Spin Casimir expectation values: ⟨S²⟩_(s=1/2) = 3ℏ²/4; ⟨S²⟩_(s=1) = 2ℏ²; ⟨S²⟩_(s=3/2) = 15ℏ²/4

Spin Casimir expectation values. From framework F9 (Casimir invariant), the total spin-squared operator S² on a spin-s irrep has eigenvalue ℏ²·s(s+1). Concrete cases: s = 1/2 ⇒ ⟨S²⟩ = (3/4)·ℏ²; s = 1 ⇒ ⟨S²⟩ = 2·ℏ²; s = 3/2 ⇒ ⟨S²⟩ =…