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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 305

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Year 0·Month 11·Week 4

Day 305: Addition of Angular Momenta

Day 305 of 2,016~12 min read

Learning Objectives

•Understand the tensor product structure of composite systems
•Master the uncoupled and coupled bases
•Derive the range of total angular momentum
•Apply to physical examples: singlet/triplet, spin-orbit coupling
•Prepare for Clebsch-Gordan coefficients

Today's Schedule (7 hours)

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Overview Learning Objectives 1 The Physical Problem Why Add Angular Momenta The Question 2 Tensor Product Space The Combined Hilbert Space Operators on Composite System Commutation Relations 3 The Uncoupled Basis Product States Example Two Spin-12 Particles Total m in Uncoupled Basis 4 The Coupled Basis A Different Set of Quantum Numbers The Key Question 5 Range of Total Angular Momentum The Triangle Rule Derivation via Counting Dimension Check 6 The Basis Change Relation Between Bases Notation Selection Rule 7 Example Two Spin-12 Particles The Setup Finding the States Summary Singlet and Triplet 8 Physical Applications Spin-Orbit Coupling Helium Atom Nuclear Physics 9 Representation Theory Perspective Tensor Product of Representations Dimension Formula Group Theory Interpretation 10 Computational Lab 11 Practice Problems Problem 1 Dimension Counting Problem 2 Three Spin-12 Problem 3 Singlet Projection Problem 4 Exchange Operator Problem 5 Spin-Orbit Splitting Summary Angular Momentum Addition Rule Two Spin-12 Singlet and Triplet Key Relations Preview Day 306

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