Spin Hall Effect And Landau-like Atomic Problem In Noncommutative Phase Space

In recent years, a research in noncommutative space and noncommutative phase space has become a hot topic, because development of noncommutative quantum field theory, the study of M theory and superstring theory showed that space-time has non-commutative property in the string scale. This caused the deeper change of space-time concept that we established on the bases of quantum mechanics. The problems, such as energy quantization and wave function of neutral particles in an electromagnetic field, spin-orbit interaction of the Dirac particle, we have discussed in noncommutative phase space, haven’t been studied before.In this thesis, we used the Bopp’s shift and star eigenvalue equation method, Schrodinger equation and the operators’time evolution equation in the Heisenberg picture to solve the problems mentioned above within the scope of noncommutative phase space. And obtained the energy levels, wave functions, and the spin-orbit interaction of Dirac particle induced by position-position and momentum-momentum noncommutativity.Our research works in this thesis mainly consist of following three sections:En-ergy quantization of neutral particles in the presence of external electromagnetic field in noncommutative phase space, spin-orbit interaction of the Dirac particle in a noncom-mutative phase space, spin hall effect on noncommutative phase space.First of all, since Neutral particles (especially atoms) under the external electro-magnetic field that perpendicular to each other will get polarization and produce electric dipole, we can study the quantization of neutral particles, taking into account the elec-tric and the magnetic dipoles. We give the Hamiltonian of electric dipoles that moving in electromagnetic field. And then by using the Bopp’s shift obtain transformation of Hamiltonian from noncommutative phase space into the general quantum mechanical space. Since what we got by variable transformation has the form of one-dimensional harmonic oscillator, we can get the energy levels and wave functions through solving the Schrodinger equation for harmonic oscillator.Second, based on the FW unitary transformation, we get the general form of the Dirac hamiltonian in the presence of external electromagnetic field and point out the spin orbit interaction terms in ordinary space. Then, we calculate hamiltonian of this system in noncommutative phase space and get the hamiltonian of the general form in noncommutative phase space. At the end, according to these informations, we give the spin orbit interaction of this system and point out the correction terms caused by the position-position and momentum-momentum noncommutativity.Finally, starting from the unitary FW transformed hamiltonian of the Dirac particle, we calculate the spin hall effect in noncommutative phase space and the results show that though the position-position and momentum-momentum noncummutate in noncommu-tative phase space, only the position-position noncommutativity does make contribution for the spin hall effect of this system.