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.

This chapter uses hydrogen as the prototype but does not repeat the exact Coulomb solution. The canonical derivation of hydrogenic wavefunctions and energies belongs to Hydrogen Atom, and the accidental Coulomb degeneracy belongs to Degeneracy of the Hydrogen Atom. Here the emphasis is the hierarchy that turns that prototype into a working description of real atoms.

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.

Why Atoms Are Tractable and Nontrivial

Atoms occupy a productive middle ground. Their microscopic Hamiltonians are compact enough to write down, rotational symmetry gives powerful exact constraints, and electromagnetic interactions are quantitatively understood. Yet all atoms beyond hydrogen contain electron–electron interactions, antisymmetry, correlation, multiple angular momenta, and several separated correction scales.

Feature	Why it helps	Why it remains difficult
Coulomb interaction	its form is known and hydrogen is exactly solvable	the many-electron Coulomb problem is not separable
Rotational and parity symmetry	states fall into angular-momentum and parity sectors	external fields and configuration mixing can reduce the useful symmetry
Small fine-structure constant	relativistic and radiative corrections often form a hierarchy	high nuclear charge and precision measurements expose higher orders
Heavy nucleus	fixed-nucleus models are often accurate at first pass	recoil, finite size, spin, and isotope dependence matter at finer resolution
Discrete bound levels	spectroscopy can isolate narrow transitions	continua, resonances, decay, collisions, and field broadening are unavoidable
Identical electrons	antisymmetry constrains allowed states	exchange and correlation couple configurations nontrivially

The subject is therefore not a sequence of unrelated corrections. It is an exercise in controlled resolution. A model that predicts the gross spectrum may be inadequate for a clock shift, while a detailed many-body calculation may be unnecessary for choosing a laser polarization.

The Atomic Hamiltonian Hierarchy

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

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

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,

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

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 exact hydrogenic 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 are the canonical homes for the machinery behind 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 $R_M$ includes the appropriate reduced-mass convention and $\delta_{\ell j}$ is a quantum defect. 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.

Example: reading a term symbol

The term ${}^3P_2$ means $S=1$, $L=1$, and $J=2$. It says nothing by itself about which configuration produced the term. Its magnetic substates have $M_J=-2,-1,0,1,2$ before additional couplings or external fields are resolved. The multiplicity is three because $2S+1=3$, not because the $J=2$ level has three magnetic substates.

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	Common good labels	Failure signal
gross structure	Coulomb binding and electron correlation	configuration, $n$, $L$, $S$	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	$F=I+J$	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. The Spin–Orbit Coupling page owns the general angular-momentum structure; this chapter applies it to atomic spectra.

Hyperfine structure

Let $\mathbf I$ be nuclear spin and $\mathbf J$ electronic angular momentum. The leading magnetic-dipole model is

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

When this model is adequate, the shift of a level labeled by $F$ is

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

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 owns the magnetic-dipole and electric-quadrupole Hamiltonians, the $F$-multiplet spectrum, and 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 form assumes that hyperfine coupling still defines $F$. 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 owns the atomic regime map, 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 owns the atomic response tensors, alkali examples, trapping, and spectroscopic interpretation; the Stark Effect Example owns the general 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.

Spectroscopy as Evidence and Tool

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.

Evaluated data and provenance

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.

