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Quantum Engineering Year 1: Quantum Mechanics Core Month 13Day 339

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Year 1·Month 13·Week 1

Day 339: Operators in Hilbert Space — The Mathematics of Physical Observables

Day 339 of 2,016~18 min read

Learning Objectives

•Define linear operators and verify linearity
•Compute the action of operators on quantum states: Â|ψ⟩
•Construct matrix representations of operators in a given basis
•Transform operator matrices under change of basis
•Define and evaluate functions of operators
•Perform operator algebra: addition, multiplication, and preview commutators

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1 Linear Operators Definition and Properties 2 Operator Action on States 3 Matrix Representation of Operators 4 Operator Expansion in Outer Products 5 Change of Basis for Operators 6 Functions of Operators 7 Operator Algebra 8 The Commutator Preview Quantum Computing Connection Operators as Quantum Gates Single-Qubit Gates as 22 Unitary Matrices Gate Composition Operator Multiplication Non-Commutativity in Circuits Universal Gate Sets

Day 338Day 339 of 2,016Day 340