M. Masili, 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**in this state the difference between ?

*state*^{k}(q) and ?

^{k1}(q) is just the phase while the amplitud is the same.

Then the propeties of the **quantum mechanical systems in the canonical ensemble**

**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

_{B}T. The

**expectation value**of any observable

**G**can be calculated as <

**G**>=Tr(

**G**exp[-?

**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 Grand canonical partition function 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 **P ² **=

**(the identity operator), so the**

*1***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 determinan**

**t**, 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.