∣ n ⟩

0 \leq n \leq 2 S

, with the specified operator ordering. Its matrix elements are

S^{z} ∣ n ⟩ = (S - n) ∣ n ⟩, S^{-} ∣ n ⟩ = (2 S - n) (n + 1) ∣ n + 1 ⟩, S^{+} ∣ n ⟩ = n (2 S - n + 1) ∣ n - 1 ⟩ .

The unavailable endpoint states have zero coefficient. These formulas directly give

[S^{z}, S^{\pm}] ∣ n ⟩ = \pm S^{\pm} ∣ n ⟩

. Moreover

[S^{+}, S^{-}] ∣ n ⟩ = [(2 S - n) (n + 1) - n (2 S - n + 1)] ∣ n ⟩ = 2 (S - n) ∣ n ⟩ = 2 S^{z} ∣ n ⟩ .

ℏ = 1

2 S + 1

, so unrestricted boson Fock space is not itself this finite-spin representation. The square root and oscillator operators must retain their order for the displayed matrix elements and endpoint conditions to agree.

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