Atomic, Molecular, and Optical Physics

Atomic, molecular, and optical (AMO) physics connects quantum states to things we can prepare, drive, and measure: atomic spectra, molecular bonds, photon counts, interference fringes, and precise frequency comparisons. This volume develops those connections from microscopic structure through light–matter interaction to laboratory control.

Required background. Core Formalism supplies states and probabilities; the hydrogen atom supplies a first atomic model; angular momentum supplies level labels and selection rules; perturbation theory explains small corrections. You can revisit these tools as each chapter needs them.

Start with a physical question

If you are new to AMO physics, begin with Models and Scales in AMO Physics, then read Atomic Physics and Spectroscopy. This gives a concrete sequence: calculate levels, find which transitions couple to light, and predict the recorded signal.

Other entry points depend on your interests:

• Atoms and precision spectroscopy: follow atomic structure, multi-electron atoms, and precision measurement. Learn how electron correlation, nuclear properties, applied fields, and radiation shift an observed line.
• Molecules and chemistry: start with Molecular Quantum Mechanics. Use the oscillator, rotor, and Born–Oppenheimer approximation to connect electronic bonding with nuclear motion.
• Light and quantum control: start with Light–Matter Interaction, then Quantum Optics. Add Fock Space for quantized modes and Open Quantum Systems for decay and photodetection.
• Numerical calculations: use Computational AMO and Quantum Chemistry alongside the physical chapter. Basis convergence, transition matrix elements, and time propagation need different checks.

The AMO Physics Roadmap and Quantum Chemistry Roadmap give longer study sequences.

From an energy level to a spectral line

An energy difference determines a candidate transition frequency,

$$E_f-E_i=h\nu_{fi}=\hbar\omega_{fi},\qquad \omega_{fi}=2\pi\nu_{fi}.$$

It does not determine whether a line is visible. The initial state must be populated, the probing interaction must connect the states, and the detector must be sensitive to the resulting signal. In the electric-dipole approximation, the relevant amplitude contains

$$\langle f|\mathbf d\cdot\boldsymbol\epsilon|i\rangle,$$

where $\mathbf d$ is the dipole operator and $\boldsymbol\epsilon$ describes the light’s polarization. A vanishing amplitude can suppress a line even when the energy difference is known accurately.

Hydrogen gives a useful example. In the nonrelativistic Coulomb model with an infinitely heavy point nucleus,

$$E_n=-\frac{E_{\mathrm h}}{2n^2},\qquad E_2-E_1=\frac{3}{8}E_{\mathrm h}.$$

Using $E_{\mathrm h}\simeq27.2\,\mathrm{eV}$ gives about $10.2\,\mathrm{eV}$, or a vacuum wavelength near $121.6\,\mathrm{nm}$. But the degenerate $2s$ and $2p$ states behave differently: a single electric-dipole photon can connect $1s$ to $2p$, while parity makes the $1s$–$2s$ dipole matrix element zero. Fine structure, recoil, and radiative corrections refine the energies; populations, polarization, and broadening refine the observed spectrum.

This distinction between level position, transition strength, and measurement signal recurs throughout the volume. Foot and Bransden–Joachain develop it for atoms; Cohen-Tannoudji, Dupont-Roc, and Grynberg develop the interaction with radiation.

Chapters and what you will learn

Chapter	Main question
Atomic Physics	How do atomic levels, quantum numbers, and external-field shifts arise?
Multi-Electron Atoms	How do antisymmetry, shells, exchange, and correlation organize real atoms?
Molecular Quantum Mechanics	How do electrons bind nuclei, and how do molecules rotate and vibrate?
Spectroscopy	What do positions, intensities, widths, and time-dependent signals reveal?
Light–Matter Interaction	How do fields drive transitions, create coherence, and cause shifts or decay?
Quantum Optics	Which properties of light require a quantum description?
Lasers	How do gain, loss, saturation, and noise determine laser operation?
AMO Platforms and Quantum Control	How are atoms, ions, molecules, and photons cooled, trapped, prepared, and measured?
Precision Measurement and Metrology	How do quantum systems become clocks, interferometers, and tests of physical laws?
Computational AMO and Quantum Chemistry	Which numerical representation and convergence checks fit a calculation?
Frontiers and Open Problems	Which physical limitations motivate current research, and where can you follow the open questions?
Reference and Data	Where can you check units, constants, notation, and evaluated spectroscopic data?

Choosing the right level of approximation

The most detailed Hamiltonian is not always the most useful starting point. Begin with the scales relevant to your observable. An atomic gross spectrum can often be understood with a nonrelativistic Coulomb model. A hyperfine transition needs nuclear spin. A clock measurement may require shifts much smaller than a visible linewidth. A short or intense pulse may couple levels omitted from a two-level model.

Three comparisons recur:

• Size versus wavelength: $ka\ll1$ motivates the electric-dipole approximation for system size $a$ and wave number $k$. Higher multipoles can still dominate when the dipole matrix element vanishes.
• Coupling versus level separation: a far-detuned state can often be eliminated when its coupling and the pulse bandwidth are small compared with its detuning. Its virtual contribution can nevertheless shift the retained levels.
• Lifetime versus observation time: coherent Rabi dynamics is useful over times short compared with relevant relaxation and dephasing times. Longer observations need the corresponding open-system dynamics.

The discussion of models and scales develops these comparisons. Detailed lessons state the assumptions next to the formulas that use them.

Exercises

1. Test an electric-dipole transition. The $1s$ and $2s$ hydrogenic orbitals are even under parity, while $2p$ is odd. The dipole operator is odd. Explain why parity forbids $1s\to2s$ but permits $1s\to2p$ in this approximation.

Solution

Under inversion, the integrand of a dipole matrix element gains the product of the final parity, the operator parity, and the initial parity. For two even states that product is $-1$, so the integral over all space vanishes. For $1s$ and $2p$ it is $+1$, so parity does not force the result to vanish. Angular-momentum and polarization conditions must also be satisfied. The forbidden one-photon dipole transition does not exclude two-photon excitation of $2s$.

2. Convert a linewidth convention. A transition has an angular-frequency full width at half maximum of $2\pi\times 1\,\mathrm{MHz}$. What is the width in ordinary frequency and in energy?

Solution

The frequency width is $1\,\mathrm{MHz}$. Its energy width is $h\times1\,\mathrm{MHz}$, equivalently $\hbar\times2\pi\times1\,\mathrm{MHz}$. These are the same interval in different units; multiplying an ordinary frequency by $\hbar$ would miss a factor of $2\pi$.

References

• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson (2003).
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom–Photon Interactions: Basic Processes and Applications, Wiley (1992).
• C. J. Foot, Atomic Physics, Oxford University Press (2005).
• A. Kramida, Yu. Ralchenko, J. Reader, and the NIST ASD Team, NIST Atomic Spectra Database, version 5.12, NIST Standard Reference Database 78 (2024), doi:10.18434/T4W30F, accessed 2026-10-13.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (1997).

For textbooks by level and specialty, continue to the AMO Bibliography and Reading Guide. For conversion factors, use a consistent adjustment of the NIST/CODATA constants and the Constants and Conversions reference.