= Axially symmetric quadratic orbital eigenfunction
{title2=$\psi=(x_3^2-r^2/3)f(r),\quad L^2\psi=6\hbar^2\psi,\quad L_3\psi=0$}
Each <orbital angular momentum> operator annihilates a radial factor and $r^2$. Direct differentiation gives $L^2x_3^2=\hbar^2(6x_3^2-2r^2)$ and $L_3x_3^2=0$. Hence the displayed trace-free quadratic polynomial, times any admissible radial factor, has orbital labels $l=2,m=0$. Its angular dependence is proportional to the <spherical harmonic> $Y_{20}$; subtracting $r^2/3$ removes the rotational scalar part.
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