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Quantum Engineering Year 0: Mathematical Foundations Month 11Day 307

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Year 0·Month 11·Week 4

Day 307: Applications — Selection Rules and the Wigner-Eckart Theorem

Day 307 of 2,016~12 min read

Learning Objectives

•State and apply the Wigner-Eckart theorem
•Derive selection rules for electric dipole transitions
•Calculate transition probabilities using reduced matrix elements
•Apply to atomic spectra and quantum information
•Connect group theory to observable physics

Today's Schedule (7 hours)

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Overview Learning Objectives 1 Tensor Operators Definition Examples 2 The Wigner-Eckart Theorem Statement Significance Selection Rules from CG Coefficients 3 Electric Dipole Transitions The Electric Dipole Operator Selection Rules Derivation Parity Selection Rule 4 Transition Probabilities Fermis Golden Rule Using Wigner-Eckart Sum Rules 5 Magnetic Dipole and Electric Quadrupole Magnetic Dipole Transitions Electric Quadrupole Transitions Comparison of Transition Strengths 6 Atomic Fine Structure Spin-Orbit Coupling Energy Levels Fine Structure Selection Rules 7 Quantum Computing Applications Qubit Operations as Rotations Selection Rules in Quantum Gates Forbidden Transitions in Quantum Error Correction 8 Computational Lab 9 Practice Problems Problem 1 Selection Rules Problem 2 Wigner-Eckart Problem 3 Reduced Matrix Element Problem 4 Polarization Problem 5 Fine Structure Summary Wigner-Eckart Theorem Electric Dipole Selection Rules Key Applications Preview Day 308

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