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Chapter Summary:

3. Particles with Spin

This 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.

• 3.1. Introduction 113
• 3.2. Classical Theory of the Magnetic Moment 115
  • 3.2.1. Magnetic moment of an extended particle 115
  • 3.2.2. The influence of an external magnetic field on a magnetic moment 116
• 3.3. The Stern-Gerlach Experiment 118
• 3.4. The Spin Operators 123
  • 3.4.1. Magnetic moment and spin 123
  • 3.4.2. The $g$-factor 123
  • 3.4.3. Electron in a constant magnetic field 125
  • 3.4.4. Properties of the spin operator 126
• 3.5. Spinor-Wave Functions 127
  • 3.5.1. A Hilbert space for a spin 1/2 particle 127
  • 3.5.2. Spin operators in the standard representation 129
  • 3.5.3. Changing representations 130
  • 3.5.4. Interpretation of spinor-wave functions 131
  • 3.5.5. Visualization of spinor-wave functions 132
    • 3.5.5.1. Visualization by vectors 132
    • 3.5.5.2. Vizualization by colors 134
• 3.6. The Pauli Equation 134
  • 3.6.1. The Pauli operator 134
  • 3.6.2. Magnetic fields with constant direction 136
  • 3.6.3. Supersymmetric structure 137
• 3.7. Solution in a Homogeneous Magnetic Field 138
  • 3.7.1. The $g$-factor of orbital motion 138
  • 3.7.2. Solutions with zero energy 138
  • 3.7.3. The spectrum of eigenvalues 141
• 3.8. Special Topic: Magnetic Ground States 142
  • 3.8.1. Two dimensions 143
  • 3.8.2. Three dimensions 145
• 3.9. The Coulomb Problem with Spin 146
  • 3.9.1. Coulomb-Hamiltonian for a particle with spin 146
  • 3.9.2. Complete set of observables 147
  • 3.9.3. The spin-orbit interaction 148
  • 3.9.4. Eigenfunctions of the spin-orbit operator 150
  • 3.9.5. The radial equation 155