Solve the radial equation with the Coulomb potential and something remarkable happens. Near the origin the solution goes as $r^{\ell+1}$, far away it must decay as $e^{-\kappa r}$, and in between it is a polynomial times these — but only if the power series in between stops. For most energies it does not, and its tail grows like $e^{+\kappa r}$, which no bound state can do. The series stops only when $\kappa$ is $1/n$ for a whole number $n$, and that is the whole spectrum: $E=-13.6\,\text{eV}/n^{2}$.

Tuning a guitar string by ear, you hear a clean note only at certain tensions; in between, the sound is wrong. Integrating the radial equation outwards, you find a clean decaying solution only at certain energies; in between, it runs away.

The $\ell=0$ radial equation of hydrogen integrated outwards from the origin, in atomic units, at three energies. At exactly $-\tfrac12$ hartree the solution $u=re^{-r}$ decays. A little above or below, it swings away to : no bound state there.

In atomic units, with $E=-\kappa^{2}/2$ for a bound state, peel off the behaviour at the origin and at infinity and expand the rest as a power series. Normalisability forces the series to be a polynomial.

• The series stops after $n_{r}$ terms only if $\kappa=1/n$ with $n=n_{r}+\ell+1$, a whole number at least .

Substituting the peeled form gives a two-term recursion for the coefficients. If it never stops, the ratio of successive coefficients approaches $2\kappa/j$, the ratio for $e^{2\kappa r}$, and $u$ grows like $e^{\kappa r}$. So some coefficient must vanish, which happens only when for a whole .

The radial equation in atomic units, for a bound state.

Take out the behaviour at the origin and at infinity.

Substitute and collect each power of $r$.

A series that never stops grows like an exponential: not normalisable.

The series stops at $j=n_{r}$ exactly when the bracket vanishes there.

The bound states of hydrogen have energies $-13.6\,\text{eV}/n^{2}$, with $n$ a whole number from 1 up.

Between the origin and infinity the solution is a power series. If it never stops, it grows like $e^{\kappa r}$, and the state cannot be normalised. Only special energies stop it.

The series stops after $n_{r}$ terms, and the energy depends only on $n=n_{r}+\ell+1$, the principal quantum number.

Since $n=n_{r}+\ell+1$ with $n_{r}$ at least zero, the orbital quantum number is at most .

Great work! You now know how to:

• peel the radial function and derive the recursion
• show why only $E=-13.6\,\text{eV}/n^{2}$ gives a normalisable state
• relate $n$, $\ell$ and the number of radial nodes