Pauli-matrix spectral projectors ($P_s=(I+s\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$)

For $H=\mathbf{d}\cdot \mathbf{\sigma }$ with $\mathbf{d}\ne 0$, the eigenvalues are $\pm |\mathbf{d}|$ and the projectors are $P_s=(I+s\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$. A normalized eigenstate has $⟨\mathbf{\sigma }⟩=s\hat{\mathbf{d}}$. At $\mathbf{d}=0$ the spectrum is degenerate and that direction is undefined.