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

Paramagnetism. Consider a system constituted by atoms located at nodes of a lattice. Let be the total kinetic moment of atom number in its fundamental state. It is known that to such a kinetic moment is associated a magnetic moment given by: where is the Bohr magneton and is the Land\'e factor. can have only semi integer values. Assume that the hamiltonian describing the system of atoms is: where is the external magnetic field. What sort of particles are the atoms in this system, discernables or undiscernables? Find the partition function of the system. Study the Ising model at two dimension. Is it possible to envisage a direct method to calculate ? Write a programm allowing to visualize the evolution of the spins with time, temperature being a parameter. Consider a gas of independent fermions. Calculate the mean occupation number of a state . The law you'll obtained is called Fermi distribution. Consider a gas of independent bosons. Calculate the mean occupation number of a state . The law you'll obtained is called Bose distribution. 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 , being the nergy of a state. Give the expression of . Find the expression binding electron number to chemical potential. Give the expression of the potential when temperature is zero.