M. Marsili, ICTP

This lesson the teacher goes deeper in the study of the **quantum mechanical systems in the microcanonical ensemble**. The number of microstates was calculated for the case of N indistinguishable<span style="font-size: 11pt; line-height: 115%; font-family: 'Calibri','sans-serif';"> </span> particles and for the three types of probability distribution of energy levels, the **Maxwell-Boltzmann** (MB), **Fermi–Dirac** (FD) and **Bose–Einstein** (BE). The computation of the **entropy** is difficult because of teh large sum over all the microstates, then an some approximation are used, first the sum is approximated by the larger possible term. Then as this problem has two constrains, the number of particles N and energy E are constant, the **Lagrange multiplier** is introduced, finally the **Stirling's approximation** for the factorial is used. The occupation number that is found to be the larger term in the sum is n^{*}_{i} = g_{i}/(Exp[?+??_{i}]+a) where g_{i} is the number of energy level ?_{i}, ?, ? are parameters to satisfy the constrains and a=1 for BE, a=-1 for FD and a=0 for MB distributions. Then the **entropy** is calculated for all the distributions.

In the second part of the lesson the case of the **canonical** and the **Grand canonical**<span class="l"> ensembles </span>were considered for the same probability distributions. for this cases the **Grand partition function** introduced in the previous lesson was calculated and it is obtained the same result that in the **microcanonical ensemble** but in a simpler way, but this results implies that the **entropy** is dominated indeed by the larger term in the sum n^{*}_{i}. Then the expression for the average of the occupation number of particles is calculated and It is shown that the distribution of the occupation number in the classical case (MB) is given by the **Poisson distribution** while for the other cases (FD, BE) is given by the **geometric distribution**. Quantum effects will be important when the occupation number is large this happen in the regime of large **chemical potential,** this condition is exactly the same that the one obtained in the previous lesson that is that at low density or high temperature quantum effects can be neglected. As an example the **Ideal gas** was considered and the pressure as a function of the energy density is obtained.