Atomic Physics

Atomic physics asks how a nucleus and its electrons acquire discrete stationary states, how those states are labeled and split, and how radiation reveals or controls them. It is practical quantum mechanics: Hilbert-space structure becomes a level diagram; angular-momentum algebra becomes polarization dependence; perturbations become frequency shifts; and transition matrix elements become intensities, lifetimes, and branching ratios.

Start from the hydrogenic prototype and ask which corrections matter for the observed spectrum. Electron interactions, nuclear properties, relativity, radiation, and applied fields each modify that starting point in a different way.

Helpful background. The Hydrogen Atom develops Coulomb wavefunctions; Angular Momentum explains the level labels; Perturbation Theory explains small shifts and mixing. Models and Scales in AMO Physics explains how the observable and required accuracy determine which interactions to retain.

What Atomic Physics Studies

An isolated atom is already a composite quantum system. A useful description must say which constituents and interactions are retained, what labels identify approximate eigenstates, and which observable is being predicted. Typical atomic-physics questions include:

• What are the bound and continuum energies of a chosen atom or ion?
• Which quantum numbers are exact, and which survive only approximately?
• How do electron spin, relativistic motion, nuclear spin, recoil, and radiative effects split a nominal level?
• How do static electric and magnetic fields shift or mix states?
• Which transitions couple to a field of specified frequency and polarization?
• What do a measured wavelength, line strength, lifetime, linewidth, or isotope shift imply about the Hamiltonian?
• When is a one-electron, central-field, configuration-interaction, relativistic, or few-level model adequate?

The word structure therefore has several resolutions. Gross structure separates configurations or principal shells. Fine structure resolves relativistic and spin-dependent corrections. Hyperfine structure resolves coupling to nuclear moments. QED corrections such as the Lamb shift resolve effects that do not arise from a bare nonrelativistic Schrödinger Hamiltonian. External fields add Zeeman, Stark, and light shifts on top of this intrinsic hierarchy.

Atomic physics also includes ions, highly excited Rydberg states, collisions, photoionization, and controlled atomic platforms. The present chapter begins with bound-state structure and spectroscopy because those ideas supply the labels and scales used throughout the volume.

The Atomic Hamiltonian Hierarchy

For a nucleus of charge $+Ze$ fixed at the origin and $N$ nonrelativistic electrons, the Coulomb Hamiltonian is

$$H_{\mathrm C} ={}\sum_{i=1}^{N} \left( \frac{\mathbf p_i^2}{2m_e} -\frac{Ze^2}{4\pi\epsilon_0 r_i} \right) +\sum_{i<j} \frac{e^2}{4\pi\epsilon_0 r_{ij}} .$$

The first line is a sum of hydrogen-like one-electron terms. The second line couples the electron coordinates and prevents exact separation for $N>1$. Antisymmetry of the total electronic state is an additional constraint, not an optional correction to this Hamiltonian.

A practical atomic Hamiltonian is better viewed as an ordered family,

$$H_{\mathrm{atom}} ={}H_{\mathrm C} +\delta H_{\mathrm{mass}} +\delta H_{\mathrm{rel}} +H_{\mathrm{hfs}} +H_{\mathrm{ext}} +\delta H_{\mathrm{rad}} +\cdots .$$

Here $\delta H_{\mathrm{mass}}$ contains reduced-mass, recoil, and mass-polarization effects; $\delta H_{\mathrm{rel}}$ contains relativistic kinetic, spin–orbit, Darwin, and related terms in a low-energy expansion; $H_{\mathrm{hfs}}$ couples electronic and nuclear moments; $H_{\mathrm{ext}}$ represents applied fields; and $\delta H_{\mathrm{rad}}$ summarizes QED radiative corrections. The partition is useful only when the retained order and conventions are stated.

