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Quantum EngineeringYear 0: Mathematical FoundationsMonth 11Day 290

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Year 0·Month 11·Week 2

Day 290: Schur's Lemma — The Fundamental Tool

Day 290 of 2,016~13 min read

Learning Objectives

  • •State and prove both parts of Schur's lemma
  • •Apply Schur's lemma to intertwining operators
  • •Derive orthogonality of matrix elements
  • •Understand the connection to selection rules
  • •Use Schur's lemma to analyze operators commuting with representations
  • •Apply the lemma to quantum mechanical problems

Today's Schedule (7 hours)

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OverviewPrerequisitesLearning Objectives1 Intertwining OperatorsDefinitionNotationExamples2 Schurs LemmaStatementProof3 Consequences of Schurs LemmaCorollary 1 Dimension of Hom SpaceCorollary 2 CommutantCorollary 3 Center of Group AlgebraCorollary 4 Abelian Groups4 Orthogonality of Matrix ElementsThe Great Orthogonality Theorem PreviewProof Sketch Using Schur5 Quantum Mechanics ConnectionSelection RulesExample Angular MomentumDegeneracy and Commuting Operators6 Worked ExamplesExample 1 Schurs Lemma for mathbbZ_nExample 2 Commuting Operators in S_3Example 3 Reducible Representation7 Computational Lab8 Practice ProblemsProblem Set A Basic ApplicationsProblem Set B Intertwining OperatorsProblem Set C Quantum Applications9 SummarySchurs LemmaKey Consequences10 Preview Day 291
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