---
title: The Hydrogen Atom
module: Central Potentials
moduleNumber: 7
lessonNumber: 2
order: 702
summary: >
  The Coulomb potential turns the radial equation into one whose bound states
  exist only for a discrete set of energies. A power-series solution truncated to
  keep the wavefunction normalizable forces the principal quantum number, and the
  energy comes out proportional to minus one over its square, recovering the
  Rydberg spectrum. The bound states are the associated Laguerre functions times
  spherical harmonics, and their energy depends on the principal number alone,
  giving an n-squared degeneracy larger than rotational symmetry can explain.
topics: [Central Potentials]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 4 — Quantum Mechanics in Three Dimensions; §4.2 The Hydrogen Atom"
  - book: Shankar
    ref: "Ch. 13 — The Hydrogen Atom"
  - book: Tipler & Llewellyn
    ref: "Ch. 7 — Atomic Physics; §7-1 The Schrödinger Equation for Hydrogen"
draft: false
---

The hydrogen atom is the only atom whose bound-state problem admits a closed
solution, and every feature of atomic structure is calibrated against it. A
single electron of charge $-e$ moves in the Coulomb field of a proton of charge
$+e$. The potential is central, so the machinery of the
[radial equation](/quantum-mechanics/central-potentials/the-schrodinger-equation-in-three-dimensions)
applies directly: the angular part is a
[spherical harmonic](/quantum-mechanics/angular-momentum/orbital-angular-momentum-and-spherical-harmonics),
and the whole problem reduces to a one-dimensional radial equation on the
half-line. What is special is what the $1/r$ potential does to the spectrum: the
energies depend only on a single principal quantum number, and the resulting
degeneracy exceeds what rotational invariance alone requires.

## The Coulomb radial equation

The electron-proton interaction is
$V(r) = -\dfrac{e^2}{4\pi\epsilon_0}\dfrac{1}{r}$ in SI units. Because the proton
is roughly $1836$ times heavier than the electron, both orbit their common
center of mass and the correct one-body mass is the reduced mass
$\mu = m_e m_p/(m_e + m_p) \approx 0.99946\,m_e$; the approximation
$\mu \approx m_e$ is used throughout and corrected where a part-per-thousand
number matters. With $u = rR$ the radial equation is

$$
-\frac{\hbar^2}{2\mu}\,\frac{\d^2 u}{\d r^2}
  + \left[-\frac{e^2}{4\pi\epsilon_0}\,\frac{1}{r}
    + \frac{\hbar^2\,\ell(\ell+1)}{2\mu r^2}\right]u = E\,u.
$$

Bound states have $E < 0$. Two abbreviations make the structure visible. Define
the positive decay constant $\kappa$ and the dimensionless radius $\rho$,

$$
\kappa = \frac{\sqrt{-2\mu E}}{\hbar},
\qquad
\rho = \kappa r,
$$

so that $\kappa$ carries the (negative) energy and $\rho$ is measured in units of
$1/\kappa$. Dividing the equation by $E = -\hbar^2\kappa^2/2\mu$ and collecting
constants,

$$
\frac{\d^2 u}{\d\rho^2}
  = \left[1 - \frac{\rho_0}{\rho} + \frac{\ell(\ell+1)}{\rho^2}\right]u,
\qquad
\rho_0 \equiv \frac{\mu e^2}{2\pi\epsilon_0\hbar^2\,\kappa}.
$$

The single dimensionless combination $\rho_0$ measures the strength of the
Coulomb attraction relative to the energy. Quantization will come from requiring
that the solution stay normalizable, which restricts $\rho_0$ to a discrete set.