Finite nuclear mass and structure

Even a one-electron atom is not literally an electron orbiting an immovable point. Separating center-of-mass and relative motion replaces $m_e$ by the reduced mass

$$\mu=\frac{m_eM}{m_e+M},$$

where $M$ is the nuclear mass. For several electrons, nuclear recoil also produces a mass-polarization term that couples electron momenta. Nuclear charge radius, magnetic dipole moment, electric quadrupole moment, and polarizability become relevant at progressively finer resolution. Isotope shifts combine mass-dependent and field-dependent contributions, so they can probe both electronic wavefunctions and nuclear structure.

One-electron and many-electron models

The exactly solvable nonrelativistic Coulomb model is the first rung, not a universal template. A useful model ladder is:

1. Hydrogenic one-electron ion: exact Coulomb eigenstates, followed by perturbative corrections when controlled.
2. Central-field atom: each electron moves in an effective spherical potential, producing orbital and shell labels.
3. Alkali-like atom: a closed-shell core plus one valence electron, often summarized by quantum defects and effective operators.
4. Independent-particle or mean-field atom: self-consistent orbitals provide a reference configuration.
5. Configuration-interaction or correlated atom: superpositions of configurations recover mixing and electron correlation.
6. Relativistic atomic structure: Dirac-based orbitals and relativistic many-body methods are used when $Z\alpha$ or the target precision requires them.
7. Few-level effective model: selected exact or approximate levels are retained for driving, cooling, clocks, or quantum control.

Moving down this list does not always mean “more accurate.” A few-level model can be the correct effective theory for a narrow experimental bandwidth, provided off-resonant states, decay channels, and field-induced mixing are bounded.

Atomic Units and Natural Scales

Atomic units expose the size of Coulombic quantities by setting

$$\hbar=m_e=e=4\pi\epsilon_0=1.$$

The corresponding length and energy units are the Bohr radius and Hartree energy,

$$\begin{aligned} a_0&=\frac{4\pi\epsilon_0\hbar^2}{m_e e^2},\\ E_{\mathrm h}&=\frac{e^2}{4\pi\epsilon_0a_0},\\ E_{\mathrm h}&=\alpha^2m_ec^2. \end{aligned}$$

The Rydberg energy is $E_{\mathrm h}/2$ in the infinite-nuclear-mass convention. Numerical work must state whether reduced-mass corrections are included and which CODATA adjustment supplies conversion factors. The dedicated Atomic Units reference gives the quick conversion table; Atomic Units and Scales derives the units and develops the distinction among energy, ordinary frequency, angular frequency, and spectroscopic wavenumber.

For a hydrogenic ion, a compact parametric hierarchy is

$$\begin{aligned} E_{\mathrm{gross}}&\sim m_ec^2(Z\alpha)^2,\\ E_{\mathrm{fine}}&\sim m_ec^2(Z\alpha)^4,\\ E_{\mathrm{rad}}&\sim \frac{\alpha}{\pi}m_ec^2(Z\alpha)^4\mathcal L,\\ E_{\mathrm{recoil}}&\sim \frac{m_e}{M}E_{\mathrm{electronic}}, \end{aligned}$$

The dimensionless factor $\mathcal L$ represents the logarithms that occur in the leading radiative expansion.

These are organizing estimates, not universal error bars. Coefficients, selection rules, cancellations, nuclear moments, and near-degeneracies can alter the observed ordering. In many-electron atoms, screening and configuration mixing also replace simple powers of $Z$ by state-dependent behavior.

External-field scales

Two especially useful comparisons are

$$\Delta E_B\sim\mu_BB, \qquad \Delta E_E\sim eEa_0,$$

with $\mu_B=e\hbar/(2m_e)$. Compare these energies with the smallest intrinsic splitting whose quantum numbers are being used. A field can be weak relative to the gross electronic structure yet strong relative to hyperfine coupling. “Weak field” is therefore a statement about a specified level manifold, not about the laboratory field alone.

Central Potentials and Quantum Numbers

For a spin-independent central potential $V(r)$,

$$H=\frac{\mathbf p^2}{2m}+V(r)$$

commutes with $L^2$ and $L_z$. Spatial eigenstates may be labeled by a radial index, $\ell$, and $m_\ell$, with parity $(-1)^\ell$. In the Coulomb problem the principal quantum number $n$ has special dynamical significance and the nonrelativistic energy depends only on $n$. A generic central potential does not possess that enlarged degeneracy.

