Introduction to Quantum Mechanics · Problem 6.17

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · rotation_angular_momentum

Problem

Consider the action of an infinitesimal rotation about the

\mathbf{n}

axis of an angular momentum eigenstate

\psi_{n\ell m}

. Show that

\hat{R}_{\mathbf{n}}(\delta)\,\psi_{n\ell m} = \sum_{m'} D_{m'm}\,\psi_{n\,\ell\,m'},

and find the complex numbers

D_{m'm}

(they will depend on

\delta

\mathbf{n}

, and

\ell

as well as

m

and

m'

). This result makes sense: a rotation doesn't change the magnitude of the angular momentum (specified by

\ell

) but does change its projection along the

z

axis (specified by

m