$$
% caption: The Coulomb effective potential $-1/r + \ell(\ell+1)/r^2$ supports a
% tower of bound levels below zero; the centrifugal barrier lifts the inner wall
% for $\ell \ge 1$ and sets the minimum radius each level can reach.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (0,-2.9) -- (0,2.2) node[anchor=south, black] {$V$};
\draw[->, black] (0,0) -- (7.2,0) node[anchor=east, black] {$r$};
% l=0: -1/r attractive
\draw[black, very thick, densely dashed, domain=0.55:7.0, samples=140, variable=\x]
  plot ({\x}, {-2.4/\x});
\node[black, anchor=north] at (4.8,-1.05) {$l = 0$};
% l=1: -1/r + b/r^2 with a well
\draw[acc, very thick, domain=0.72:7.0, samples=160, variable=\x]
  plot ({\x}, {-2.4/\x + 1.1/(\x*\x)});
\node[acc, anchor=west] at (2.55,-0.15) {$l = 1$};
% bound levels as dashed horizontal lines (l=0 well)
\draw[black, dashed] (0.55,-1.55) -- (4.0,-1.55) node[anchor=west, black, font=\scriptsize] {$n=1$};
\draw[black, dashed] (1.4,-0.7) -- (5.6,-0.7) node[anchor=west, black, font=\scriptsize] {$n=2$};
\draw[black, dashed] (2.4,-0.38) -- (6.8,-0.38) node[anchor=west, black, font=\scriptsize] {$n=3$};
\end{tikzpicture}
$$

## Asymptotic behavior and the series solution

The solution is peeled apart by extracting its behavior at the two ends of the
half-line, exactly as for the
[harmonic oscillator](/quantum-mechanics/wave-mechanics-1d/operators-expectation-values-and-the-harmonic-oscillator).

- **As $\rho \to \infty$** the bracket approaches $1$, so
  $u'' \approx u$ and the normalizable solution is $u \sim e^{-\rho}$.
- **As $\rho \to 0$** the centrifugal term dominates, giving
  $u'' \approx \ell(\ell+1)\rho^{-2}u$ and the regular solution
  $u \sim \rho^{\ell+1}$, the origin behavior established for every central
  potential.

Factor out both limits and let a power series carry the rest:

$$
u(\rho) = \rho^{\ell+1}\,e^{-\rho}\,v(\rho),
\qquad
v(\rho) = \sum_{j=0}^{\infty} c_j\,\rho^{\,j}.
$$

Substituting into the radial equation and cancelling the common factor
$\rho^{\ell}e^{-\rho}$ produces a second-order equation for $v$,

$$
\rho\,\frac{\d^2 v}{\d\rho^2}
  + 2(\ell + 1 - \rho)\,\frac{\d v}{\d\rho}
  + \big[\rho_0 - 2(\ell+1)\big]\,v = 0.
$$

Matching powers of $\rho$ term by term gives a two-term recursion for the
coefficients,

> **Theorem (Laguerre recursion).** The series coefficients of $v$ obey
> $$
> c_{j+1} = \frac{2(j + \ell + 1) - \rho_0}{(j+1)(j + 2\ell + 2)}\,c_j.
> $$
> The ratio $c_{j+1}/c_j \to 2/j$ for large $j$, the same growth as the series
> for $e^{2\rho}$. If the series does not terminate, $v \sim e^{2\rho}$ and
> $u \sim \rho^{\ell+1}e^{+\rho}$ diverges, so a normalizable state exists only
> when the series truncates.

## Quantization of the energy

Truncation requires the numerator of the recursion to vanish at some finite
$j = j_{\max}$, killing $c_{j_{\max}+1}$ and all higher terms:

$$
2(j_{\max} + \ell + 1) = \rho_0.
$$

The integer combination on the left is the **principal quantum number**,

$$
n \equiv j_{\max} + \ell + 1,
\qquad
\rho_0 = 2n,
\qquad
n = 1, 2, 3, \dots
$$

Since $j_{\max} \ge 0$, the angular momentum is bounded by $\ell \le n - 1$: a
level with principal number $n$ admits $\ell = 0, 1, \dots, n-1$. The truncation
condition $\rho_0 = 2n$ feeds straight back into the definition of $\rho_0$ and
$\kappa$. Solving $\dfrac{\mu e^2}{2\pi\epsilon_0\hbar^2\kappa} = 2n$ for
$\kappa$ gives $\kappa = \dfrac{1}{n a_0}$ with the natural length scale

$$
a_0 = \frac{4\pi\epsilon_0\hbar^2}{\mu e^2}
   = 5.29177 \times 10^{-11}\ \text{m},
$$

the **Bohr radius**. Because $E = -\hbar^2\kappa^2/2\mu$, the energy is

> **Theorem (Bohr energy spectrum).** The bound-state energies of hydrogen are
> $$
> E_n = -\frac{\mu}{2\hbar^2}\!\left(\frac{e^2}{4\pi\epsilon_0}\right)^{\!2}\frac{1}{n^2}
>   = -\frac{E_1}{n^2},
> \qquad
> E_1 = 13.606\ \text{eV},
> $$
> depending only on $n$ and not on $\ell$ or $m$. The binding energy of the
> ground state is $13.6\ \text{eV}$, and the levels accumulate toward the
> ionization threshold $E = 0$ as $n \to \infty$.