Electron spin adds $s=1/2$ and $m_s$. When spin–orbit coupling is relevant while rotational invariance remains, $\mathbf J=\mathbf L+\mathbf S$ is the conserved electronic angular momentum and states are more naturally labeled by $j$ and $m_j$. The Angular Momentum Algebra and Addition of Angular Momentum explain how to construct these labels.

Exact labels, approximate labels, and basis labels

Three kinds of label should not be conflated:

• An exact quantum number labels an eigenspace of an operator commuting with the full Hamiltonian under discussion.
• An approximate quantum number remains useful because symmetry breaking or mixing is weak relative to relevant separations.
• A basis label identifies a component used to expand a state, even when the corresponding operator is not conserved.

For example, $m_J$ is not generally conserved in an electric field whose direction differs from the chosen quantization axis. A configuration label may still identify the dominant component of a correlated eigenvector without being an exact observable. Spectroscopic databases often report leading configurations and term assignments with explicit uncertainty or mixing information; those labels should not be promoted to exact identities.

Quantum defects

Outside a closed-shell core, an alkali valence electron sees an approximately Coulombic potential at large radius but a screened, non-Coulombic potential near the core. A common spectral representation is

$$E_{n\ell j} \simeq E_{\mathrm{ion}} -\frac{hcR_M}{(n-\delta_{\ell j})^2},$$

where $R_M$ is a reduced-mass Rydberg constant in inverse-length units and $\delta_{\ell j}$ is a dimensionless quantum defect. Treating the defect as constant is a useful first approximation; accurate series fits allow energy dependence. Low-$\ell$ orbitals penetrate the core more strongly and usually have larger defects. The formula is an empirical effective description of a Rydberg series, not a claim that the core has disappeared.

Angular Momentum and Term Labels

For several electrons, define total orbital and spin angular momenta

$$\begin{aligned} \mathbf L&=\sum_i\mathbf l_i,\\ \mathbf S&=\sum_i\mathbf s_i,\\ \mathbf J&=\mathbf L+\mathbf S. \end{aligned}$$

In the Russell–Saunders, or $LS$, coupling regime, an atomic term is conventionally written

$${}^{2S+1}L_J,$$

where $2S+1$ is the spin multiplicity, $L=0,1,2,3,\ldots$ is denoted by $S,P,D,F,\ldots$, and $J$ ranges from $|L-S|$ to $L+S$. Parity is an independent label; odd parity is often marked with a superscript degree sign in spectroscopic notation.

The symbol is a compressed set of angular-momentum labels, not a complete wavefunction. It omits the dominant electron configuration unless that is written separately, and it does not specify radial correlations. Moreover, $LS$ coupling becomes less accurate when one-electron spin–orbit interactions compete strongly with residual electrostatic couplings. In a $jj$ description, individual $\mathbf j_i=\mathbf l_i+\mathbf s_i$ are coupled instead. Intermediate coupling is common in real spectra, so calculated eigenvectors and measured transition patterns may be needed to justify an assignment.

Perturbations and Splittings

Atomic level structure is organized by comparing each interaction with the splittings already present. Diagonalizing all named terms at once can obscure which labels and approximations are physically controlled.

