the hydrogen atom
The hydrogen atom -- a single electron bound to a single proton -- is the crown jewel of introductory quantum mechanics: the one realistic atom whose Schrodinger equation can be solved exactly, in closed form. Its solution is the moment abstract postulates turn into the periodic table. Out of one equation come the discrete energy levels that Bohr had to assume, the shapes of the orbitals that chemists draw, and the quantum numbers that label every electron in every atom.
The electron sits in the attractive Coulomb potential V(r) = -e^2/(4 pi epsilon_0 r). Because the potential is spherically symmetric, the wavefunction separates into a radial part and an angular part: psi_{n l m}(r, theta, phi) = R_{n l}(r) Y_l^m(theta, phi), where the angular factors are the spherical harmonics and the radial factors R_{n l} involve associated Laguerre polynomials. The bound-state energies depend only on the principal quantum number n: E_n = -13.6 eV / n^2, or equivalently E_n = -(m e^4)/(2 (4 pi epsilon_0)^2 hbar^2 n^2). The characteristic size is the Bohr radius a_0 = 4 pi epsilon_0 hbar^2/(m e^2) is approximately 0.053 nm. Each level n allows l = 0, 1, ..., n-1 and, for each l, m from -l to +l, giving a degeneracy of n^2 (2 n^2 counting spin).
That the energy depends on n alone -- not on l -- is a special, 'accidental' degeneracy of the pure 1/r Coulomb potential, tied to a hidden symmetry (the conserved Runge-Lenz vector); it is not generic, and any deviation from a perfect 1/r potential lifts it. Real hydrogen shows exactly such deviations: relativistic and spin-orbit effects split the levels into fine structure, the proton's magnetic moment adds hyperfine structure (the 21 cm line that maps our galaxy), and QED shifts them further (the Lamb shift). The exact solution is thus not the end of the story but the clean baseline on which perturbation theory builds the real spectrum.
The photon emitted when hydrogen falls from n=3 to n=2 has energy E_3 - E_2 = -13.6/9 - (-13.6/4) = 1.89 eV, corresponding to a wavelength near 656 nm -- the red H-alpha line, the brightest line in the Balmer series and the glow that gives ionized-hydrogen nebulae their characteristic red colour.
The n=3 to n=2 transition gives the 656 nm red H-alpha line of the Balmer series.
The l-independence of E_n is special to the exact 1/r potential (an 'accidental' degeneracy from a hidden symmetry), not a general feature of atoms; in every multi-electron atom, screening already makes energy depend on l. The exact solution also ignores fine structure, hyperfine structure, and the Lamb shift, which real hydrogen exhibits.