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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 306

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Year 0·Month 11·Week 4

Day 306: Clebsch-Gordan Coefficients

Day 306 of 2,016~10 min read

Learning Objectives

•Define and interpret Clebsch-Gordan coefficients
•Derive symmetry relations and selection rules
•Calculate coefficients using recursion relations
•Apply to physical problems
•Use standard tables and computational tools

Today's Schedule (7 hours)

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Overview Learning Objectives 1 Definition and Notation The Transformation Alternative Notations Inverse Transformation 2 Selection Rules The m-Rule The Triangle Rule Integer Constraint 3 Symmetry Relations Exchange of j_1 and j_2 Reversal of m Values Sign Conventions Condon-Shortley 4 Orthogonality Relations First Orthogonality Second Orthogonality 5 Recursion Relations Starting Point Lowering Recursion Raising Recursion 6 Special Cases j_2 12 Adding Spin-12 j_2 1 Adding Spin-1 Two Spin-12 Particles 7 The Wigner 3j Symbol Definition Advantages Cyclic Symmetry Exchange Symmetry 8 Explicit Formula Racah The General Formula 9 Computational Lab 10 Practice Problems Problem 1 Direct Calculation Problem 2 Orthogonality Problem 3 Symmetry Problem 4 3j Symbol Problem 5 d-Electron Summary Clebsch-Gordan Coefficients Key Properties Standard Sources Preview Day 307

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