Atoms in Magnetic and Electric Fields: Hyperfine Structure, Stark Effect and Perturbation Theory

Overview

This lecture from MIT OpenCourseWare examines how atoms respond to external magnetic and electric fields, tying together hyperfine and fine structure with magnetic and electric field interactions. The instructor builds intuition through limiting cases, discusses the vector model and rapid precession, and explains how to treat non commuting Hamiltonian terms by first handling the strongest interaction, then the weaker ones.

In the electric field part, the talk introduces the Stark effect, parity arguments that suppress permanent dipole moments for non-degenerate states, and how perturbation theory yields polarizability and a quadratic energy shift. The session also uses a gravity analogy to illuminate internal energy contributions and discusses the connection to experimental observables such as g-factors and energy level ladders.

Overview

The lecture explores atoms in external fields, focusing first on magnetic fields and then on electric fields. It presents a pedagogical framework for understanding how two non commuting parts of the Hamiltonian – the hyperfine interaction and the magnetic field coupling – can be treated in different limiting regimes. The two main regimes are: weak magnetic fields where hyperfine structure dominates and strong magnetic fields where the field dominates and the hyperfine coupling is treated perturbatively.

Atoms in Magnetic Field

In the weak field limit, the hyperfine structure sets the eigenbasis. The total angular momentum F, the quantum numbers J and I, and their coupling to the magnetic field are used to determine how the system behaves as the field changes. In this regime, one uses perturbation theory to add the magnetic interaction to the hyperfine Hamiltonian and derives quantities like the Landé g-factor for hyperfine levels. In the strong field limit, the magnetic field dominates and the electron and nuclear spins couple individually to the field. New good quantum numbers become M_J and M_I, and the hyperfine coupling is treated as a perturbation on top of the diagonalized Zeeman-like Hamiltonian. The vector model, rapid precession, and the concept of first treating the strongest term before the weaker ones are emphasized as powerful tools for intuition.

Vector Model and Perturbation Theory

The vector model provides a simplified view of how spins precess about the magnetic field axis, with cross terms treated perturbatively. The method yields analytic expressions for quantities like the hyperfine splitting in different field regimes and clarifies why certain quantities are conserved or not when the Hamiltonian terms do not commute. The discussion also notes that in modern computations one can diagonalize the full Hamiltonian numerically, but the limiting-case analysis remains essential for building intuition.

Atoms in Electric Field and Stark Effect

Moving to electric fields, the lecture expands the electrostatic energy into monopole, dipole, and polarizability terms. For neutral atoms, the permanent dipole moment vanishes in non-degenerate states due to parity, so the linear Stark effect is typically absent and the dominant shift is quadratic, governed by the polarizability alpha. Perturbation theory shows that the energy shift scales as alpha E^2, and the induced dipole moment is proportional to the field. The speaker draws a gravity-like analogy to illustrate how the energy shift splits into an external field contribution and a cost in internal excitation energy, with a factor of one half appearing in the energy expression when considering the induced dipole and the associated perturbation terms. A discussion of how degeneracy can enable linear Stark effects is also included, along with unit analysis and the hydrogenic estimate for alpha.

Q&A and Conceptual Touchstones

The session includes clicker questions that revisit electronic structure scaling with principal quantum number, hydrogenic wavefunction density at the origin, and the origin of singlet-triplet splitting in helium. Topics such as the Lamb shift, proton radius effects, and Dirac theory degeneracies are discussed, linking perturbation theory to real-world spectroscopic observations. The Positronium examples extend the analysis to two spin-1/2 particles, showing how degeneracies evolve in high versus low magnetic fields and how the four hyperfine states behave in each regime.

Takeaways

Across magnetic and electric field problems, the central themes are the hierarchy of Hamiltonian terms, the choice of a convenient limiting basis, and the use of perturbation theory to quantify corrections. The DC Stark effect is highlighted as a canonical example where the polarizability governs the energy shift, while the internal excitation energy accounts for the factor of two that appears in the quadratic Stark energy. The material culminates in a unifying perspective on how to analyze atoms in fields using a combination of limiting-case reasoning, vector-model intuition, and rigorous perturbative calculations.