Checkpoint: Hydrogen in Depth

This checkpoint pulls together the quantitative tools of the module: the energy spectrum, ionization energies, and the counting of degenerate states. Work each piece, then combine them.

What you need

The spectrum. $E_n = -13.6\,\text{eV}/n^2$. The level $n=2$ sits at $E_2 = -3.4\,\text{eV}$.

Ionization energy. Removing the electron means raising it to the unbound threshold $E = 0$. The energy required from level $n$ is

So from the ground state $I_1 = 13.6\,\text{eV}$, and from $n=2$, $I_2 = 3.4\,\text{eV}$.

Degeneracy. The number of distinct states at level $n$, including the electron's two spin orientations, is

For $n = 2$ this is $g_2 = 8$: one $2s$ orbital and three $2p$ orbitals, each holding two spins.

Putting it together

These two numbers describe two independent facets of the same level — its binding (an energy) and its multiplicity (a count). The exercise asks you to compute both for $n=2$ and report their product, $I_2 \cdot g_2$, as a single check that you can produce each quantity correctly.

The takeaway

You now hold the core quantitative results of the hydrogen atom: a $-13.6\,\text{eV}/n^2$ spectrum, $+13.6\,\text{eV}/n^2$ ionization energies, and $2n^2$-fold degenerate levels. Together they account for hydrogen's spectrum and the shell structure that organizes the periodic table.

Try it

This is a numerical exercise — return a number. For the $n=2$ level, compute the ionization energy $I_2$ (in eV) and the total degeneracy $g_2 = 2n^2$ (including spin), then return their product $I_2 \cdot g_2$.

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