# Atomic Physics

An atom is a bound quantum system whose electrons occupy sharply defined
energy levels. Almost everything atomic physics measures — colors of light,
tiny splittings, the ticking of a clock — is a difference between two of
those levels, which is why the field is the most precisely tested corner of
quantum mechanics.


_Figure 001 — The hydrogenic ladder: bound levels crowd toward the ionization limit, and
an electron cascades down, emitting a photon on each jump.
_

The starting point is the one-electron atom. Solving the Coulomb problem
gives the levels **En = −R/n²**, and every downward
transition radiates a photon of energy _Ei − Ef_
— the discrete lines Bohr first explained.


_Figure 002 — An emission series: lines pile up geometrically as they converge on the
series limit.
_

Group those transitions by their lower level and you get the named series —
Lyman, Balmer, Paschen — each a fingerprint of hydrogen written in light.
The pattern of a spectrum is a direct readout of the level structure behind
it.


The gross structure is only the first approximation. Electron spin coupled
to its orbital motion, together with relativistic corrections, splits each
level into a **fine structure** — the small doublets that
turned single lines into pairs and forced quantum theory to grow.


_Figure 003 — Spin–orbit coupling lifts the degeneracy of a level into a close doublet.
_

Finer still, the nucleus has its own magnetic moment; its coupling to the
electron produces **hyperfine structure**, the origin of the
21-cm line and the definition of the second. Placing the atom in an external
field splits the levels further — the Zeeman and Stark effects.


Radiative transitions have rates and rules. Selection rules say which jumps
are allowed, line shapes encode the atom's environment, and stimulated
emission — one photon provoking an identical twin — is the amplifying step
that makes a **laser** possible.


_Figure 004 — Stimulated emission: an incoming photon triggers a second, identical,
in-phase photon.
_

Beyond hydrogen, the Pauli principle stacks electrons into shells, and the
shapes of the orbitals they fill — spherical _s_, two-lobed
_p_, and beyond — set the structure of the periodic table and the
logic of chemistry.


_Figure 005 — Orbital shapes: the spherical s cloud and the two-lobed p orbital.
_

The modern frontier turns these transitions into tools. Laser cooling and
trapping bring atoms to a standstill, and driving a single narrow line
builds an **optical clock** — the most accurate measurement
humans have ever made.


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## Contents

### 1. Early Atomic Models and the Old Quantum Theory

1. [Atomic Spectra and Rutherford's Nucleus](/atomic-physics/early-models-and-old-quantum-theory/atomic-spectra-rutherford)
2. [The Bohr Model of Hydrogen](/atomic-physics/early-models-and-old-quantum-theory/bohr-model-hydrogen)
3. [X-Ray Spectra and the Franck-Hertz Experiment](/atomic-physics/early-models-and-old-quantum-theory/x-ray-spectra-franck-hertz)
4. [The Bohr-Sommerfeld Old Quantum Theory](/atomic-physics/early-models-and-old-quantum-theory/bohr-sommerfeld-old-quantum-theory)
5. [Limits of the Old Quantum Theory and the WKB Bridge](/atomic-physics/early-models-and-old-quantum-theory/old-quantum-theory-limits-wkb)

### 2. The Quantum Hydrogen Atom

1. [The Schrödinger Equation in Three Dimensions and Hydrogen](/atomic-physics/quantum-hydrogen-atom/schrodinger-3d-hydrogen)
2. [Hydrogen Wave Functions and Orbitals](/atomic-physics/quantum-hydrogen-atom/hydrogen-wave-functions)
3. [Solving the Radial Equation in Full](/atomic-physics/quantum-hydrogen-atom/radial-equation-in-full)
4. [Accidental Degeneracy and the Runge-Lenz Symmetry](/atomic-physics/quantum-hydrogen-atom/symmetry-degeneracy-runge-lenz)
5. [Expectation Values, the Virial Theorem, and Scaling](/atomic-physics/quantum-hydrogen-atom/expectation-values-virial)
6. [Quantum Defects and Alkali Spectra](/atomic-physics/quantum-hydrogen-atom/quantum-defects-alkali-spectra)
7. [Rydberg Atoms](/atomic-physics/quantum-hydrogen-atom/rydberg-atoms)