The constant $E_1$ is the **Rydberg energy**, $\text{Ry} = 13.6057\ \text{eV}$;
it can be written $E_1 = \tfrac12\alpha^2\mu c^2$ with $\alpha \approx 1/137$ the
fine-structure constant, exposing hydrogen binding as an order-$\alpha^2$
correction to the electron rest energy. The spectral lines follow from energy
differences: a transition from $n_i$ to $n_f$ emits or absorbs a photon of energy

$$
E_\gamma = E_1\!\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right),
$$

the Rydberg formula, with $n_f = 1$ the Lyman series (ultraviolet), $n_f = 2$ the
Balmer series (visible), $n_f = 3$ the Paschen series (infrared).

$$
% caption: Hydrogen levels $E_n = -13.6\,\text{eV}/n^2$ crowd toward the
% ionization limit; each level $n$ carries the angular-momentum values
% $\ell = 0,\dots,n-1$, and downward transitions form the named spectral series.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% energy axis
\draw[->, black] (0,-5.4) -- (0,0.7) node[anchor=south, black] {$E$};
\draw[black, dashed] (0,0) -- (7.2,0);
\node[anchor=west, black, font=\scriptsize] at (7.25,0) {$E = 0$ (ionized)};
% levels at -13.6/n^2 scaled by /2.6 : n1 -13.6->-5.23 ; n2 -3.4->-1.31 ; n3 -1.51->-0.58 ; n4 -0.85->-0.33
\draw[acc, very thick] (0.7,-5.23) -- (6.6,-5.23);
\node[anchor=west, black, font=\scriptsize] at (6.7,-5.23) {$n=1$};
\node[anchor=east, black, font=\scriptsize] at (0.6,-5.23) {-13.6 eV};
\draw[acc, very thick] (0.7,-1.31) -- (6.6,-1.31);
\node[anchor=west, black, font=\scriptsize] at (6.7,-1.31) {$n=2$};
\node[anchor=east, black, font=\scriptsize] at (0.6,-1.31) {-3.40 eV};
\draw[acc, very thick] (0.7,-0.58) -- (6.6,-0.58);
\node[anchor=west, black, font=\scriptsize] at (6.7,-0.58) {$n=3$};
\node[anchor=east, black, font=\scriptsize] at (0.6,-0.58) {-1.51 eV};
\draw[acc, very thick] (0.7,-0.33) -- (6.6,-0.33);
\node[anchor=west, black, font=\scriptsize] at (6.7,-0.33) {$n=4$};
% transition arrows
\draw[->, acc, thick] (1.6,-1.31) -- (1.6,-5.10);
\node[acc, anchor=west, font=\scriptsize] at (1.5,-3.4) {Lyman};
\draw[->, black, thick, densely dashed] (3.4,-0.58) -- (3.4,-1.24);
\node[black, anchor=west, font=\scriptsize] at (3.5,-0.95) {Balmer};
\end{tikzpicture}
$$

## The bound-state wavefunctions

The truncated series $v$ is, up to normalization, an **associated Laguerre
polynomial**. With the standard definitions

$$
L_q^{p}(x) = (-1)^p\,\frac{\d^p}{\d x^p}\,L_{p+q}(x),
\qquad
L_q(x) = \frac{e^x}{q!}\,\frac{\d^q}{\d x^q}\big(e^{-x}\,x^q\big),
$$

the radial factor is $v(\rho) \propto L_{n-\ell-1}^{2\ell+1}(2\rho)$, a polynomial
of degree $n - \ell - 1$. Restoring $u = \rho^{\ell+1}e^{-\rho}v$ and
$R = u/r$, and normalizing so that
$\int_0^\infty |R_{n\ell}|^2 r^2\,\d r = 1$, gives

$$
R_{n\ell}(r)
  = \sqrt{\left(\frac{2}{n a_0}\right)^{\!3}\frac{(n-\ell-1)!}{2n\,(n+\ell)!}}\;
    e^{-r/n a_0}
    \left(\frac{2r}{n a_0}\right)^{\!\ell}
    L_{n-\ell-1}^{2\ell+1}\!\left(\frac{2r}{n a_0}\right).
$$

