= Solution
The <orbital angular momentum> of a particle is the operator $\mathbf L=\mathbf r\times\mathbf p$, with $\mathbf p=-i\hbar\nabla$ in the <position> representation. Its <angular momentum commutation relations> are
$$
[L_i,L_j]=i\hbar\epsilon_{ijk}L_k,\qquad [L^2,L_i]=0.
$$
Thus one may specify $L^2$ and one component, conventionally $L_z$, simultaneously. One generally cannot assign sharp values to all three components, because they do not commute. The common angular <eigenfunctions> are the <spherical harmonics>:
$$
\boxed{L^2Y_l^m=\hbar^2l(l+1)Y_l^m,\qquad L_zY_l^m=\hbar mY_l^m,\qquad l=0,1,2,\ldots,\quad m=-l,\ldots,l.}
$$
For a scalar orbital <wavefunction>, single-valuedness in the azimuthal angle gives integer $m$, and regularity on the sphere and termination of the ladder give integer $l\geq|m|$. Half-integer angular momenta describe intrinsic <spin> rather than scalar orbital motion.
The <ladder operators> $L_\pm=L_x\pm iL_y$ obey $[L_z,L_\pm]=\pm\hbar L_\pm$. Their normalized action is
$$
L_\pm|l,m\rangle=\hbar\sqrt{l(l+1)-m(m\pm1)}\,|l,m\pm1\rangle.
$$
The coefficient vanishes at the top or bottom of the multiplet. Its squared magnitude follows from $L_\mp L_\pm=L^2-L_z^2\mp\hbar L_z$, and positivity of the resulting <norm> constrains the allowed $m$ values.
For a <central potential>, the <Hamiltonian (quantum mechanics)> commutes with $L^2$ and every component of $\mathbf L$. Stationary states can be separated as $R_{nl}(r)Y_l^m(\theta,\phi)$, and rotational symmetry makes the <energy> independent of $m$, giving a $2l+1$ <quantum degeneracy> at fixed radial label and $l$. Additional degeneracy between distinct $l$ values requires additional structure, as in the <Coulomb potential>. The <parity> of $Y_l^m$ is $(-1)^l$. These results organize atomic orbitals and their <selection rules>. They describe quantized rotation and symmetry, without assigning the electron a classical orbit or a simultaneously definite three-component <angular momentum> vector.
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