## ABSTRACT

This chapter considers the quantum charged particle beam optics for spin- 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315232515/65736064-9446-4cb4-b9c7-60895bcd97bb/content/inequ6_1.tif"/> particles. For electrons, or for any spin- 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315232515/65736064-9446-4cb4-b9c7-60895bcd97bb/content/inequ6_1.tif"/> particle, the proper equation to be the basis for the theory of quantum beam optics is the Dirac equation. This chapter presents a general formalism of spinor quantum charged particle beam optics based on the Dirac equation, and the Dirac-Pauli equation which takes into account the anomalous magnetic moment. This formalism is used to discuss the image formation in an electron microscope with axially symmetric magnetic lens, focusing and defocusing by magnetic quadrupoles, and the bending of a charged particle beam by the dipole magnet. The main framework for studying the spin dynamics and beam polarization in accelerator physics is essentially based on the quasiclassical Thomas-Frenkel-Bargmann-Michel-Telegdi equation (widely known as the Thomas-BMT equation). The spinor quantum charged particle beam optics, at the level of single particle dynamics, gives a unified account of orbital motion, Stern-Gerlach effect, and the Thomas-BMT spin evolution for a paraxial beam of Dirac particles with anomalous magnetic moment. Nonrelativistic approximation of the relativistic spinor quantum charged particle beam optics leads to the nonrelativistic spinor quantum charged particle beam optics based on the nonrelativistic Schrödinger-Pauli equation.