= Solution
{c}
Neglect <hyperfine structure>, <spin-orbit coupling> and the diamagnetic term, and set $b=\mu_BB$, $\varepsilon=\alpha a_0^2$. The <Zeeman energy> is $b(L_{\widehat B}+2S_{\widehat B})$. The <electron spin> is free, so every spatial result below is duplicated with an added $\pm b$.
For a <magnetic field> parallel to $z$, the old <magnetic quantum number> $m$ remains good and the first-order <energy levels> are
$$
\boxed{\begin{aligned}
E_{1s,\pm}&=-\mathrm{Ry}+\tfrac32\varepsilon\pm b,\\
E_{2p_0,\pm}&=-\mathrm{Ry}/4+3\varepsilon\pm b,\\
E_{2s,\pm}&=-\mathrm{Ry}/4+21\varepsilon\pm b,\\
E_{2p_m,\pm}&=-\mathrm{Ry}/4+21\varepsilon+mb\pm b,\quad m=\pm1.
\end{aligned}}
$$
The two $s$ <orbitals> have zero <orbital angular momentum>, so only their <electron spin> contributes at this order.
For a <magnetic field> parallel to $x$, the expectation of $L_x$ in each individual $p_m$ <orbital> vanishes. One must still check the degenerate $m=\pm1$ subspace: its projected $L_x$ is zero as well. Thus \b[there is no first-order orbital splitting in a sufficiently weak transverse <magnetic field>]. Spin still contributes $\pm b$.
The <transverse orbital Zeeman mixing> can be seen without assuming its size. Put $E_0=-\mathrm{Ry}/4+3\varepsilon$, $E_t=-\mathrm{Ry}/4+21\varepsilon$, and $\Delta=E_t-E_0=18\varepsilon$. In the real $p_x,p_y,p_z$ <orbital> basis, $p_x$ is unaffected while $p_y,p_z$ have off-diagonal <matrix elements> of modulus $b$. Diagonalizing this two-dimensional <Hamiltonian> gives the spatial <energy eigenvalues>
$$
\boxed{E_t,\qquad \frac{E_t+E_0}{2}\pm\sqrt{\left(\frac\Delta2\right)^2+b^2}.}
$$
When $b\ll\Delta$, their shifts relative to the earlier levels are $0,+b^2/\Delta,-b^2/\Delta$, with $\pm b$ added for <electron spin>. The displayed square-root expression also handles $b$ comparable to $\Delta$, provided coupling to other principal shells stays negligible. The $2s$ <orbital> retains zero orbital correction.
These quadratic orbital estimates keep the enhanced contribution from mixing the split $2p$ levels. Minimal <magnetic minimal coupling> also produces $e^2B^2r_\perp^2/(8m_e)$; its <expectation value> is positive and must be added for a full order-$B^2$ calculation. For example it gives $e^2B^2a_0^2/(4m_e)$ for $1s$. It is parametrically smaller than the $2p$ mixing term when $\varepsilon\ll\mathrm{Ry}$.
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