Take and multiply the four matrices of

PHYSICS · Multi-part · 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

Take

\alpha=\beta=0.02\ \mathrm{rad}

and multiply the four matrices of

\mathsf{M}=\mathsf{R}_y(\alpha)\mathsf{R}_z(\beta)\mathsf{R}_y(-\alpha)\mathsf{R}_z(-\beta)

. The result has

\mathsf{M}_{zy}=+4.00\times10^{-4}

and

\mathsf{M}_{yz}=-4.00\times10^{-4}

, diagonal entries within

10^{-7}

of 1, and entries of about

4\times10^{-6}

in the

x

row and column. ((a)) Which single rotation is

\mathsf{M}

, to this accuracy? Give its axis and angle. ((b)) Use

e^Ae^Be^{-A}e^{-B}=\mathsf{1}+[A,B]+\cdots

with

A=-i\alpha J_y

and

B=-i\beta J_z

to deduce

[J_y,J_z]

from these numbers.