For the spin angular momentum about for example the y -axis we just replace L z with S y = ℏ ℏ 2 σ σ y (where σ σ y is the Pauli Y matrix) and we get the spin rotation operator D ⁡ ⁡ ( y , t ) = exp ⁡ ⁡ ( − − i t 2 σ σ y ) .