= Solution
{c}
Write $k=e^2/(4\pi\epsilon_0)$ and retain the nuclear <Coulomb potential>. In <spherical coordinates>, the total <potential energy> is
$$
V=-\frac{k}{r}+r^2[\alpha+(\beta-\alpha)\cos^2\theta]
=-\frac{k}{r}+\frac{\alpha r^2}{2}[1-2P_2(\cos\theta)]
$$
when $\beta=-\alpha/2$; $P_2(u)=(3u^2-1)/2$ is a <Legendre polynomial>. Treat the added term using <quadratic anisotropic perturbation of hydrogen> and <degenerate perturbation theory> in the $n=2$ <hydrogen energy eigenbasis>. It preserves <parity> and $m$, so it cannot mix $2s$ with $2p$, or distinct $2p$ <magnetic sublevels>.
For a <hydrogen atom>, the <hydrogen radial second moment> gives
$$
\langle r^2\rangle_{nl}=\frac{a_0^2n^2}{2}[5n^2+1-3l(l+1)],
\qquad
\langle r^2\rangle_{1s}=3a_0^2,\quad
\langle r^2\rangle_{2s}=42a_0^2,\quad
\langle r^2\rangle_{2p}=30a_0^2.
$$
The angular <expectation values> of $\cos^2\theta$ are $1/3$ for an $s$ <orbital>, $3/5$ for $p_0$, and $1/5$ for $p_{\pm1}$, obtained by integrating the squared <spherical harmonics>. Therefore the five lowest spatial <quantum states>, counting <statistical weight>, have first-order <energy levels>
$$
\boxed{\begin{array}{c|c|c}
\text{state}&E\text{ to first order}&\text{spatial multiplicity}\\\hline
1s&-\mathrm{Ry}+\tfrac32\alpha a_0^2&1\\
2p_0&-\mathrm{Ry}/4+3\alpha a_0^2&1\\
2s,\ 2p_{+1},\ 2p_{-1}&-\mathrm{Ry}/4+21\alpha a_0^2&3
\end{array}}
$$
Each spatial <quantum state> has two <electron spin> states when <spin-orbit coupling> and the <magnetic field> are omitted. Thus these are five spatial states but only three distinct first-order <energies>; five distinct <energies> would be an incorrect reading. The accidental $2s,2p_{\pm1}$ equality can be lifted at higher order. Quantitatively the added shifts must also be small compared with neighboring principal-shell gaps; $\alpha\ll k/a_0^3$ is an asymptotic condition, not a guarantee for arbitrary large $n$.
\Image[/past-exam-of-the-atomic-and-laser-physics-course-of-the-university-of-oxford/2021-quadratic-hydrogen-levels.png]
{title=First-order spatial levels of locally perturbed hydrogen; the gap between principal shells is compressed}
For general $\beta$, the $1s$ <expectation value> is
$$
\Delta E_{1s}^{(1)}=(2\alpha+\beta)a_0^2,\qquad
\boxed{\beta=-2\alpha\text{ cancels the first-order shift.}}
$$
It does \b[not] cancel every order. At $\beta=-2\alpha$, $V_{\rm ext}=-2\alpha r^2P_2(\cos\theta)$ couples $1s$ to $d$ <orbitals>. For example its angular <matrix element> with $Y_{20}$ is $1/\sqrt5$, and the $3d$ radial <matrix element> is nonzero. The <second-order nondegenerate perturbation theory> sum has negative denominators, so this is a concrete nonzero higher-order contribution.
There is also a global limitation: for either negative $\beta$ above, $V(0,0,z)=-k/|z|+\beta z^2\to-\infty$. A localized wavepacket translated far along $z$ has fixed <kinetic energy> but arbitrarily negative <potential energy>, proving there is no true lowest <energy level>. The results describe local perturbative <resonance poles>, or a potential completed to a confining form far from the <atomic nucleus>. No globally exact ground-state cancellation exists for the potential as printed.
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