Angular momentum of a homogeneous harmonic polynomial (${\hat{L}}^2{H}_l=l(l+1){\hslash }^2{H}_l$)

For a homogeneous harmonic polynomial $H_l$ of degree $l$ in three dimensions, the Euler theorem for homogeneous functions gives $D{H}_l=l{H}_l$, where $D=\mathbf{x}\cdot \nabla$. The orbital angular momentum identity ${\hat{L}}^2=-{\hslash }^2(r^2\Delta -D^2-D)$ therefore gives ${\hat{L}}^2{H}_l=l(l+1){\hslash }^2{H}_l$. This constructs angular momentum eigenstates directly from polynomials.