice. Let $J_{i}$ be the total kinetic moment of atom number $i$ in its fundamental state. It is known that to such a kinetic moment is associated a magnetic moment given by:

$\mu _{i}=-g\mu _{B}J_{i}$

where $\mu _{B}$ is the Bohr magneton and $g$ is the Land\'e factor. $J_{i}$ can have only semi integer values.

Assume that the hamiltonian describing the system of $N$ atoms is:

$H=\sum -\mu _{i}B_{0}$

where $B_{0}$ is the external magnetic field. What sort of particles are the atoms in this systme, discernables or undiscernables? Find the partition function of the system.

Exercice:

Study the Ising model at two dimension. Is it possible to envisage a direct method to calculate $Z$ ? Write a programm allowing to visualize the evolution of the spins with time, temperature being a parameter.

Exercice:

Consider a gas of independent fermions. Calculate the mean occupation number ${\bar {N}}_{\lambda }$ of a state $\lambda$. The law you'll obtained is called Fermi distribution.

Exercice:

Consider a gas of independent bosons. Calculate the mean occupation number ${\bar {N}}_{\lambda }$ of a state $\lambda$. The law you'll obtained is called Bose distribution.

Exercice:

Consider a semi--conductor metal. Free electrons of the metal are modelized by a gas of independent fermions. The states are assumed to be described by a sate density $\rho (\epsilon )$, $\epsilon$ being the nergy of a state. Give the expression of $\rho (epsilon)$. Find the expression binding electron number ${\bar {N}}$ to chemical potential. Give the expression of the potential when temperature is zero.