Using for the generator of rotations about 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

Using (Jk)ij=−i ϵkij(J_k)_{ij} = -\mathrm i\,\epsilon_{kij} for the generator of rotations about axis kk, write down the 3×33\times3 Hermitian matrix JyJ_y (axis 22) explicitly. Exponentiate the one-parameter subgroup Ry(θ)=exp⁡(−iθJy)\mathsf R_y(\theta) = \exp(-\mathrm i\theta J_y) by summing the series, as done for JzJ_z, and evaluate the resulting active rotation matrix at θ=90∘\theta = 90^\circ.

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