| Since the spin of electron was discovered, investigations on the description of spin polarization have been becoming an important subject in both fundamental quantum theory and spintronics theory. Whereas, the knowledge on the spin polarization is not sufficient, there exist some questions to be considered deeply, which may help us interpret some subtle spin-dependent phenomena including the well-known spin Hall effect and Imbert-Fedorov effect.The polarization of light beams is a local quantity, similarly, the polarization of electron beams is generally local except for linear-polarized electron beams, it exhibits inhomogeneous distribution. In view of the above, we may predict kinds of electron beams with novel polarization including"cylindrically symmetric electron beams".Furthermore, due to relativistic effect, the physical behavior of electron in non-relativistic and relativistic cases will exhibits great distinctions. Therefore, in this thesis, we develop a theory of electron spin polarization, taking into account of both non-relativistic and relativistic electrons, and apply this theory to describe the spin polarization of free electron. We also make some effort to the investigations on symmetry properties and spatial displacement. The main results and innovations contain:(1) First, new degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the Pauli vector allows us to use the azimuthal angle of a second direction, denoted by unit vector I, with respect to the quantization direction to characterize the spin polarization. Upon utilizing this approach to a free electron and letting the quantization direction for each plane wave be the wave vector, we arrive at a representation in which the unit vector I functions as an independent index to characterize the spin polarization.(2) Second, we investigate the important role of unit vector I(?) to describe the spatial symmetry of single free electron systems; Making use of numerical simulation, it is shown that electron systems with different unit vector have different symmetry properties; We have also illustrated the close relation between azimuthal angleΦI of unit vector and spatial rotation of system along z axis.(3) Third, cylindrically symmetric electron beams in spin polarization are reported for the first time. They are shown to be the eigen-states of total angular momentum in z direction. But they are neither the eigen-states of spin nor the eigen-states of orbital angular momentum in that direction.(4) Fourth, we briefly review several important mechanical operators which can be served as spin angular momentum of relativistic electron. Making use of polarization operator preferably, we propose a method to construct spin polarization bases of free relativistic electron, and find that, besides the direction of spin quantization, there exists another degree of freedom---unit vector to characterize these bases. Similar discussions can be carried out by referring to those in chapter 2 for non-relativistic electron, just replacing Pauli vector by polarization operator.(5) Fifth, when relativistic electron transmit a non-symmetric square single potential structure, the spin polarization of transmitting electron will change compared with the spin polarization of incident electron, we also discuss the spin-dependent spatial displacements of transmitting electron beams, which also depend on the energy, incident angle and spin polarization of incident electron.These phenomena can not occur for non-relativistic electron. These results may provide necessary theoretical instructions and references for the application of high-energy electron beam in particle physics, high-energy physics and condensed matter physics, et al., and the investigation and development of"relativistic spin electronics"in future. |