The full stationary state is the product with the angular harmonic,

$$
\psi_{n\ell m}(r,\theta,\varphi) = R_{n\ell}(r)\,Y_\ell^m(\theta,\varphi),
$$

labelled by the three quantum numbers $(n, \ell, m)$: principal, azimuthal, and
magnetic. The lowest few radial functions, with $a \equiv a_0$:

$$
R_{10} = \frac{2}{a^{3/2}}\,e^{-r/a},
\qquad
R_{20} = \frac{1}{\sqrt{2}\,a^{3/2}}\!\left(1 - \frac{r}{2a}\right)e^{-r/2a},
$$
$$
R_{21} = \frac{1}{\sqrt{24}\,a^{3/2}}\,\frac{r}{a}\,e^{-r/2a},
\qquad
R_{30} = \frac{2}{\sqrt{27}\,a^{3/2}}\!\left(1 - \frac{2r}{3a} + \frac{2r^2}{27a^2}\right)e^{-r/3a}.
$$

The ground state $\psi_{100} = (\pi a^3)^{-1/2}e^{-r/a}$ is nodeless, spherically
symmetric, and peaks at the origin. Each increase of $n$ at fixed $\ell$ adds one
radial node; the polynomial degree $n - \ell - 1$ equals the number of interior
nodes.

### Radial probability and characteristic radii

The probability of finding the electron in a shell of thickness $\d r$ at radius
$r$ is the **radial distribution**

$$
P_{n\ell}(r)\,\d r = |R_{n\ell}(r)|^2\,r^2\,\d r,
$$

the extra $r^2$ from the volume element competing against the exponential decay
to place the density at a finite radius. For the ground state,
$P_{10}(r) = \dfrac{4}{a^3}r^2 e^{-2r/a}$; setting $P_{10}'(r) = 0$ gives the most
probable radius $r = a_0$, reproducing the Bohr radius as the peak of the
quantum-mechanical distribution rather than a sharp orbit. The mean radius is
larger,

$$
\langle r \rangle_{n\ell}
  = \frac{a_0}{2}\big[\,3n^2 - \ell(\ell+1)\,\big],
$$

growing as $n^2$: highly excited states are physically large, the wave-mechanical
content of the classical scaling of orbit size with energy.

$$
% caption: Radial distributions $r^2|R_{n\ell}|^2$ for the 1s, 2s, and 2p states.
% The 1s peaks at the Bohr radius; the 2s carries an interior node and a small
% inner lobe; the 2p is nodeless in the radius but centered farther out.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% axes: r/a0 from 0 to ~12 mapped to x 0..6.8 (r/a0 = 1.7778*\x)
\draw[->, black] (0,0) -- (7.1,0) node[anchor=north, black] {$r / a_0$};
\draw[->, black] (0,0) -- (0,3.3) node[anchor=south, black] {$P(r)$};
% ticks
\foreach \xr/\lab in {1.125/2, 2.8125/5, 5.625/10}
  \draw[black] (\xr,0.06) -- (\xr,-0.06) node[anchor=north, black, font=\scriptsize] {\lab};
% horizontal variable \x maps to r/a0 = 1.7778*\x. Products written out (the ^
% operator fails at r=0 via ln 0), with exp interleaved so intermediates stay in
% the pgfmath fixed-point range (< 16384). Domain capped at r ~ 10.7 a0 so the
% exp does not underflow into fixed-point noise at the tail.
% P_10 = 4 r^2 e^{-2r}; scale *4.4.
\draw[acc, very thick, domain=0:5.6, samples=320, variable=\x]
  plot ({\x}, {4.4*4*(1.7778*\x)*exp(-2*1.7778*\x)*(1.7778*\x)});
\node[acc, anchor=south] at (0.95,2.45) {1s};
% P_20 = (1/2) r^2 (1 - r/2)^2 e^{-r}; scale *4.4
\draw[black, very thick, densely dashed, domain=0:5.6, samples=340, variable=\x]
  plot ({\x}, {4.4*0.5*(1.7778*\x)*(1.7778*\x)*exp(-1.7778*\x)*(1-0.8889*\x)*(1-0.8889*\x)});
\node[black, anchor=south] at (3.55,1.35) {2s};
% P_21 = (1/24) r^4 e^{-r}; scale *4.4
\draw[black, very thick, densely dotted, domain=0:5.6, samples=340, variable=\x]
  plot ({\x}, {4.4*(1/24)*(1.7778*\x)*(1.7778*\x)*exp(-1.7778*\x)*(1.7778*\x)*(1.7778*\x)});
\node[black, anchor=south] at (2.7,1.0) {2p};
\end{tikzpicture}
$$

