Spin angular momentum, Ŝ, is even under parity, just like orbital angular momentum L̂:

Acting on a spinor written in the standard basis (Equation 4.139), the parity operator becomes a 2 x 2 matrix. Show that, due to Equation 6.27, this matrix must be a constant times the identity matrix. As such, the parity of a spinor isn’t very interesting since both spin states are parity eigenstates with the same eigenvalue. We can arbitrarily choose that parity to be + 1, so the parity operator has no effect on the spin portion of the wave function.

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ιŝĤ = ŝ or [A. s] = 0. (6.27)