= Quadratic anisotropic perturbation of hydrogen
{title2=$V'=\alpha(x^2+y^2)+\beta z^2$}
An axially symmetric quadratic <potential energy> perturbation preserves <parity> and the $z$ component of <orbital angular momentum>. For $1s,2s,2p_0,2p_{\pm1}$, its first-order <energy> shifts are respectively $(2\alpha+\beta)a_0^2$, $14(2\alpha+\beta)a_0^2$, $(12\alpha+18\beta)a_0^2$, and $(24\alpha+6\beta)a_0^2$. They follow by multiplying the <hydrogen radial second moment> by the angular average of $\alpha+(\beta-\alpha)\cos^2\theta$, whose $\cos^2\theta$ averages are $1/3,1/3,3/5,1/5$. Negative $\beta$ makes the globally extended potential unbounded below; the shifts then describe local perturbative <resonance poles> or a potential stabilized at large distance.
Back to article page
