Theorem: sl(2) quadratic Casimir C_2 = j(j+1) evaluates to 3/4 at spin j = 1/2

Theorem (sl(2) spin-1/2 Casimir canonical): for spin-j irreducible representation of sl(2), the quadratic Casimir C_2 = H^2/2 + EF + FE acts as the scalar j(j+1) (Schur's lemma). At spin j = 1/2 (fundamental rep, 2x2 Pauli matrices): C_2 =…