THREE-DIMENSIONAL ROTATION GROUP

The following sections are included:

• Three-dimensional Rotations

• Fundamental Concept of a Lie Group

  • The Composition Functions of a Lie Group

  • The Local Property of a Lie Group

  • Generators and Differential Operators

  • The Adjoint Representation of a Lie Group

  • The Global Property of a Lie Group

• The Covering Group of SO(3)

  • The Group SU(2)

  • Homomorphism of SU(2) onto SO(3)

  • The Group Integral

• Irreducible Representations of SU(2)

  • Euler Angles

  • Linear Representations of SU(2)

  • Spherical Harmonics Functions

• The Lie Theorems

• Clebsch–Gordan Coefficients of SU(2)

  • Direct Product of Representations

  • Calculation of Clebsch–Gordan Coefficients

  • Applications

  • Sum of Three Angular Momentums

• Tensors and Spinors

  • Vector Fields

  • Tensor Fields

  • Spinor Fields

  • Total Angular Momentum Operator

• Irreducible Tensor Operators and Their Application

  • Irreducible Tensor Operators

  • Wigner–Eckart Theorem

  • Selection Rule and Relative Intensity of Radiation

  • Landé Factor and Zeeman Effects

• An Isolated Quantum n-body System

  • Separation of the Motion of Center-of-Mass

  • Quantum Two-body System

  • Quantum Three-body System

  • Quantum n-body System

• Exercises