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Zeeman Perturbation Matrices in the Coupled Basis: Instructor's Guide

The weak field Zeeman effect requires students to work in the coupled basis, where the fine structure is diagonal, but the Zeeman perturbation is not diagonal. This activity shows students the power of the Clebsch-Gordan coefficients.

Main Ideas

  • Angular momentum eigenvalue equations
  • Uncoupled and coupled basis systems

Students' Task

  • Students write down angular momentum matrices in the uncoupled basis by inspection.
  • Students use Clebsch-Gordan coefficients to calculate angular momentum matrices in the coupled basis.

Estimated Time: 45 minutes

Prerequisite Knowledge

  • Coupling or addition of angular momentum.
  • Uncoupled and coupled bases.


  • Small Whiteboard with markers
  • Tabletop Whiteboard with markers
  • A handout for each student with Clebsch-Gordan coefficients and blank matrices in both bases (needed to have consistent row-column labeling).

Activity: Introduction

The Zeeman effect occurs when an external magnetic field is applied to an atom. The system we wish to study is the hydrogen atom in the 2p state. To do the Zeeman effect properly, we must include the electron spin, but we can safely neglect the proton spin. For this problem, we assume that the magnetic field is weak (i.e., smaller than the fine structure), meaning that we must include the fine structure in the zeroth-order Hamiltonian and treat the Zeeman effect as a perturbation. The fine structure is diagonal in the coupled basis, while the Zeeman perturbation is diagonal in the uncoupled basis. But we must use the coupled basis, because perturbation theory requires us to use the zeroth-order basis. So we must understand both bases for this problem. (See p. 397 of QM text for further details.)

Activity: Student Conversations

Activity: Wrap-up


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