## The n-squared degeneracy

For each principal number $n$, the allowed values are $\ell = 0, 1, \dots, n-1$,
and for each $\ell$ there are $2\ell + 1$ magnetic sublevels. Summing,

$$
g_n = \sum_{\ell=0}^{n-1}(2\ell + 1) = n^2.
$$

The $n^2$ states with the same $n$ all share the single energy $E_n$. Part of
this is expected: the $(2\ell+1)$-fold $m$-degeneracy holds for **any** central
potential, being a direct consequence of rotational invariance — nothing in
$H$ singles out a direction, so states differing only in the orientation of
$\vec L$ must be degenerate. The remainder is not expected. That levels of
different $\ell$ (say $2s$ and $2p$) coincide is a property of the Coulomb $1/r$
potential specifically; a spherical box or a screened potential splits them.

> **Definition (Accidental degeneracy).** A degeneracy not forced by the manifest
> symmetry of the Hamiltonian. In hydrogen the $\ell$-degeneracy at fixed $n$ is
> accidental with respect to rotational symmetry, and it signals a larger hidden
> symmetry — the $\mathrm{SO}(4)$ generated by angular momentum together with the
> conserved [Runge–Lenz vector](/quantum-mechanics/central-potentials/the-isotropic-oscillator-and-hidden-symmetry).

$$
% caption: The $n^2$ hydrogen states of the $n = 3$ shell, arranged by
% subshell $\ell = 0,1,2$ (columns $s,p,d$) and by $m$ (dots stacked in a
% column). Rotational symmetry explains each column's $2\ell+1$ height; the
% equal energy across columns is the accidental Coulomb degeneracy.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% n=3 shell: l=0 (1 state), l=1 (3), l=2 (5) -> 9 = 3^2
\node[anchor=south, black, font=\small] at (4.0,3.4) {shell $n = 3$: $1 + 3 + 5 = 9$ states};
% column subshell labels (Greek/ell kept out of nodes)
\node[anchor=north, black!70, font=\scriptsize] at (1.2,-0.55) {$s$};
\node[anchor=north, black!70, font=\scriptsize] at (4.0,-0.55) {$p$};
\node[anchor=north, black!70, font=\scriptsize] at (6.8,-0.55) {$d$};
% l=0: one dot at m=0
\fill[acc] (1.2,0) circle (3pt);
% l=1: m=-1,0,1
\foreach \y in {-1,0,1} \fill[acc] (4.0,\y) circle (3pt);
% l=2: m=-2..2
\foreach \y in {-2,-1,0,1,2} \fill[acc] (6.8,\y) circle (3pt);
% m axis labels on left
\node[anchor=east, black, font=\scriptsize] at (0.3,2) {$m=+2$};
\node[anchor=east, black, font=\scriptsize] at (0.3,0) {$m=0$};
\node[anchor=east, black, font=\scriptsize] at (0.3,-2) {$m=-2$};
\draw[black] (0.5,-2.4) -- (0.5,2.4);
\end{tikzpicture}
$$

### Orbital shapes

The probability density $|\psi_{n\ell m}|^2$ has an angular shape from
$|Y_\ell^m|^2$ and a radial envelope from $|R_{n\ell}|^2$. The $s$ states
($\ell = 0$) are spherically symmetric; the $p$ states ($\ell = 1$) have a
two-lobe angular profile; the $d$ states ($\ell = 2$) have four lobes or the
pinched $d_{z^2}$ shape. Multiplying by the radial nodes of $R_{n\ell}$ produces
the shells-within-lobes structure familiar from chemistry.

