The Hydrogen Atom & Atomic Spectra
Bound states of the Coulomb problem and the discrete spectrum of one-electron atoms.
The non-relativistic Schrödinger equation for an electron in the Coulomb field of a proton separates in spherical coordinates. Solutions are labeled by three quantum numbers $(n, \ell, m)$ with $n \geq 1$, $0 \leq \ell \leq n-1$, $-\ell \leq m \leq \ell$. Energies depend only on $n$:
$$E_n = -\frac{13.6 \text{ eV}}{n^2}.$$Wavefunctions $\psi_{n\ell m} = R_{n\ell}(r)\, Y_\ell^m(\theta, \varphi)$ factor into a radial part (Laguerre polynomials × $e^{-r/na_0}$) and a spherical harmonic. The Bohr radius $a_0 = \hbar^2 / (m_e e^2) \approx 0.529$ Å sets the atomic length scale.
Transitions emit photons at frequencies given by the Rydberg formula:
$$\frac{1}{\lambda} = R \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \quad R \approx 1.097 \times 10^7 \text{ m}^{-1}.$$Series: Lyman ($n_1=1$, UV), Balmer ($n_1=2$, visible), Paschen ($n_1=3$, IR). Hyperfine, fine, and Lamb shifts split lines further — windows into relativistic and QED corrections.