Degeneracy and Quantum Numbers

Several distinct quantum states of hydrogen share the same energy. Counting them — the degeneracy of each level — explains the structure of the periodic table and the capacities of electron shells.

The three spatial quantum numbers

A bound state is specified by three integers:

• Principal $n = 1, 2, 3, \ldots$ — sets the energy $E_n = -13.6\,\text{eV}/n^2$.
• Orbital (azimuthal) $\ell = 0, 1, \ldots, n-1$ — sets the magnitude of angular momentum, $|\mathbf{L}| = \hbar\sqrt{\ell(\ell+1)}$.
• Magnetic $m = -\ell, \ldots, +\ell$ — sets the $z$-component, $L_z = \hbar m$.

The constraint $\ell \le n-1$ comes from the node count $n_r = n - \ell - 1 \ge 0$, and $|m| \le \ell$ comes from the spherical harmonics.

Counting orbital states

For a fixed $\ell$ there are $2\ell + 1$ values of $m$. Summing over all allowed $\ell$ at a given $n$:

So the level $n$ has an orbital degeneracy of $n^2$. The $n=1$ level has $1$ orbital state, $n=2$ has $4$ ($2s$ plus three $2p$), $n=3$ has $9$, and so on. The energy is independent of $\ell$ (an "accidental" Coulomb degeneracy) and independent of $m$ (a consequence of rotational symmetry, true for any central potential).

Adding spin

The electron also carries spin $s = \tfrac{1}{2}$, with two states $m_s = \pm\tfrac{1}{2}$. Each spatial orbital therefore holds two electrons, doubling the count. The total degeneracy of level $n$ is

This is exactly the pattern of the periodic table: the $K$, $L$, $M$ shells hold $2, 8, 18$ electrons — that is $2 \cdot 1^2$, $2 \cdot 2^2$, $2 \cdot 3^2$.

The takeaway

Hydrogen states are labelled by $(n, \ell, m)$ with $0 \le \ell \le n-1$ and $|m| \le \ell$. The orbital degeneracy of level $n$ is $\sum_\ell (2\ell+1) = n^2$, and including the electron's two spin states gives a total degeneracy $g_n = 2n^2$.

Try it

This is a numerical exercise — return a number. Count the total number of states (including spin) for the $n = 3$ hydrogen level by summing $2\ell+1$ over the allowed $\ell$ and then multiplying by the two spin states.

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