$$
% caption: Angular shapes of the probability density: the isotropic 1s, the
% two-lobe 2p aligned with the polar axis, and the four-lobe 3d profile, drawn
% as polar plots of $|Y_\ell^m|^2$ against the polar angle.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% --- 1s: circle ---
\begin{scope}[xshift=0cm]
  \draw[->, black] (0,-1.5) -- (0,1.8) node[anchor=south, black, font=\scriptsize] {$z$};
  \draw[acc, very thick, fill=acc!10] (0,0) circle (1.0);
  \node[anchor=north, font=\scriptsize] at (0,-1.65) {1s};
\end{scope}
% --- 2p_z: two lobes ---
\begin{scope}[xshift=4.2cm]
  \draw[->, black] (0,-1.9) -- (0,2.0) node[anchor=south, black, font=\scriptsize] {$z$};
  \draw[acc, very thick, fill=acc!10]
    (0,0) .. controls (1.4,0.4) and (0.85,1.6) .. (0,1.6)
          .. controls (-0.85,1.6) and (-1.4,0.4) .. (0,0);
  \draw[acc, very thick, fill=acc!10]
    (0,0) .. controls (1.4,-0.4) and (0.85,-1.6) .. (0,-1.6)
          .. controls (-0.85,-1.6) and (-1.4,-0.4) .. (0,0);
  \node[anchor=north, font=\scriptsize] at (0,-2.0) {2p};
\end{scope}
% --- 3d: four lobes ---
\begin{scope}[xshift=8.6cm]
  \draw[->, black] (0,-1.9) -- (0,2.0) node[anchor=south, black, font=\scriptsize] {$z$};
  \foreach \a in {45,135,225,315} {
    \draw[acc, very thick, fill=acc!10]
      (0,0) .. controls ({1.6*cos(\a-16)},{1.6*sin(\a-16)}) and ({1.6*cos(\a+16)},{1.6*sin(\a+16)}) .. (0,0);
  }
  \node[anchor=north, font=\scriptsize] at (0,-2.0) {3d};
\end{scope}
\end{tikzpicture}
$$

## The size of the correction terms

The solution here treats the electron as a spinless particle in a static Coulomb
field, and it reproduces the gross spectrum to within a part in $10^4$. The
neglected effects each carry a definite scale.

| Effect | Origin | Fractional size |
| --- | --- | --- |
| Reduced mass | finite proton mass | $\sim m_e/m_p \approx 5\times 10^{-4}$ |
| Fine structure | relativistic + spin–orbit | $\sim \alpha^2 \approx 5\times 10^{-5}$ |
| Lamb shift | quantum-electrodynamic | $\sim \alpha^3\ln\alpha$ |
| Hyperfine | proton magnetic moment | $\sim (m_e/m_p)\alpha^2$ |

The reduced-mass correction is included exactly by using $\mu$ in $a_0$ and
$E_n$. The others enter as
[perturbations](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
on the degenerate $n^2$ multiplets and break the accidental degeneracy: the
[fine structure](/quantum-mechanics/approximation-methods/fine-structure-and-the-real-hydrogen-atom)
splits levels of different $j$, and external fields produce the
[Zeeman and Stark](/quantum-mechanics/approximation-methods/the-zeeman-and-stark-effects)
effects. The unperturbed spectrum derived here is the platform every one of those
corrections is measured against.

The exactness of the Coulomb solution and its oversized degeneracy point to a
structure beyond rotational symmetry. The
[next lesson](/quantum-mechanics/central-potentials/the-isotropic-oscillator-and-hidden-symmetry)
identifies the conserved Runge–Lenz vector whose algebra with $\vec L$ closes
into $\mathrm{SO}(4)$ and forces exactly the $n^2$ degeneracy found here.[^gs-hydrogen][^shankar-hydrogen][^codata]

[^gs-hydrogen]: Griffiths & Schroeter, _Introduction to Quantum Mechanics_, 3rd ed. (Cambridge, 2018), §4.2 — the Coulomb radial equation, the power-series solution and its truncation to the associated Laguerre polynomials, the Bohr spectrum $E_n = -13.6\,\text{eV}/n^2$, and the $n^2$ degeneracy. Publisher: https://doi.org/10.1017/9781316995433
[^shankar-hydrogen]: Shankar, _Principles of Quantum Mechanics_, 2nd ed. (Springer, 1994), Ch. 13 — the hydrogen atom, radial wavefunctions, and the degeneracy structure. Publisher: https://doi.org/10.1007/978-1-4757-0576-8
[^codata]: CODATA 2018 recommended values: Bohr radius $a_0 = 5.29177210903(80)\times 10^{-11}\ \text{m}$ and Rydberg energy $\mathrm{Ry} = 13.605693122994(26)\ \text{eV}$. NIST, https://physics.nist.gov/cuu/Constants/