A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of p and L must be defined carefully. Typically, the operators for the Cartesian components A s are defined using a symmetrized (Hermitian) product, A s = − − m k r ^ ^ s + 1 2 ∑ ∑ i = 1 3 ∑ ∑ j = 1 3 ε ε s i j ( p i ℓ ℓ j + ℓ ℓ j p i ) , Once this is done, one can show that the quantum LRL operators satisfy commutations relations exactly analogous to the Poisson bracket relations in the previous section—just replacing the Poisson bracket with 1 / ( i ℏ ℏ ) times the commutator.