Particles with Spin

Abstract

Chapter summaryThis chapter introduces the spin as an important intrinsic property of electrons. We discuss, in particular, some results that are relevant for atomic physics. More results will be presented from a slightly more abstract point of view in Chapter 4.We start by describing the connection between angular momentum and magnetic moment according to classical electrodynamics. Then we proceed to discuss the Stern-Gerlach experiment, which can only be explained by assuming that the state of an electron is characterized by the eigenvalues of a spin operator S. This spin operator has the properties of an angular momentum, but half-integer eigenvalues.In Section 3.5, we describe the mathematical consequences of this assumption. We construct a Hilbert space for particles with spin 1/2 and define the operators describing the components of the spin.In Section 3.6, we define the Pauli operator, that is, the Hamiltonian for a spin-1/2 particle in an external field. We discuss the solutions in a constant, homogeneous magnetic field, thereby generalizing results from Book One in Section 3.7. An important difference from the results without spin is the occurence of bound states with zero energy. This phenomenon also occurs for nonhomogeneous magnetic fields and for certain situations in three dimensions (Section 3.8).The spin is most important for understanding finer details of the spectrum of hydrogenic atoms. In Section 3.9, we introduce the spin-orbit coupling and describe the spinor eigenfunctions of the hydrogen atom and the structure of the energy spectrum.