A spin one-half state is to be rotated actively by about the -axis

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Invariance, Ch Lie groups and Lie algebras: rotations, SU(2) and the double cover

Problem

A spin one-half state is to be rotated actively by

\theta = 120^\circ

about the

x

-axis. Using

U = \cos(\theta/2)\,\mathsf 1 - \mathrm i\sin(\theta/2)\,\hat{\mathbf n}\cdot\boldsymbol\sigma

with

\hat{\mathbf n} = (1,0,0)

, write down the

2\times2

unitary matrix

U

representing this rotation in

SU(2)