Griffiths Introduction to Elementary Particles · Problem 4.21

Bound States — David J. Griffiths

Physics — QFT · rotation-operator-spin-half

Problem

(a) Show that

e^{i\pi\sigma_z/2} = i\sigma_z

. (b) Find the matrix

U

representing a rotation by

180°

about the

y

axis, and show that it converts ``spin up'' into ``spin down'', as we would expect. (c) More generally, show that

U(\boldsymbol{\theta}) = \cos\frac{\theta}{2} - i(\hat{\boldsymbol{\theta}} \cdot \boldsymbol{\sigma}) \sin\frac{\theta}{2}

where

U(\boldsymbol{\theta})

is given by equation (4.28),

\theta

is the magnitude of

\boldsymbol{\theta}

, and

\hat{\boldsymbol{\theta}} = \boldsymbol{\theta}/\theta

. [Hint: Use Problem 4.20, part (c).]