A vector observable transforms under rotations as the spin- representation of

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

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

A vector observable transforms under rotations as the spin-

1

representation of

\mathfrak{su}(2)

, with character

\chi_1(\alpha)=\dfrac{\sin(3\alpha/2)}{\sin(\alpha/2)}

. A rotation by

\alpha=60^\circ

about a fixed axis is applied. ((a)) Evaluate

\chi_1(60^\circ)

from the character formula. ((b)) The same rotation, written as a

3\times3

real orthogonal matrix, has trace

1+2\cos\alpha

(Chapter 9). Evaluate this trace at

\alpha=60^\circ

and check it agrees with part (a).