Canonical Commutation Relation - Angular Momentum Operators

Angular Momentum Operators

where is the Levi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin operators.

All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations, involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators A and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being (ΔA)2≡〈(A−<A>)2〉, etc.

Then

where ≡ AB−BA is the commutator of A and B, and {A,B} ≡AB + BA is the anticommutator.

This follows through use of the Cauchy–Schwarz inequality, since |〈A2〉| |〈B2〉| ≥ |〈AB〉|2, and AB = ( + {A,B}) /2 ; and similarly for the shifted operators A−〈A〉 and B−〈B〉. (cf. Uncertainty principle derivations.) Judicious choices for A and B yield Heisenberg's familiar uncertainty relation, for x and p, as usual; or, here, L_x and L_y, in angular momentum multiplets, ψ = |l,m〉, useful constraints such as l (l+1) ≥ m (m+1), and hence l ≥ m, among others.