= Solution
{c}
Measure each <energy level> by the <wavenumber> $\widetilde E=(E-E_g)/(hc)$ in $\mathrm{m^{-1}}$. An upward <electronic transition> of vacuum <wavelength> $\lambda$ adds $1/\lambda$. Treat the quoted <wavelengths> as vacuum values to their limited accuracy; air corrections would be smaller than the precision justified here.
The strong ground-state <absorption> must be $5s^2\,{}^1S_0\to5s5p\,{}^1P_1$ at $461\,\mathrm{nm}$. The weak ground-state <electronic transition> at $689\,\mathrm{nm}$ must reach $5s5p\,{}^3P_1$: the <electric-dipole selection rule> on $J$ excludes the other two <triplet state> <energy levels>, while weak <spin-orbit coupling> admixes <singlet state> character and relaxes the approximate <spin selection rule>.
Assume the usual ordering $E({}^3P_0)<E({}^3P_1)<E({}^3P_2)$, as expected from a positive effective <spin-orbit coupling> constant. The three nearby strong <spectral lines> then join these <energy levels> to a common $5s6s\,{}^3S_1$ <energy level>. Larger lower-state <energy> means a smaller gap and longer <wavelength>, so $679,688,707\,\mathrm{nm}$ correspond to $J=0,1,2$ respectively. The remaining $1.12\,\mu\mathrm m$ <spectral line> joins $5s5p\,{}^1P_1$ and $5s6s\,{}^1S_0$. Assume the latter is the upper <energy level>. These assignments and orderings agree with independently tabulated <strontium> <energy levels>.<Energy Levels of Neutral Strontium by NIST>[Entries for 5s², 5s5p and 5s6s; the tabulated energies use cm⁻¹ rather than m⁻¹]{ref}
Starting with the two ground-state gaps, the construction is
$$
\begin{aligned}
\widetilde E({}^1P_1)&=(461\,\mathrm{nm})^{-1},\\
\widetilde E({}^3P_1)&=(689\,\mathrm{nm})^{-1},\\
\widetilde E({}^3S_1)&=\widetilde E({}^3P_1)+(688\,\mathrm{nm})^{-1},\\
\widetilde E({}^3P_0)&=\widetilde E({}^3S_1)-(679\,\mathrm{nm})^{-1},\\
\widetilde E({}^3P_2)&=\widetilde E({}^3S_1)-(707\,\mathrm{nm})^{-1},\\
\widetilde E(5s6s\,{}^1S_0)&=\widetilde E({}^1P_1)+(1.12\,\mu\mathrm m)^{-1}.
\end{aligned}
$$
The resulting <energy levels>, rounded to avoid suggesting spectroscopic precision, are
$$
\boxed{\begin{array}{c|r}
\text{level}&\widetilde E\ (\mathrm{m^{-1}})\\\hline
5s^2\,{}^1S_0&0\\
5s5p\,{}^3P_0&1.432\times10^6\\
5s5p\,{}^3P_1&1.451\times10^6\\
5s5p\,{}^3P_2&1.490\times10^6\\
5s5p\,{}^1P_1&2.169\times10^6\\
5s6s\,{}^3S_1&2.905\times10^6\\
5s6s\,{}^1S_0&3.062\times10^6
\end{array}}
$$
\Image[/past-exam-of-the-atomic-and-laser-physics-course-of-the-university-of-oxford/2022-strontium-levels.png]
{title=Strontium level assignments inferred from the supplied wavelengths; vertical spacings are schematic and numerical energies are in inverse metres}
\b[Both $461\,\mathrm{nm}$ and $1.12\,\mu\mathrm m$ exhibit the <normal Zeeman effect>.] Each connects two <singlet state> <energy levels>, with <Landé g-factor> $g_J=1$ for the $J=1$ <energy level> and no first-order shift for $J=0$. Their <frequency> shifts are $0,\pm\mu_BB/h$. The weak $689\,\mathrm{nm}$ <spectral line> also has three components, but $g({}^3P_1)=3/2$ gives shifts $0,\pm3\mu_BB/(2h)$; a three-component pattern alone does not establish the <normal Zeeman effect>. The $679\,\mathrm{nm}$ <spectral line> likewise has three components with $g({}^3S_1)=2$.
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