Spin operator ($\hat{\mathbf{S}}$)

A spin operator describes intrinsic angular momentum. Its components obey angular-momentum commutators; for spin one half, ${\hat{S}}_j=\hslash {\sigma }_j/2$ in terms of the Pauli matrices.

For spin one-half, the Pauli representation makes every diagonal quadratic component a multiple of the identity. Thus $A{S}_z^2+B(S_x^2-S_y^2)=A{\hslash }^{2}I/4$ cannot split the doublet or cause spin reversal. This conclusion is special to this quadratic Hamiltonian and does not exclude linear Zeeman dynamics.