The classical definition of angular momentum as L=r× p depends on six numbers: rx, ry, rz, px, py, and pz. Translating this into quantum-mechanical terms, the Hiesienberg uncertainity principle tells us that it is not possible for all six of these numbers to be measured simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.
Mathematically, angular momentum in quantum mechanics is defined like momentum- not as a quantity but as an operator on the wave function:
L=r× p where r and p are the position and momentum operators respectively. In particular, for a single particle with no electric charge and no spin, the angular momentum operator can be written in the position basis as
L=-iλ(r×Δ) where Δ is the gradient operator, read as "del," "grad," or "nabla". This orbital angular momentum operator is the most commonly encountered form of the angular momentum operator, though not the only one. It satisfies the following commutation relations
,[Li,LJ>=iλ εijk Lk
where εijk is the (antisymmetric) Levi-Civita symbol.