A Practical Atomic-Structure Workflow

1. Specify the system. State element, isotope when relevant, ionization stage, charge state, and external environment.
2. Specify the observable and accuracy. A gross configuration, a MHz-scale interval, and a clock-level shift require different Hamiltonians.
3. Choose a reference Hamiltonian. Decide whether the starting point is hydrogenic, central-field, mean-field, configuration-interaction, relativistic, or effective few-level.
4. Identify exact symmetries. Record total angular momentum, parity, exchange symmetry, and any remaining axial symmetry.
5. Estimate omitted scales. Compare recoil, relativistic, correlation, hyperfine, radiative, and field energies with the target uncertainty.
6. Choose the coupling scheme. Test whether $LS$, $jj$, hyperfine-coupled, or uncoupled labels are justified by scale separation.
7. Compute both energies and matrix elements. A plausible level diagram does not guarantee correct intensities or lifetimes.
8. Propagate to the measured signal. Include populations, polarization, broadening, detection response, and environmental shifts.
9. Validate against limits and data. Check one-electron limits, zero-field limits, angular-momentum sums, gauge or basis convergence where relevant, and evaluated spectroscopy.
10. Report provenance and uncertainty. Separate measured inputs, fitted effective parameters, calculated corrections, and neglected effects.

This workflow is deliberately observable-first. It prevents an elaborate calculation from being mistaken for a complete prediction when preparation, line shape, or detection dominates the comparison.

From Atomic Structure to AMO Platforms

The same hierarchy supports modern controlled systems:

• Alkali atoms approximate a closed-shell core plus one valence electron, giving accessible optical cycling transitions and hyperfine qubits.
• Alkaline-earth-like atoms and ions offer narrow intercombination or clock transitions, but their multi-electron structure requires additional care.
• Rydberg atoms amplify size, polarizability, and interatomic interactions through large principal quantum number.
• Trapped ions combine discrete internal levels with quantized center-of-mass modes.
• Optical clocks interrogate transitions selected for narrow linewidth and low sensitivity, then account for systematic shifts.
• Ultracold gases and optical lattices use atomic internal states, collisions, and light shifts as engineered many-body parameters.

These platforms do not erase atomic complexity. They exploit it selectively. A useful effective model records how preparation, leakage, spontaneous emission, collisions, and field noise connect the retained levels to the discarded atomic spectrum. AMO Platforms and Quantum Control develops that complete preparation–control–measurement cycle.

Chapter Map

Page	Canonical 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?

The neighboring Multi-Electron Atoms chapter owns antisymmetry, configurations, self-consistent fields, exchange, correlation, and detailed coupling schemes. Spectroscopy owns line strengths, line shapes, assignments, and inference. Light–Matter Interaction owns driven dynamics. Keeping those homes distinct lets this index remain a map rather than a duplicate textbook.

Common Mistakes

Treating hydrogenic labels as universally exact

The label $n\ell m_\ell$ is adapted to a spin-independent central potential. Electron correlation, spin-dependent interactions, and external fields can change which labels are conserved. State the Hamiltonian before declaring a quantum number good.

Calling orbitals electron trajectories

An orbital is a one-electron state or basis function. In a correlated many-electron state, no unique set of occupied orbitals need represent literal individual electrons. Observables follow from the many-electron state and its reduced densities, not from assigning classical paths.

Confusing multiplicity with magnetic degeneracy

In ${}^{2S+1}L_J$, the multiplicity $2S+1$ concerns total spin. A particular $J$ level has $2J+1$ magnetic substates before symmetry-breaking fields. These numbers can coincide accidentally but encode different things.

Using “weak field” without naming a comparison scale

A magnetic field may be weak relative to fine structure and strong relative to hyperfine structure. Compare $\mu_BB$ or the appropriate Stark energy with the relevant zero-field interval.

Treating a selection rule as an absolute prohibition

A rule applies to a specified interaction and symmetry limit. Mixing, higher multipoles, hyperfine coupling, or relativistic terms can produce a smaller nonzero amplitude.

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

Exercise 1: Hydrogenic scaling

Ignore reduced-mass and radiative corrections. Compare a hydrogen atom and a hydrogenic ion of nuclear charge $Z$. How do the characteristic orbital radius, binding energy, and fine-structure scale change with $Z$ at fixed $n$?

