Quantum Mechanics Demystified 2nd Edition David Mcmahon -

For a particle (e.g., electron, proton, neutron), the eigenvalues of (\hatS^2) are (\hbar^2 s(s+1)) with (s = 1/2), and eigenvalues of (\hatS_z) are (\pm \hbar/2).

[ \sigma_x |\psi\rangle = \beginpmatrix 0&1\1&0 \endpmatrix \frac1\sqrt2 \beginpmatrix 1\ i \endpmatrix = \frac1\sqrt2 \beginpmatrix i \ 1 \endpmatrix. ] [ \langle \psi | \sigma_x | \psi \rangle = \frac1\sqrt2 \beginpmatrix 1 & -i \endpmatrix \cdot \frac1\sqrt2 \beginpmatrix i \ 1 \endpmatrix = \frac12 (i - i) = 0. ] So (\langle S_x \rangle = 0). Quantum Mechanics Demystified 2nd Edition David McMahon

These operators satisfy the fundamental commutation relations: For a particle (e