Rotation Group

Quantum Mechanics · Part 5

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Table of Contents

  1. 1.Rotation Group SO(3)
  2. 2.Lie Algebra so(3)
  3. 3.SU(2) Group
  4. 4.Lie Algebra su(2)
  5. 5.The 2-to-1 Homomorphism
  6. 6.Spin-1/2 and Spinors
  7. 7.Representations
  8. 8.Wigner D-Matrices
  9. 9.Physical Implications
  10. 10.Why Nature Uses SU(2), Not SO(3)

Key Equations

$$\text{SO}(3) = \{R \in \mathbb{R}^{3\times 3} : R^T R = \mathbb{I}, \det R = 1\}$$
$$[L_i, L_j] = i\epsilon_{ijk} L_k$$
$$J_1 = \frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad J_2 = \frac{1}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad J_3 = \frac{1}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
$$\text{SU}(2) / \mathbb{Z}_2 \cong \text{SO}(3)$$

Equations are rendered with MathJax in the PDF with professional LaTeX typesetting.

Course Context

This PDF is part of the Quantum Mechanics course on CoursesHub.World. Master quantum mechanics from mathematical foundations to advanced topics. Covers Hilbert spaces, wave functions, angular momentum, perturbation theory, scattering, and path integrals with 450+ pages ...

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