Quantum Mechanics (1) · Problem 5.2

Rotations and angular momentum

Physics — Quantum Mechanics · orthogonal-transformations-rotations

Problem

Prove the following theorem: The most general mapping

\mathbf{x} \to \mathbf{x}'

R^3

into itself, such that the origin is mapped into the origin, and such that all distances are preserved, is a linear, real orthogonal transformation

Q

\mathbf{x}' = Q\mathbf{x}, \quad \text{where} \quad Q^T Q = I, \quad \text{and} \quad Q^* = Q. \tag{5.23}