Termodinamica di bosoni confinati e interagenti

The Bose-Einstein condensation (BEC) is a quantum phenomenon which was theorized in 1920s, but it was realized for the first time in the JILA Laboratory (University of Colorado, Boulder) by Eric Cornell and Carl Wieman only in 1995. Since 1995, lots of experiments on BEC have been realized using different gases and various conditions. One can give a correct interpretation of the results of these works only by considering the role of interaction between particles on relevant physical quantities, such as critical temperature and condensed fraction. The purpose of our work is deducing the main thermodynamic properties of trapped and interacting Bose gases using the tools of "traditional" Quantum Mechanics, avoiding the introduction of "second quantization" (which often recurs in litterature). The first chapter shows a derivation of the famous Gross-Pitaevskii equation, starting from the many-bodies Dirac action and applying the principle of least action. Moreover, two approximate solutions of this equation are discussed as well: the gaussian variational ansatz for weakly interacting particles, and the Thomas-Fermi approximation, for strongly interacting particles. In chapter 2 we derive the Bogoliubov dispersion relation for excited states (or quasiparticle energy spectrum) with the tools of Hamiltonian Mechanics and semiclassical approximation. Furthermore, we discuss some approximations for this formula (especially the Hartree-Fock spectrum for weakly interacting bosons) and we use the Bose-Einstein distribution to find the thermal cloud density. In chapter 3 we use the previous results to calculate explicit expressions for critical temperature and condensed fraction (where possible) in four particular cases: ideal free and trapped Bose gas, interacting free and trapped Bose gas. We also discuss the main results obtained by Giorgini, Pitaevskii and Stringari, who managed to explain accurately Cornell and Wieman's experimental plots. Finally we conclude that traditional Quantum Mechanics is quite efficient to derive the main equations of BEC and that introducing second quantization is not essential. Nevertheless, some issues remain unresolved in this context: for example the phenomenon of quantum depletion and the generalisation of the Gross-Pitaevskii equation at non-zero temperature.