Angular momentum operator

Definition

Let $\hbar > 0$, and on $L^2(\R^3)$ let $x = (x_1, x_2, x_3)$ be the position operator and $p = -i\hbar\nabla$ the momentum operator. The orbital angular momentum operator is $L = x \times p$, that is, $$L_1 = -i\hbar \left( x_2 \partial_3 - x_3 \partial_2 \right), \quad L_2 = -i\hbar \left( x_3 \partial_1 - x_1 \partial_3 \right), \quad L_3 = -i\hbar \left( x_1 \partial_2 - x_2 \partial_1 \right).$$ Each $L_k$ is the closure of its action on smooth compactly supported functions, and it is self-adjoint. More generally, an angular momentum operator on a Hilbert space is a triple $J = (J_1, J_2, J_3)$ of self-adjoint operators satisfying, on a common invariant dense domain, $$[J_1, J_2] = i\hbar J_3, \qquad [J_2, J_3] = i\hbar J_1, \qquad [J_3, J_1] = i\hbar J_2.$$ Its square is $J^2 = J_1^2 + J_2^2 + J_3^2$.