Solution

Rescale the Coulomb Schrödinger equation with $r=(a_0/Z)\rho$. The characteristic radius scales as

$$r_n\sim\frac{n^2a_0}{Z},$$

while the nonrelativistic energy is

$$E_n=-\frac{Z^2E_{\mathrm h}}{2n^2}.$$

Thus the atom contracts as $Z^{-1}$ and the binding grows as $Z^2$. The leading fine-structure scale behaves parametrically as $m_ec^2(Z\alpha)^4$, so at fixed $n$ it grows as $Z^4$ before state-dependent coefficients are included. This last scaling also warns that a low-order nonrelativistic expansion becomes less controlled as $Z\alpha$ approaches unity.

Exercise 2: 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.

Exercise 3: 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.

Exercise 4: Choose labels through a magnetic crossover

An atom has hyperfine interaction $A\mathbf I\cdot\mathbf J$ 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 $g_J\mu_BB$, with the hyperfine interval set by $A$. When the Zeeman energy is much smaller, hyperfine coupling first forms $\mathbf F=\mathbf I+\mathbf J$, and $F,m_F$ are useful labels. When it is much larger, electronic and nuclear projections decouple and $m_J,m_I$ are more useful.

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 $m_F=m_I+m_J$. 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.

Exercise 5: What does one spectral line establish?

An experiment reports a narrow peak at frequency $\nu_0$ with an integrated intensity. List at least four additional pieces of information needed before treating $h\nu_0$ as an unperturbed atomic level difference or the intensity as an intrinsic transition strength.

Solution

A defensible interpretation should identify, at minimum:

• the element, isotope, and ionization stage;
• the initial-state populations and preparation procedure;
• applied magnetic and electric fields, including probe-induced AC Stark shifts;
• laser or spectrometer calibration and its uncertainty;
• polarization and observation geometry;
• Doppler, collision, transit-time, power, and instrumental broadening;
• saturation or optical-pumping effects;
• unresolved hyperfine, Zeeman, or isotope components;
• detector response and collection efficiency for an intensity comparison.

The peak position is an observable of the full apparatus and environment. Recovering an isolated-atom interval requires correcting or extrapolating the relevant shifts. Likewise, an integrated signal combines a transition matrix element with population, driving, branching, and detection factors.

Cross-Links

• AMO Bibliography and Reading Guide compares introductory, graduate, specialist, historical, and evaluated-data sources for atomic physics.
• Atomic, Molecular, and Optical Physics
• Conceptual Overview
• Multi-Electron Atoms
• AMO Physics Roadmap
• Central-Field Approximation
• Alkali Atoms
• Atomic Orbitals Revisited
• Fine Structure
• Lamb Shift Overview
• Hyperfine Structure
• Zeeman Effect in Atoms
• Stark Effect in Atoms
• Atomic Selection Rules
• Atomic Term Symbols
• Rydberg Atoms Basics
• Hydrogen Atom
• Atomic Units
• Addition of Angular Momentum
• Spin–Orbit Coupling
• Dipole Transitions
• Atomic Spectra Experiment Entry

References

1. C. J. Foot, Atomic Physics, Oxford University Press, 2005.
2. B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson, 2003.
3. H. Friedrich, Theoretical Atomic Physics, 4th ed., Springer, 2017.
4. I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer, 1992.
5. W. R. Johnson, Atomic Structure Theory: Lectures on Atomic Physics, Springer, 2007.
6. I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory and Computation, Springer, 2007.
7. C. E. Moore, Atomic Energy Levels, National Bureau of Standards Circular 467, Vols. I–III, 1949–1958.
8. A. Kramida, Yu. Ralchenko, J. Reader, and the NIST ASD Team, NIST Atomic Spectra Database, version 5.12, National Institute of Standards and Technology, 2024, DOI: 10.18434/T4W30F, accessed 2026-07-21.
9. J. E. Sansonetti, W. C. Martin, and S. L. Young, Handbook of Basic Atomic Spectroscopic Data, National Institute of Standards and Technology.
10. P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA recommended values of the fundamental physical constants: 2022,” Reviews of Modern Physics 97, 025002 (2025), DOI: 10.1103/RevModPhys.97.025002.