The Hydrogen Atom

The hydrogen atom is the crowning exactly-solvable problem of quantum mechanics: a single electron in the Coulomb potential of a proton, whose bound-state spectrum reproduces the observed atomic line series and fixes the whole qualitative shell structure of the periodic table. It combines the angular machinery (already solved for any central potential) with a Coulomb-specific radial equation, and it exhibits an "accidental" degeneracy that signals a hidden symmetry.

We treat the idealized non-relativistic problem: an electron of mass $m$ (strictly, the reduced mass) in

$$V(r)=-\frac{e^2}{4\pi {\epsilon }_{0}r}\equiv -\frac{\kappa }{r}.$$

Fine structure, the Lamb shift, and hyperfine splitting are perturbative corrections on top of this.

1. Separation of Variables

Because $V$ is central, the wavefunction separates:

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

and the angular part is already solved by the spherical harmonics. Substituting the ${\hat{L}}^2$-eigenvalue ${\hslash }^2\ell (\ell +1)$, the radial equation for $u(r)\equiv rR(r)$ is a 1D Schrödinger equation with an effective potential:

$$-\frac{{\hslash }^2}{2m}\frac{d^{2}u}{d{r}^2}+\underset{V_{\text{eff}}(r)}{\underbrace{[-\frac{\kappa }{r}+\frac{{\hslash }^2\ell (\ell +1)}{2m{r}^2}]}}u=Eu.$$

The second term is the repulsive centrifugal barrier; it pushes higher-$\ell$ states away from the origin.

2. The Bound-State Spectrum

Requiring $u\to 0$ at both $r=0$ and $r\to \infty$ (normalizability) quantizes the bound-state energies. The solutions are associated Laguerre polynomials times a decaying exponential, and the energies depend only on the principal quantum number $n$:

$$\boxed{E_n=-\frac{m{\kappa }^2}{2{\hslash }^2}\frac{1}{n^2}=-\frac{13.6\text{eV}}{n^2},n=1,2,3,\dots }$$

The characteristic length scale is the Bohr radius $a_0={\hslash }^2/(m\kappa )\approx 0.529\text{A˚}$. The quantum numbers are nested:

$$n=1,2,\dots ;\ell =0,1,\dots ,n-1;m=-\ell ,\dots ,+\ell .$$

Transitions between levels emit/absorb photons at $E_n-E_{n^{\prime }}$, reproducing the Lyman, Balmer, and Paschen series — the empirical anchor of early quantum theory.

3. Degeneracy and the Hidden $SO(4)$ Symmetry

Counting states of energy $E_n$:

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

The $(2\ell +1)$-fold $m$-degeneracy is expected from rotational symmetry (true for any central potential). But the additional degeneracy across different $\ell$ at the same $n$ is special to the $1/r$ potential and is called "accidental." It is not an accident: the Coulomb problem has an extra conserved vector, the quantum Laplace–Runge–Lenz vector

$$\hat{\mathbf{A}}=\frac{1}{2m}(\hat{\mathbf{p}}\times \hat{\mathbf{L}}-\hat{\mathbf{L}}\times \hat{\mathbf{p}})-\kappa \frac{\hat{\mathbf{r}}}{r},$$

which commutes with $\hat{H}$. Together $\hat{\mathbf{L}}$ and (rescaled) $\hat{\mathbf{A}}$ close into the algebra of $SO(4)$, the symmetry group of the bound-state problem, whose irreducible representations have exactly dimension $n^2$. The extra degeneracy is thus a consequence of a larger symmetry than rotations alone — and it is lifted by any perturbation (relativistic corrections, external fields) that breaks $SO(4)$ down to $SO(3)$.

4. Wavefunctions and Orbitals

The full stationary states are

$${\psi }_{n\ell m}(r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell }^m(\theta ,\varphi ),R_{n\ell }(r)\propto \left(\frac{r}{a_0}{)}^{\ell }{e}^{-r/n{a}_0}{L}_{n-\ell -1}^{2\ell +1}\right(\frac{2r}{n{a}_0}).$$

The ground state ${\psi }_{100}={\pi }^{-1/2}{a}_0^{-3/2}{e}^{-r/a_0}$ is a spherically symmetric ($s$) orbital; $R_{n\ell }$ has $n-\ell -1$ radial nodes. These are the familiar atomic orbitals, and — filled according to the Pauli principle with two spin states each — they generate the shell structure of the periodic table.

5. Beyond the Idealization

Real hydrogen departs from the pure Coulomb spectrum through effects that split the degeneracy: fine structure (relativistic kinetic correction + spin–orbit coupling, $\sim {\alpha }^2{E}_n$), the Lamb shift (a QED radiative correction), and hyperfine structure (electron–proton spin coupling, the 21 cm line). Each is computed by perturbation theory on this solution; the spin–orbit term requires addition of angular momenta to diagonalize.

See also

• Orbital angular momentum — the separated angular problem.
• Time-independent perturbation theory — fine structure, Zeeman, Stark.
• Identical particles — filling orbitals, the periodic table.
• Symmetries & conservation — degeneracy from symmetry.