İki ve üç boyutta model kare kuyu potansiyeli etkileşimiyle BCS-BEY geçişi

Superfluidity is one of the most remarkable phenomena that we come across in many distinct fields of physics. It was first observed by H. K. Onnes in 1911 when he cooled a mercury sample below 4.2 Kelvin. He found that the sample conducts electricity without any dissipation. Both bosonic and fermionic systems can exhibit superfluidity. We focus on the fermionic superfluidity in this thesis. The crossover from weak coupling Bardeen-Cooper-Schrieffer (BCS) pairing to a Bose-Einstein condensate (BEC) of tightly bound pairs, as a function of the attractive interaction has long been of interest to theoretical physicists. The past decade has seen a series of remarkable experimental developments in ultracold Fermi gases that has realized the BCS-BEC crossover in the laboratory, bringing with it fresh new insights into the very strongly interacting unitary regime in the middle of this crossover. In this thesis we study the crossover from the BCS state to BEC in a Fermionic gas interacting with a model square well potential at zero temperature by tuning the interaction between fermions and using mean field approximation. The possibility to change the interaction strength allows the ground state of the system to evolve from a weakly interacting BCS limit to the strongly interacting BCS limit. The thesis consists of two parts. In the first part we examine the the BCS-BEC crossover in a $3$-dimensional fermionic gas. We show that there is a crossover between two limits and calculate the physical quantities such as momentum distribution, energy per particle, excitation energies, changing of the chemical potential and superconducting gap parameter with respect to the interaction parameter or scattering length. In the second part we repeat the all process for the two-dimensional and quasi-two-dimensional system. We show that the crossover occurs at weaker interaction strength with respect to the three-dimensional system. We obtain strictly two-dimensional and three-dimensional limits by changing the thickness of the quasi-two-dimensional system. Within the developed formalism and numerical implementation it is possible to study different systems having different interaction or dispersion relations in which there is different interaction potential and linear energy dispersion with the numerical implementation given in the thesis.