Atomic Physics Applications

Symmetry makes atomic spectra easier to read. It identifies which angular-momentum labels survive a coupling, which degeneracies a field can split, and which transition amplitudes must vanish. The labels depend on the full Hamiltonian and the experimental regime: the same atom can be most naturally described in a coupled basis at weak field and an approximately uncoupled basis at strong field.

Helpful background. Central potentials supplies orbital labels. Coupling schemes combines angular momenta, and selection rules explains how the probe restricts transitions.

Atomic levels and conserved angular momenta

For the spinless nonrelativistic Coulomb model,

$$H_0=\frac{\mathbf P^2}{2\mu}-\frac{Ze^2}{4\pi\epsilon_0R},$$

rotations allow simultaneous eigenstates of $H_0,L^2,L_z$, labeled $|n\ell m\rangle$. They have wavefunctions $R_{n\ell}(r)Y_\ell^m(\hat{\mathbf r})$. Rotations explain the $2\ell+1$ degeneracy across $m$; the equality of energies for different $\ell$ at the same $n$ uses the additional Coulomb structure. See hydrogen and its extra degeneracy.

For one electron, a central spin–orbit term $\xi(r)\mathbf L\cdot\mathbf S$ preserves $L^2,S^2,J^2,J_z$, with $\mathbf J=\mathbf L+\mathbf S$. It generally does not preserve $L_z$ and $S_z$ separately. Thus $\ell$ and $s$ remain valid labels in this model; it is the separate projections that are replaced by $j,m_j$. In many-electron atoms, $LS$ labels can be only approximate because several interactions compete.

Within a fixed orbital multiplet, write an effective coupling as $H_{\rm SO}=\lambda\mathbf L\cdot\mathbf S/\hbar^2$, where $\lambda$ has energy units and contains the radial expectation. For $\ell=1,s=1/2$,

$$\Delta E_{j=3/2}=\lambda/2, \qquad \Delta E_{j=1/2}=-\lambda.$$

The splitting is $3\lambda/2$. Angular algebra supplies these factors; it does not determine the radial coefficient. Fine structure also includes relativistic kinetic and Darwin corrections. It should not be identified with spin–orbit coupling alone.

Hyperfine coupling and the field scale

Couple nuclear spin $\mathbf I$ to electronic $\mathbf J$ to form $\mathbf F=\mathbf I+\mathbf J$. In a chosen electronic level, the magnetic-dipole hyperfine model is

$$H_{\rm hfs}=\frac A{\hbar^2}\mathbf I\cdot\mathbf J, \qquad E_F=\frac A2[F(F+1)-I(I+1)-J(J+1)].$$

Here $A$ is an energy, and higher hyperfine multipoles are omitted. If $I=J=1/2$, the singlet has energy $-3A/4$ and the triplet $A/4$.

A field along $z$ adds couplings to $J_z$ and $I_z$ with generally different coefficients. The total projection $m_F$ remains exact in this axial model, but $F$ generally does not at nonzero field. At weak field, mixing between well-separated hyperfine levels is small, so $F$ remains a useful approximate label. When the Zeeman scale dominates the hyperfine coupling while remaining small compared with the relevant electronic-level separations, $m_J,m_I$ become approximately good labels. “Strong field” therefore always means strong relative to a specified coupling, not arbitrarily large field.

Use coupling schemes and the Zeeman perturbation example to choose the appropriate basis before computing shifts.

A spectral line needs an energy difference and a matrix element

In the electric-dipole approximation, the transition amplitude contains $\langle f|\mathbf d\cdot\boldsymbol\epsilon|i\rangle$, with a specified absorption or emission polarization convention. For hydrogenic orbital states, the vector character and odd parity give

$$\Delta\ell=\pm1,\qquad \Delta m=0,\pm1.$$

For coupled total angular momenta, use the corresponding rank-one triangle condition, excluding $j=0\to j'=0$, together with parity and any remaining spin or nuclear restrictions. A nonzero allowed angular factor still multiplies a reduced matrix element. Radial cancellations, preparation, populations, and line broadening affect the observed intensity.

A transition forbidden at electric-dipole order can occur through magnetic-dipole, electric-quadrupole, multiphoton, or symmetry-breaking mechanisms. The mechanism must be specified. Spin–orbit mixing can relax an approximate spin selection rule while preserving the exact parity restriction in a parity-invariant atom. Atomic selection rules and transition-rate methods develop the calculation.

Electric fields and parity mixing

A static electric field contributes $-\mathbf d\cdot\mathbf E$. It is odd under inversion of the atom with the external field held fixed, so it mixes opposite-parity states. For an isolated nondegenerate parity eigenstate, the diagonal matrix element vanishes and the leading Stark energy shift is ordinarily quadratic. An exactly degenerate opposite-parity manifold can instead split linearly after the perturbation is diagonalized within it.

Ideal hydrogen provides the familiar degenerate example; fine structure and Lamb shifts can change which field range behaves approximately that way. The Stark example separates the nondegenerate polarizability calculation from degenerate mixing.

Exercises

1. Derive the two $\ell=1,s=1/2$ spin–orbit shifts above. Check that the degeneracy-weighted mean shift is zero.

Solution

Use $\mathbf L\cdot\mathbf S=(J^2-L^2-S^2)/2$. The dimensionless factors are $(15/4-2-3/4)/2=1/2$ and $(3/4-2-3/4)/2=-1$. The multiplet dimensions are four and two, so $4(\lambda/2)+2(-\lambda)=0$. This is also the trace of $\lambda\mathbf L\cdot\mathbf S/\hbar^2$ on the product multiplet.

2. Two opposite-parity states have equal unperturbed energy $E_0$ and dipole coupling $\langle+|d_z|-\rangle=d$, with real $d$. Find their energies in a weak $z$-directed electric field within this two-state model.

Solution

The diagonal dipoles vanish by parity and the Hamiltonian is $E_0I-dE\sigma_x$. The states $(|+\rangle\pm|-\rangle)/\sqrt2$ have energies $E_0\mp dE$. The pair is split by $2|dE|$, linearly in the field magnitude. A nonzero initial energy separation would instead suppress mixing at sufficiently small field and give quadratic leading shifts.

References

• H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer, 1957.
• B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed., Pearson, 2003.
• G. W. F. Drake, ed., Springer Handbook of Atomic, Molecular, and Optical Physics, 2nd ed., Springer, 2023.
• C. J. Foot, Atomic Physics, Oxford University Press, 2005.
• I. I. Sobelman, Atomic Spectra and Radiative Transitions, 2nd ed., Springer, 1992.