Structure	Dominant origin	Useful labels	When the description needs refinement
gross structure	Coulomb binding and electron correlation	dominant configuration and approximate $L,S$	strong configuration mixing
fine structure	relativistic motion and spin-dependent interactions	$J$, parity	high-$Z$ or near-degenerate mixing requires a relativistic treatment
hyperfine structure	nuclear magnetic dipole and electric quadrupole moments	$F,m_F$ from coupling $I$ and $J$	Zeeman energy becomes comparable to hyperfine intervals
Zeeman structure	coupling to an applied magnetic field	weak-field $F,m_F$ or stronger-field uncoupled labels	avoided crossings and nonlinear shifts appear
Stark structure	electric-dipole coupling and polarizability	field-dressed labels, often $m$ about the field axis	opposite-parity levels mix appreciably
radiative structure	electron self-energy, vacuum polarization, and related QED effects	labels inherited from the reference Hamiltonian	required precision exceeds the included QED and nuclear terms

Fine structure

In a low-energy expansion for a one-electron Coulomb problem, fine structure includes the relativistic kinetic correction, spin–orbit coupling, and Darwin term. Treating these pieces consistently reproduces the expansion of the Dirac spectrum at the corresponding order. In multi-electron atoms, spin–other-orbit, spin–spin, and relativistic two-body terms can also matter. Spin–Orbit Coupling explains the angular-momentum structure, and Fine Structure applies it to atomic spectra.

Hyperfine structure

Here use dimensionless angular-momentum operators $\mathbf I$ and $\mathbf J$, whose physical angular momenta are $\hbar\mathbf I$ and $\hbar\mathbf J$. The leading magnetic-dipole model is

$$H_{\mathrm{hfs}}=A\,\mathbf I\cdot\mathbf J, \qquad \mathbf F=\mathbf I+\mathbf J.$$

The coefficient $A$ has energy units in this convention. Within a fixed electronic $J$ manifold, the allowed total quantum numbers are $F=|I-J|,|I-J|+1,\ldots,I+J$. The shift of a level labeled by $F$ is

$$\begin{aligned} \Delta E_F&=\frac{A}{2}K, \\ K&=F(F+1)-I(I+1) -J(J+1). \end{aligned}$$

For $I\geq1$ and $J\geq1$, an electric-quadrupole term may contribute. The constants $A$ and $B$ depend on both nuclear moments and the electronic field at the nucleus; they are not universal constants of an isotope independent of electronic state.

Hyperfine Structure derives the multiplet energies and adds the electric-quadrupole term, with hydrogen and alkali examples.

Zeeman structure

In a weak magnetic field, an effective level with angular momentum $F$ often shifts to first order as

$$\Delta E_Z\simeq g_F\mu_Bm_FB.$$

This first-order formula uses approximately unmixed zero-field $F$ states. At nonzero field, $F$ is generally not an exact conserved quantum number. When the electronic and nuclear Zeeman energies compete with or exceed the hyperfine interaction, $F$ ceases to be a good label and the Hamiltonian must be diagonalized in a more appropriate basis. Zeeman Effect in Atoms explains the Breit–Rabi crossover, polarization patterns, and spectroscopic interpretation. The general perturbative method is developed in Zeeman Effect Example.

Stark structure

For a static electric field $\mathbf E$, the leading interaction is

$$H_E=-\mathbf d\cdot\mathbf E.$$

A nondegenerate state of definite parity has no permanent electric-dipole expectation value, so its leading DC shift is usually quadratic. Degenerate opposite-parity subspaces can instead show a linear Stark effect after degenerate perturbation theory. Frequency-dependent fields produce dynamic polarizabilities and AC Stark shifts. Stark Effect in Atoms develops response tensors, alkali examples, and trapping; the Stark Effect Example gives the perturbation calculation.

Lamb shift and radiative corrections

The Lamb shift is a spectroscopic signature of radiative and recoil effects beyond the simple Dirac–Coulomb picture. Phrases such as “vacuum fluctuations cause the Lamb shift” are useful only as qualitative orientation: precision theory organizes gauge-invariant QED contributions, recoil, nuclear size, and other effects order by order. This volume provides the atomic interpretation and experimental role; a full derivation belongs to quantum electrodynamics.

What an atomic spectrum tells you

Spectroscopy compares level differences with radiation frequencies. For two stationary levels,

$$h\nu=E_f-E_i,$$

but a measured spectral feature contains more than this difference. Its position, polarization, intensity, width, shape, and response to fields can test different parts of the model.

