= Solution
<orbital angular momentum> generates spatial rotations and is $\mathbf L=\mathbf r\times\mathbf p$. With $\mathbf p=-i\hbar\nabla$, its components satisfy
$$
[L_i,L_j]=i\hbar\epsilon_{ijk}L_k,\qquad [L^2,L_i]=0.
$$
Thus $L^2$ and one component, conventionally $L_z$, have simultaneous <eigenstates>, while all three components cannot generally have sharp values. On ordinary single-valued spatial <wavefunctions>,
$$
L^2Y_{\ell m}=\hbar^2\ell(\ell+1)Y_{\ell m},\qquad
L_zY_{\ell m}=\hbar mY_{\ell m},
\quad\ell=0,1,\ldots,\quad m=-\ell,\ldots,\ell.
$$
The <spherical harmonics> form an <orthonormal> angular <basis>. The integer restriction is specific to orbital <wavefunctions>; half-integer spin is an intrinsic degree of freedom, not the angular dependence of a scalar spatial <wavefunction>.
The <angular momentum ladder operators> obey
$$
L_\pm Y_{\ell m}=\hbar\sqrt{\ell(\ell+1)-m(m\pm1)}\,Y_{\ell,m\pm1}.
$$
The ladder ends at $m=\pm\ell$, producing $2\ell+1$ states. A state with specified $m$ has a sharp projection $\hbar m$, but the <angular momentum> magnitude is $\hbar\sqrt{\ell(\ell+1)}$, not $\hbar\ell$. No classical picture of a vector with simultaneously fixed Cartesian components reproduces all these measurement properties.
In a <central potential>, <rotational symmetry> gives $[H,L_i]=0$. <separation of variables> yields $\psi(\mathbf r)=R_\ell(r)Y_{\ell m}(\theta,\phi)$. The radial <Schrödinger equation> includes a <centrifugal potential> $\hbar^2\ell(\ell+1)/(2mr^2)$, which suppresses small-radius amplitude for nonzero $\ell$. The <energy> is independent of $m$ when <rotational symmetry> is intact, but need not be independent of $\ell$. A <magnetic field> can break that degeneracy. Spatial inversion gives $Y_{\ell m}(-\widehat{\mathbf r})=(-1)^\ell Y_{\ell m}(\widehat{\mathbf r})$, connecting <orbital angular momentum> to <parity> and to the even/odd rotational restriction for the <hydrogen> spin isomers.
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