Modern Physics · Upper Division

Atomic Physics & Spectroscopy

Atoms are the best-understood quantum systems. From the hydrogen spectrum to many-electron atoms, from laser cooling to precision measurements, atomic physics provides the most stringent tests of quantum electrodynamics and probes for new physics beyond the Standard Model.

Prerequisites

Quantum mechanics (Ch. 20) \cdot Spin & angular momentum (Ch. SP) \cdot Perturbation theory (Ch. P

T) \cdot Electromagnetism (Ch. 14–15

Learning Goals

Apply the Dirac fine-structure formula to compute energy splittings and compare them to the Lamb shift from QED.
Use Hund's rules and term symbols to determine the ground-state configuration of many-electron atoms.
Identify allowed electric dipole transitions using E1 selection rules and explain why forbidden lines still appear.
$Derive the Doppler cooling force and the Doppler temperature limit T_{D} = \hbarΓ/(2k_B) for a g$ iven atomic transition.
$Explain how Ramsey spectroscopy achieves the linewidth \Delta\nu = 1/(2T) and how optical lattice$ $clocks reach 10^{-18} fractional uncertainty$

AT.1 Hydrogen Atom — Complete Solution

The hydrogen Hamiltonian H = p²/(2m) − e²/(4πε₀r) has exact solutions in spherical coordinates. The energy levels:

E_{n} = -13.6 eV / n^{2} \qquad (n = 1, 2, 3, \cdots) \qquad (hydrogen energy levels)

(AT.1)

The wavefunctions ψ_nlm = R_nl(r) Y_l^m(θ,φ) where: n = 1, 2, 3, ... (principal), l = 0, 1, ..., n−1 (angular), m = −l, ..., +l (magnetic). Degeneracy: n² (including spin: 2n²).

The n² degeneracy in energy (independence of l) is the hidden SO(4) symmetry — the Laplace-Runge-Lenz vector is conserved. This degeneracy is broken by fine structure.

Fine structure splits levels with the same n but different l,j:

E_{nj} = -13.6 eV/n^{2} \times [1 + (\alpha^{2}/n^{2})(n/(j+\frac{1}{2}) - 3/4)] \qquad (Dirac formula)

(AT.2)

where α = e²/(4πε₀ℏc) ≈ 1/137 is the fine structure constant and j = l ± ½. The 2p_(1/2) and 2s_(1/2) states are predicted to be degenerate by the Dirac equation — but they differ by the Lamb shift (1057 MHz, due to QED vacuum fluctuations).

AT.2 Many-Electron Atoms

For atoms with Z electrons, the Hamiltonian includes electron-electron repulsion: H = Σᵢ [p_i²/(2m) − Ze²/rᵢ] + Σ_(i<j) e²/rᵢⱼ (atomic units: ℏ=m=e=1).

Hartree-Fock method: approximate the many-electron wavefunction as a Slater determinant (antisymmetrized product of single-particle orbitals). Each electron moves in the mean field of all others. Self-consistent field (SCF) iteration gives the HF ground state. Accuracy: 99% of total energy (1% = correlation energy, crucial for chemistry).

Hund's rules determine the ground state of atoms with partially filled shells: (1) Maximum S (highest spin multiplicity). (2) Maximum L (for same S). (3) J = |L−S| for less-than-half-filled shell; J = L+S for more-than-half-filled.

Term symbols: ²S+¹L_J. Example: carbon ground state (2p²): S = 1, L = 1, J = 0 → ³P₀. Iron (3d⁶): S = 2, L = 2, J = 4 → ⁵D₄.

AT.3 Selection Rules and Spectroscopy

Electric dipole transitions (strongest, determines most spectroscopy):

\Deltal = \pm1 \qquad \Deltam = 0, \pm1 \qquad \Deltaj = 0, \pm1 \qquad \Deltas = 0 \qquad (E1 selection rules)

(AT.3)

These follow from parity conservation and angular momentum conservation of the photon. Transitions violating these rules are forbidden at electric dipole order but occur via magnetic dipole (M1), electric quadrupole (E2), or two-photon processes — much weaker.

The 21 cm hyperfine line of hydrogen: the spin-flip transition 1s (F=1) → 1s (F=0) at 1420 MHz = 21.1 cm. Forbidden by M1 rules, occurs only via hyperfine interaction. Einstein A coefficient: A = 2.87×10⁻¹⁵ s⁻¹ (τ ≈ 10 Myr!). Despite its extreme weakness, it is the most important line in radio astronomy — maps the distribution of neutral hydrogen in galaxies.

Example AT.1 — Zeeman Effect in Sodium D-Lines

$Sodium's D-lines are 3s \to 3p transitions at 589 nm. In a magnetic field B, compute the Zee$ man splitting.

