M. Marsili, ICTP

This lesson continues the study of Quantum statistics that was started in the last part of the previous lesson, and the main definitions, for example a microstate is described by a wave function ?(q), an ensemble a collection of N wave functions ?^k(q) k=1,...,N and the Density matrix (?). The expectation value of any observable G can be calculated as <G>=Tr(?G) where Tr represents the trace of an operator and the bold letters the operators. This expression for <G> means that this values does not depend on the basis { ?^k(q) }. It is shown that ? is a function of the Hamiltonian H because its time derivative is equal to the commutator with H ( i hbar d?/dt = [H,?]) a second consequence is that in the energy representation ? is diagonal.

The quantum mechanical systems in the microcanonical ensemble were then studied, for this case the Density matrix in the energy representation is just a constant by the Kronecker delta ?n,m=C_n?_n,m where C_n is the probability of finding the system in a given eigenstate with energy E_n. The definition of pure state (?^k(q) =?⁰(q) for all k) was given and as an example was taken the ground state of a quantum system where C_n=1, the entropy S=0 and ?²=?. If C_n<1, S>0, the state is called mixed state in this state the difference between ?^k(q) and ?^k1(q) is just the phase while the amplitud is the same.

Then the propeties of the quantum mechanical systems<span class="toctext"> in the canonical ensemble</span> were studied, here the Density matrix ?=exp[-?H]/Q_N where Q_N is the partition function Q_N=Tr(exp[-?H]) and ?=1/K_BT. The expectation value of any observable G can be calculated as <G>=Tr(Gexp[-?H])/Q_N. So one you have calculated Q_N you can calculate all thermodynamics properties.

The quantum mechanical systems in the Grand canonical ensemble were also studied. In these systems the number of particles fluctuates. For these systems the Density matrix is also a function of the number of particles and it commutes with N, [?, N] = 0. In analogy with the classical case we have that ? = exp[??N- ?H]/Q_N where the <span class="l">Grand canonical partition function</span> Q_N=Tr(exp[??N-?H]) and ? is the chemical potential. The equation for the calculus of expectation value is the same that in the previous case.

In the second part of the lesson systems composed by identical particles were considered for which the Hamiltonian is just the sum of a Hamiltonian of a single particle. Thus the solution for this problem can be constructed by a linear combination of the solutions for the one particle problem but you have to take into account the occupation number of a single state ? n_?. The definition of Identical particles is introduced with the help of the permutation operator P and the Exchange symmetry. We have that P is both Hermitian and unitary and that P² = 1 (the identity operator), so the eigenvalues of P are +1 and ?1. The corresponding eigenvectors are the symmetric and antisymmetric states. Particles with symmetric wave functions are called Bosons and are described by the Bose–Einstein statistics. Particles with antisymmetric wave functions are called Fermions and are described by the Fermi–Dirac statistics. A particular way to construct antisymmetric wave functions from a single particle eigenstate that is called the Slater determinant, that takes care for the Pauli Exclusion principle. Finally an example for the case of two identical free particles at a temperature T was considered and the symmetric and antisymmetric states obtained it was obtained the thermal de Broglie wavelength and it was showed that at low density or high temperature quantum effects can be neglected.