2.2 Angular Momentum

2.2 Angular Momentum#

Overview#

This section addresses how angular momentum quantization emerges from commutation relations and generalizes the spin-\(\tfrac{1}{2}\) system studied previously. The core inquiry: what constrains the possible values of angular momentum, and how do higher-spin systems work?

Topics#

Topic

Title

Core Question

2.2.1

Angular Momentum Algebra

How does quantization (\(j = 0, \tfrac{1}{2}, 1, \ldots\)) follow from commutation relations alone?

2.2.2

Spin Representations

How do spinors differ from vectors under rotations?

2.2.3

Addition of Angular Momenta

How do two angular momenta combine, and why is the coupled basis essential?

Key Concepts#

  • Angular momentum commutation relations as the Lie algebra \(\mathfrak{su}(2)\).

  • Casimir operator \(\hat{J}^2\), simultaneous eigenstates \(\vert j,m\rangle\), ladder operators \(\hat{J}_\pm\).

  • Spin as intrinsic angular momentum; spin-statistics theorem.

  • Spinor rotations and the \(2\pi\) sign flip (\(SU(2)\) double cover of \(SO(3)\)).

  • Clebsch-Gordan coefficients, triangle rule, singlet and triplet states.

  • Spin-orbit coupling and fine structure.

Learning Objectives#

  • Derive angular momentum quantization algebraically without solving differential equations.

  • Construct explicit matrix representations for spin-\(\tfrac{1}{2}\) (Pauli) and spin-\(1\).

  • Apply spinor rotation formulas and explain the \(2\pi\) vs \(4\pi\) distinction.

  • Combine angular momenta using Clebsch-Gordan coefficients and construct singlet/triplet states.

Project#

Project: Ultracold Fermi Gases and the BCS-BEC Crossover#

Objective: Model the continuous evolution from weakly-paired fermions to bosonic molecules by tuning interactions.

Suggested Approach:

  • Literature survey of mean-field BCS theory

  • Derive gap equations from the pairing Hamiltonian

  • Numerical self-consistent iteration for the gap and chemical potential

  • Plot phase diagrams showing gap and coherence length across the crossover

  • Analyze fermionic vs. bosonic excitations

  • Examine angular momentum constraints on pairing

Expected Deliverable: Research report (7–10 pages) with theory, numerical solutions, phase diagrams, excitation spectra analysis, and experimental comparisons from ultracold Fermi gas laboratories.