### 3. Fine Structure and the Dirac Atom

1. [The Relativistic Kinetic-Energy Correction](/atomic-physics/fine-structure-and-the-dirac-atom/relativistic-kinetic-correction)
2. [Spin-Orbit Coupling and Thomas Precession](/atomic-physics/fine-structure-and-the-dirac-atom/spin-orbit-thomas-precession)
3. [The Darwin Term and the Fine-Structure Formula](/atomic-physics/fine-structure-and-the-dirac-atom/darwin-term-fine-structure-formula)
4. [The Dirac Equation for Hydrogen](/atomic-physics/fine-structure-and-the-dirac-atom/dirac-equation-hydrogen)

### 4. QED Corrections and Hyperfine Structure

1. [The Lamb Shift and QED Radiative Corrections](/atomic-physics/qed-corrections-and-hyperfine-structure/lamb-shift-qed)
2. [Hyperfine Structure and the 21 cm Line](/atomic-physics/qed-corrections-and-hyperfine-structure/hyperfine-structure-21cm)
3. [Nuclear Size, Moments, and Isotope Shifts](/atomic-physics/qed-corrections-and-hyperfine-structure/nuclear-effects-isotope-shift)

### 5. Many-Electron Atoms

1. [The Periodic Table and Atomic Spectra](/atomic-physics/many-electron-atoms/periodic-table-atomic-spectra)
2. [The Central-Field Approximation and the Self-Consistent Field](/atomic-physics/many-electron-atoms/central-field-self-consistent)
3. [Exchange, Slater Determinants, and Hartree-Fock](/atomic-physics/many-electron-atoms/identical-particles-hartree-fock)
4. [Helium: the Prototype Two-Electron Atom](/atomic-physics/many-electron-atoms/helium-two-electron-atom)
5. [LS and jj Coupling; Term Symbols](/atomic-physics/many-electron-atoms/ls-jj-coupling-term-symbols)
6. [Hund's Rules and Ground-State Terms](/atomic-physics/many-electron-atoms/hund-rules-ground-terms)

### 6. Atoms in External Fields

1. [The Zeeman Effect](/atomic-physics/atoms-in-external-fields/zeeman-effect)
2. [The Paschen-Back and Intermediate-Field Regimes](/atomic-physics/atoms-in-external-fields/paschen-back-intermediate)
3. [The Stark Effect and Field Ionization](/atomic-physics/atoms-in-external-fields/stark-effect-polarizability)

### 7. Radiative Transitions and Spectral Lines

1. [Time-Dependent Perturbation Theory and the Golden Rule](/atomic-physics/radiative-transitions-and-line-shapes/time-dependent-perturbation-golden-rule)
2. [The Dipole Approximation and Einstein Coefficients](/atomic-physics/radiative-transitions-and-line-shapes/dipole-approximation-einstein-coefficients)
3. [Selection Rules and Forbidden Transitions](/atomic-physics/radiative-transitions-and-line-shapes/selection-rules-forbidden-transitions)
4. [Lifetimes, Line Widths, and Line Shapes](/atomic-physics/radiative-transitions-and-line-shapes/lifetimes-and-line-shapes)

### 8. Lasers and Spectroscopy

1. [Population Inversion, Gain, and the Laser](/atomic-physics/lasers-and-spectroscopy/laser-principles)
2. [Spectroscopic Techniques and Frequency Combs](/atomic-physics/lasers-and-spectroscopy/spectroscopy-techniques)
3. [Reading Real Spectra with the NIST Database](/atomic-physics/lasers-and-spectroscopy/line-catalog-nist-asd)

### 9. Modern Atomic Physics

1. [Laser Cooling and Optical Molasses](/atomic-physics/modern-atomic-physics/laser-cooling-doppler)
2. [Sub-Doppler Cooling and Atom Traps](/atomic-physics/modern-atomic-physics/sub-doppler-trapping)
3. [Bose-Einstein Condensation of Atomic Gases](/atomic-physics/modern-atomic-physics/bose-einstein-condensation)
4. [Optical Atomic Clocks and Precision Measurement](/atomic-physics/modern-atomic-physics/optical-clocks-precision)