= Solution
Write the trap <angular frequency> as $\omega$ and the single-particle <oscillator length> as $\ell=\sqrt{\hbar/(m\omega)}$. The <center of mass> coordinate $X=(x_1+x_2)/2$ and relative coordinate $r=x_1-x_2$ give independent <quantum harmonic oscillators>, both of frequency $\omega$, with masses $2m$ and <reduced mass> $m/2$. Label their excitation numbers by $N$ and $n$. The unperturbed <energy eigenvalue> is $\hbar\omega(N+n+1)$.
Particle exchange leaves $X$ unchanged and sends $r$ to $-r$. The <exchange symmetry of two identical spin-one-half fermions> therefore requires even $n$ to accompany the <spin-one-half singlet state>, and odd $n$ to accompany a <spin-one-half triplet state>. For the <contact shifts of two spin-one-half fermions in a harmonic trap>, the <delta potential> depends only on $r$. For odd $n$, the relative <wavefunction> vanishes at $r=0$, so its contact shift is zero. Within the first-excited manifold, both perturbations are diagonal in the labels $(N,n,S,M)$: this is a <perturbation diagonal within a degenerate eigenspace>. For $n=0$, the relative <oscillator length> is $\sqrt2\ell$, giving the <first-order nondegenerate perturbation theory> shift
$$
U=\lambda|\varphi_{\mathrm{rel},0}(0)|^2
=\frac{\lambda}{\sqrt{2\pi}\ell}.
$$
The <Zeeman splitting> is $2\mu_BB_zM$ for total <spin> projection $\hbar M$. Thus the five lowest states, counted with multiplicity, have
$$
\boxed{\begin{array}{c|c|c|c}
(N,n)&S&M&E\text{ to first order}\\ \hline
(0,0)&0&0&\hbar\omega+U\\
(0,1)&1&-1,0,1&2\hbar\omega+2\mu_BB_zM\\
(1,0)&0&0&2\hbar\omega+U
\end{array}}
$$
under the assumed conditions $|U|,2\mu_B|B_z|\ll\hbar\omega$. The three <triplet states> have slopes $-2\mu_B,0,2\mu_B$, while both <singlet states> are field independent at this order. The three <triplet state> branches coincide at zero <magnetic field>. Their crossing with the excited <singlet state> is allowed because the <total-spin sectors> do not mix. The odd relative <wavefunctions> are in fact unaffected by this <delta potential> at any coupling; the displayed <singlet state> shifts retain only first order.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2022-q3-two-fermion-levels.png]
{title=Five lowest two-fermion states in a weakly interacting harmonic trap}
{description=Original plot for repulsive contact shift $U=0.12\hbar\omega$, with $b=2\mu_BB_z/(\hbar\omega)$. The two horizontal <singlet state> branches and three <triplet state> branches count five states; coincident energies count as fewer distinct <energy levels>. An attractive coupling reverses the sign of $U$ and moves both <singlet state> branches downward.}
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