Electronic structure of atoms: hydrogen spectrum (Rydberg formula), fine and hyperfine structure, selection rules, Zeeman and Stark effects, Lamb shift. Bridges quantum mechanics to chemistry at Layer 2.
atomic-physics
Rydberg formula
Spectral lines of hydrogen-like atoms: 1/λ = R(1/n₁² − 1/n₂²). Predicted by Bohr model, recovered exactly from Schrödinger eq. for hydrogen.
Fine structure
Splitting of atomic spectral lines by relativistic corrections and spin-orbit coupling. Characterized by the fine-structure constant α ≈…
Selection rules (atomic transitions)
Constraints on allowed electromagnetic transitions from symmetry of the transition matrix element: Δℓ = ±1 (electric dipole), Δm = 0, ±1,…
Zeeman effect
Splitting of atomic spectral lines in an external magnetic field. Anomalous Zeeman effect reveals electron spin.
Stark effect
Splitting and shifting of atomic spectral lines in an external electric field — the electric-field analogue of the Zeeman effect.
Lamb shift
Small splitting between 2S₁/₂ and 2P₁/₂ levels of hydrogen (≈ 1058 MHz), predicted by QED radiative corrections. Historical cornerstone of…
Hyperfine structure
Splitting of atomic energy levels by the interaction of the electron's magnetic moment with the nuclear magnetic moment; ~μeV for hydrogen…
Bohr model of hydrogen
Pre-QM semi-classical atom with quantised angular momentum L = nℏ; predicts E_n = −13.6 eV/n² and the Rydberg spectrum. Superseded by…
Exchange interaction
Effective spin-dependent Coulomb interaction between identical fermions arising from wavefunction antisymmetrisation; drives Hund's rules…
Laser cooling & magneto-optical trap
Doppler cooling + polarization-gradient sub-Doppler. MOT combines confining B + 3D beams. Cohen-Tannoudji/Chu/Phillips Nobel 1997. Enables…
Optical lattice clocks
Sr/Yb atoms in magic-wavelength lattice; Q = f/Δf > 10¹⁸. Redefine of second pending. Probes of fundamental constants, gravitational…
Rydberg atoms & quantum gates
Highly-excited (n>40) atoms with km-range interactions via dipole blockade. Neutral-atom quantum computers (QuEra, Pasqal). Rydberg-dressed…
Bose-Einstein condensate
Macroscopic occupation of ground state below Tc (nλdB³>2.612). Cornell/Wieman/Ketterle 1995 Rb/Na BECs; Nobel 2001. Probe of superfluidity,…
Atomic parity violation
Weak-force-induced electric-dipole transitions in heavy atoms (Cs, Fr, Ra+). Measures Weinberg angle at low Q². APV + EDM searches…
Quantum defect theory
Nonhydrogenic energies E = -R/(n-δ_ℓ)² with ℓ-dependent defect δ capturing core penetration; Seaton's multichannel extension.
Fine structure (Dirac prediction)
Dirac's 1928 derivation of the hydrogen fine-structure splitting from the relativistic Dirac equation: energy levels depend on principal n…
Hyperfine structure shift
A I·J interaction from nuclear spin-electron coupling; 21-cm H line (1420 MHz) from F=1→F=0; Cs clock uses 9.192 GHz transition.
Cesium atomic clock (SI second)
SI second defined as 9 192 631 770 cycles of ¹³³Cs hyperfine transition; fountain clocks achieve δf/f ~ 10⁻¹⁶.
Doppler laser cooling
Red-detuned beams preferentially scatter from counter-propagating atoms; T_D = ℏΓ/(2k_B) limit; basis of MOT.
Sub-Doppler (Sisyphus) cooling
Polarization gradients + Stark shifts give T < T_D, limited only by recoil kT ~ ℏ²k²/m; 1997 Nobel (Cohen-Tannoudji, Phillips, Chu).
Magneto-optical trap (MOT)
Quadrupole B-field + σ⁺/σ⁻ red-detuned beams give position-dependent force; traps 10⁶-10¹⁰ neutral atoms at μK.
BEC experimental realization
Evaporative cooling of ⁸⁷Rb, Na vapors below T_c ~ 100 nK; Cornell, Wieman, Ketterle 1995 (2001 Nobel). Vortex lattices, solitons,…
Degenerate Fermi gases (⁶Li, ⁴⁰K)
Sub-Fermi-temperature ultracold Fermions; BCS-BEC crossover via Feshbach resonance; unitary regime with universal constants.
Feshbach resonance
Magnetically tunable scattering length a(B) = a_bg(1 - Δ/(B-B₀)); allows interactions from strong repulsive to strong attractive.
Single-atom optical tweezers
Tightly focused dipole trap holds one atom; programmable arrays up to ~10³ atoms for quantum simulation, Rydberg-blockade gates.
Quadratic Stark & dipole shifts
ΔE = -½α E² for nondegenerate levels; gives dynamical AC polarizability; magic wavelengths where differential shift vanishes.
Autoionization & Fano profiles
Discrete state coupled to continuum yields asymmetric lineshape σ = σ₀(q+ε)²/(1+ε²); Fano-Beutler resonances.
Electron EDM search
Precession of polar molecules (ThO, HfF⁺) in strong E-fields; d_e < 4×10⁻³⁰ e·cm (ACME 2018); constrains CP-violating BSM physics.
Atomic parity violation
Weak Z⁰-exchange admixes opposite-parity states; Cs PNC gives weak charge Q_W(Cs); tests of SM Weinberg angle at low Q².
Autler–Townes (AC Stark) splitting
Strong resonant drive on one transition dresses states, splitting probe transition by Ω_R; basis of EIT, STIRAP.
Electromagnetically induced transparency
Destructive quantum interference in 3-level Λ system removes absorption at line center; slow light (v_g < m/s); storage of photons.
Atom interferometry
Mach–Zehnder for matter waves via π/2-π-π/2 laser pulses; precision measurements of g, G, ℏ/m, GR tests.
Helium variational ground state (Hartree–Fock / atomic-physics)
Atomic-physics application of L0 Newton's method and Gaussian quadrature. The simplest variational treatment of the helium ground state…
Slater determinant for multi-electron atoms (atomic-physics)
Atomic-physics application of L0 LU factorisation and Gaussian elimination. The antisymmetric N-electron wavefunction is a Slater…