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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 292

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Year 0·Month 11·Week 2

Day 292: The Great Orthogonality Theorem

Day 292 of 2,016~12 min read

Learning Objectives

•State and prove the Great Orthogonality Theorem
•Derive character orthogonality as a corollary
•Apply the GOT to construct projection operators
•Understand the completeness relations
•Connect the GOT to Fourier analysis on finite groups
•Use the GOT in quantum mechanics applications

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Overview Prerequisites Learning Objectives 1 Statement of the Great Orthogonality Theorem Matrix Element Orthogonality Interpretation Character Orthogonality as Corollary 2 Proof of the Great Orthogonality Theorem Proof Strategy Step 1 Construct the Averaging Operator Step 2 Show A is an Intertwining Operator Step 3 Apply Schurs Lemma Step 4 Extract Matrix Elements 3 Completeness Relations The Dimension Formula Completeness of Matrix Elements 4 Projection Operators Projection onto Irreducible Components Character Projection 5 Fourier Analysis on Finite Groups The Group Algebra Fourier Transform Inverse Fourier Transform Plancherel Theorem 6 Quantum Mechanics Connection Selection Rules from GOT Wigner-Eckart Theorem Physical Implications 7 Worked Examples Example 1 Verify GOT for S_3 Example 2 Projection Operator for mathbbZ_3 Example 3 Fourier Transform on mathbbZ_4 8 Computational Lab 9 Summary The Great Orthogonality Theorem Key Corollaries 10 Preview Day 293

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