Angular momentum of a homogeneous harmonic polynomial
= Angular momentum of a homogeneous harmonic polynomial
{title2=$\hat L^2H_l=l(l+1)\hbar^2H_l$}
For a homogeneous <harmonic polynomial> $H_l$ of degree $l$ in three dimensions, the <Euler theorem for homogeneous functions> gives $DH_l=lH_l$, where $D=\mathbf x\cdot\nabla$. The <orbital angular momentum> identity $\hat L^2=-\hbar^2(r^2\Delta-D^2-D)$ therefore gives $\hat L^2H_l=l(l+1)\hbar^2H_l$. This constructs <angular momentum eigenstates> directly from polynomials.