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Quantum Engineering Year 0: Mathematical Foundations Month 5Day 114

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Year 0·Month 5·Week 1

Day 114: Hermitian Operators — The Mathematics of Quantum Observables

Day 114 of 2,016~12 min read

Learning Objectives

•Define Hermitian (self-adjoint) operators
•Prove that Hermitian operators have real eigenvalues
•Prove that eigenvectors of distinct eigenvalues are orthogonal
•State and apply the spectral theorem for Hermitian operators
•Understand why quantum observables must be Hermitian
•Work with Hermitian matrices computationally

Today's Schedule (7 hours)

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On this page

1 Definition of Hermitian Operators 2 Examples of Hermitian Matrices Example 1 Diagonal with Real Entries Example 2 General 22 Hermitian Example 3 Pauli Matrices Example 4 NOT Hermitian 3 The Central Theorem Real Eigenvalues 4 Orthogonality of Eigenvectors 5 The Spectral Theorem 6 Properties of Hermitian Operators 7 Characterizations of Hermitian Matrices 8 Positive Definite and Semidefinite Quantum Mechanics Connection The Measurement Postulate Measurement Process Example Spin Measurement The Heisenberg Uncertainty Principle

Day 113Day 114 of 2,016Day 115