In the electric-dipole approximation, the transition amplitude contains

$$\mathcal M_{fi}^{(E1)} \propto \langle f|\mathbf d\cdot\boldsymbol\epsilon|i\rangle,$$

where $\boldsymbol\epsilon$ is the field polarization. Angular-momentum and parity constraints can force this matrix element to vanish within an idealized model. For one-electron orbital labels, the familiar electric-dipole rules include $\Delta\ell=\pm1$ and a parity change. For total angular momentum, $\Delta J=0,\pm1$ with $J=0\not\leftrightarrow J'=0$, subject to the actual coupling scheme and additional quantum numbers. Polarization resolves changes in magnetic projection.

“Forbidden” means that a specified leading matrix element vanishes under stated approximations. Magnetic-dipole, electric-quadrupole, relativistic, hyperfine-induced, or field-mixed amplitudes may remain. Weak transitions are scientifically valuable because their long lifetimes and sensitivity to small perturbations support clocks and precision tests.

Atomic Selection Rules is the atom-specific working guide for electronic and hyperfine labels, polarization, higher multipoles, metastability, and intensity borrowing. The canonical symmetry derivations are in Dipole Transitions and Atomic Spectra Applications. Time-dependent amplitudes and rates are developed in Selection Rules and Transition Rates.

What different observables constrain

Observable	Primary information	Frequent confounders
line center	level difference and shifts	calibration, Doppler shift, pressure shift, field shifts
relative intensity	populations and transition strengths	detector response, optical pumping, saturation
polarization	angular-momentum pathways and geometry	imperfect polarization, unresolved sublevels
natural linewidth	radiative lifetime and open channels	power, transit-time, collision, and Doppler broadening
isotope shift	recoil and nuclear-size dependence	unresolved hyperfine components and abundance weighting
field dependence	moments, polarizabilities, and mixing	field gradients, tensor shifts, avoided crossings

An observed wavelength is not automatically the unperturbed transition frequency. A trustworthy comparison records the isotope, ionization stage, environment, field conditions, line-shape model, calibration, and uncertainty budget.

Finding reliable atomic data

The NIST Atomic Spectra Database is an evaluated starting point for atomic energy levels, wavelengths, classifications, and transition probabilities. Its current interface distinguishes observed wavelengths from Ritz wavelengths inferred from optimized energy levels, and it links entries to source bibliographies. Database values should be cited with the database version and access date, then traced to primary sources when a claim depends on experimental method or uncertainty interpretation.

Choose the next lesson

Page	Question
Atomic Units and Scales	Which natural units and parametric hierarchies organize atomic calculations?
Hydrogen as Atomic Prototype	How should the exact Coulomb result be interpreted as atomic structure and spectroscopy?
Central-Field Approximation	How does a many-electron problem become an effective orbital model, and where does it fail?
Alkali Atoms	Why does a closed-shell core plus one valence electron support useful effective models?
Atomic Orbitals Revisited	What is an orbital, what does it visualize, and what is basis-dependent?
Fine Structure	How do relativistic and spin-dependent terms split electronic levels?
Lamb Shift Overview	Which atomic discrepancy opens the bridge to QED?
Hyperfine Structure	How do nuclear moments couple to electronic angular momentum?
Zeeman Effect in Atoms	How do magnetic fields shift, mix, and relabel atomic states?
Stark Effect in Atoms	How do electric fields generate linear, quadratic, and dynamic shifts?
Atomic Selection Rules	Which transition amplitudes vanish, and under what approximations?
Atomic Term Symbols	How are configuration, multiplicity, angular momentum, and parity encoded?
Rydberg Atoms Basics	How do large-$n$ scaling laws create strongly interacting, controllable atoms?

For a deeper treatment of antisymmetry, configurations, self-consistent fields, and electron correlation, continue to Multi-Electron Atoms. For line strengths and profiles, use Spectroscopy.

Common Mistakes

Equating line position with an isolated-atom interval

Doppler, pressure, recoil, Zeeman, Stark, AC Stark, and instrumental shifts can move or distort a feature. Extracting an unperturbed interval requires a measurement model and uncertainty budget.

