Abstarct: About the advent of the noncommutative theory it should be mentioned that the interference of ultra-violent and infra-red waves was one of the mathematical problems of quantum mechanics. To overcome this problem, the idea of noncommutative quantum mechanics was reviewed by some people. The subject was first introduced by Heisenberg and then successfully approved by Pauli and Pierlz. In 1933, Pierlz worked on the application of quantum mechanics in descxription of an particle in constant magnetic field. Snider, in 1946, introduced the idea of noncommutative quantum mechanics in such a way that the problem of interference of ultra-violent and infrared waves was removed. In this thesis, we first investigate the commutative and noncommutative structures and their applications. We then obtain the thermal properties of spin-1/2 fermions in commutative and noncommutative spaces using the Chiral operators. In addition, we obtain the thermal properties of nonrelativistic particles in the presence of magnetic field in commutative and noncommutative spaces by solving the Schrödinger equation and obtaining the energy eigenvalues. The time evolution of the system is as well studied in both commutative and noncommutative spaces.