States:

3s: l=0, j=1/2, m_{j}=\pm1/2. 3p: l=1, j=3/2 (^{2}P_{3/2}) or j=1/2 (^{2}P_(1/2

Energy shift:

\DeltaE = g_{J} \mu_B B m_{j} where g_{J} is the Landé g-factor: g_{J} = 1 + [J(J+1)+S(S+1)-L(L+1)]/(2J(J+

1)).

g-factors:

For ^{2}S_{1/2} (3s): L=0, S=1/2, J=1/2 \to g_{J} = 2. For ^{2}P_{1/2}: L=1, S=1/2, J=1/2 \to g_{J} = 2/

3. For ^{2}P_{3/2}: L=1, S=1/2, J=3/2 \to g_{J} = 4/3

D₁ line (3s→³P_(1/2)):

Transitions: m_{j} = +1/2 \to m_{j} = +1/2, -1/2 (\Deltam_{j} = 0, -1); m_{j} = -1/2 \to m_{j} = +1/2, -1/2. Split

ting: \Delta\nu = \mu_B B(g_{J}(upper)-g_{J}(lower)) \times m_{j}/(h) — gives 4 lines (anomalous Zeeman

Scale:

\mu_B B/h = 1.4\times10^{10} B Hz. For B = 1 T: splitting \approx 14 GHz (0.016 nm at 589 nm). With B = 10

mT (lab magnet): 140 MHz \to easily resolved by a spectrometer

AT.4 Laser Cooling and Trapping

Doppler cooling: an atom moving toward a red-detuned laser (ω < ω_0) sees the photon blue-shifted into resonance (ω + kv ≈ ω_0). Each absorbed photon transfers momentum −ℏk (opposing motion). Spontaneously emitted photons go in random directions — average to zero. Net force: F = −αv (friction, Doppler cooling force).

Doppler cooling limit: T_D = ℏΓ/(2k_B) where Γ is the natural linewidth. For Na (Γ/2π = 10 MHz): T_D = 240 μK. Achieved in 1985 (Chu, Cohen-Tannoudji, Phillips; Nobel 1997).

Sub-Doppler cooling (polarization gradient, Sisyphus): T_min = ℏΓ/(2k_B) × (Γ/4Ω)² where Ω is the Rabi frequency. Can reach T ~ μK for weak fields.

Bose-Einstein condensation (BEC) of ultracold atoms: achieved when the de Broglie wavelength λ_dB = h/√(2πmk_BT) exceeds the interparticle spacing n^(1/3). Transition temperature T_BEC ≈ (ℏ²/mk_B)(n/ζ(3/2))^(2/3) ≈ 100 nK – 1 μK for typical alkali densities. First achieved in Rb⁸⁷ (Cornell and Wieman) and Na²³ (Ketterle), 1995 (Nobel 2001).

AT.5 Precision Measurements and Atomic Clocks

Atomic clocks use the hyperfine transition of ¹³³Cs at 9,192,631,770 Hz (defined). The second is defined by this frequency. Accuracy: 10⁻¹⁶ (one second per 300 million years).

Optical lattice clocks (Sr, Yb, Hg): use optical transitions at ∼10¹⁵ Hz, achieving 10⁻¹⁸ fractional uncertainty — better than the Cs standard by 100×. This is so precise it can measure gravitational redshift due to a 1 cm height difference (ΔGR/c² ≈ 1.1×10⁻¹⁸/cm). Applications: relativistic geodesy, dark matter searches, variation of fundamental constants.

The quantum electrodynamics test: the electron g-factor g/2 = 1.001159652181643(764) (theory) vs 1.00115965218059(13) (Harvard 2023). Agreement to 10 significant figures — the most precise test of any theory in science.

Definition AT.1 — Common Traps

Selection rules are symmetry rules: forbidden transitions are suppressed, not always impossible.
Fine and hyperfine structure have different origins: spin-orbit coupling is not nuclear spin coupling.
Linewidths encode lifetimes: shorter-lived states have broader natural lines.
Perturbations split degeneracies: Zeeman and Stark effects reveal hidden quantum numbers.

Exercises — AT.1–AT.5 Atomic Physics

Calculate the fine structure splitting between the 2p_{3/2} and 2p_{1/2} states of hydroge

n. How does the Lamb shift compare?

Straightforward

Estimate the ground state energy of helium using the Hartree-Fock method. What is the correlation energy and why is it important in chemistry?

Intermediate

Derive the Doppler cooling force and the Doppler temperature limit T_{D} = \hbarΓ/(2k_B) for rub

idium (Γ = 2\pi \times 6.07 MHz). What temperature is achieved in practice

Intermediate

Describe Ramsey spectroscopy and derive the linewidth \Delta\nu = 1/(2T). How do modern atomic cl

ocks achieve 10^{-18} accuracy? What is the ultimate limit

Challenging

Key Takeaways

$Hydrogen: E_{n} = -13.6/n^{2} eV. Degeneracy n^{2} from SO(4) symmetry. Fine structure \propto \alpha^{2}$ .
$Fine structure formula: E_{nj} from Dirac. Lamb shift (1057 MHz) from QED — g-factor test$ .
Hund's rules: maximize S, then L, then J. Determine ground state of many-electron atoms.
$E1 selection rules: \Deltal = \pm1, \Deltaj = 0,\pm1, \Deltam = 0,\pm1. Higher multipoles are weaker$ .
$Doppler cooling: F = -\alphav. Limit T_{D} = \hbarΓ/(2k_B). BEC at T_{BEC} ~ 100 nK–1 \muK$ .
$Atomic clocks: Cs hyperfine at 9.19 GHz (defines second). Sr optical: 10^{-18} — measures 1 c$ m redshift.