Mixing energy, frequency, angular frequency, and wavenumber

The same interval may be quoted as $E$, $\nu=E/h$, $\omega=E/\hbar$, or $\widetilde\nu=E/(hc)$. Factors of $2\pi$ and $c$ are physical conversion factors, not typographical conventions.

Adding corrections without checking consistency

Relativistic, recoil, radiative, and nuclear terms can overlap when imported from different effective Hamiltonians. A correction list is not reliable unless its order, reference Hamiltonian, conventions, and possible double counting are clear.

Exercises

1. Decode a term symbol

Interpret ${}^3P_2$. Give $L$, $S$, $J$, spin multiplicity, the number of $M_J$ substates, and what cannot be inferred about parity from the term symbol alone.

Solution

The superscript gives $2S+1=3$, hence $S=1$. The letter $P$ means $L=1$, and the subscript gives $J=2$. The level has $2J+1=5$ magnetic substates with $M_J=-2,-1,0,1,2$ in zero field.

The symbol as written does not specify the electron configuration or parity. For a many-electron configuration, parity is the product $(-1)^{\sum_i\ell_i}$ and must be supplied separately, often by an odd-parity marker in spectroscopic notation. It is therefore unsafe to infer odd parity merely from the letter $P$ in a many-electron term.

2. Read a quantum defect

Two Rydberg series converge to the same ionization limit. One is nearly hydrogenic with $\delta=0$; the other has $\delta=3.0$. At $n=10$, compare the magnitudes of their binding energies in the simple quantum-defect formula.

Solution

The magnitude of the binding relative to the series limit is proportional to

$$|E_n-E_{\mathrm{ion}}| \propto\frac{1}{(n-\delta)^2}.$$

For the nearly hydrogenic series the denominator is $10^2=100$. For $\delta=3.0$ it is $7^2=49$. The ratio is therefore

$$\frac{|E|_{\delta=3}}{|E|_{\delta=0}} =\frac{100}{49}\simeq2.04.$$

At the same nominal $n$, the penetrating state is about twice as strongly bound in this model. Comparing states by effective principal quantum number $n^*=n-\delta$ is often more informative than comparing by $n$ alone.

3. Choose labels through a magnetic crossover

An atom has hyperfine interaction $A\mathbf I\cdot\mathbf J$ within a fixed $J$ manifold, with the dimensionless operators defined above, and is placed in a magnetic field along $z$. Explain how you would decide whether $|F,m_F\rangle$ or $|m_I,m_J\rangle$ is the better basis for interpretation. What remains conserved throughout the crossover if no transverse perturbation is present?

Solution

Compare a characteristic Zeeman energy, such as $|g_J|\mu_B|B|$, with the actual hyperfine interval (which contains both $A$ and angular factors). When the Zeeman energy is much smaller, $F$ is an approximate label and $m_F$ is exact under axial symmetry. When it is much larger, but still small compared with separations to other electronic $J$ manifolds, $m_J,m_I$ become useful approximate labels.

In the intermediate regime neither limiting label is exact. Diagonalize

$$H=A\mathbf I\cdot\mathbf J +g_J\mu_BBJ_z -g_I\mu_NBI_z$$

in blocks of fixed $m_F=m_I+m_J$. Here $g_I$ is defined using the nuclear magneton, with nuclear moment $g_I\mu_N\mathbf I$; the electronic moment has the opposite sign. Axial symmetry preserves the total projection $m_F$ even while $F$ is mixed. Tracking eigenvectors continuously with $B$ gives an unambiguous connection between the two limits and exposes avoided crossings between states of the same conserved labels.

References

• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson (2003).
• C. J. Foot, Atomic Physics, Oxford University Press (2005).
• H. Friedrich, Theoretical Atomic Physics, 4th ed., Springer (2017).
• I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, Springer (2007).
• W. R. Johnson, Atomic Structure Theory: Lectures on Atomic Physics, Springer (2007).
• 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.
• I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer (1992).