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Quantum Engineering Year 1: Quantum Mechanics Core Month 18Day 494

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Year 1·Month 18·Week 3

Day 494: Exchange and Spin States in Helium

Day 494 of 2,016~18 min read

Learning Objectives

•**Construct** antisymmetric wave functions from spatial and spin components
•**Distinguish** between singlet ($S=0$) and triplet ($S=1$) spin states
•**Calculate** the exchange integral and explain its physical origin
•**Explain** why triplet states have lower energy than singlets
•**Describe** the spectroscopic differences between parahelium and orthohelium
•**Apply** selection rules to predict allowed transitions

Today's Schedule (7 hours)

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Overview Schedule Learning Objectives 1 Symmetry Requirements Review The Pauli Principle Factorization of Wave Function Ground State Revisited 2 Spin States for Two Electrons Building Blocks The Four Two-Spin States Symmetry Under Exchange Physical Interpretation 3 Excited States of Helium Configuration One Electron Excited Complete Wave Functions Why Different Energies 4 The Exchange Integral Energy Expectation Value Direct and Exchange Integrals Physical Meaning Energy for Each Spin State The Exchange Splitting 5 Why Does the Triplet Have Lower Energy The Fermi Hole Reduced Repulsion No Fermi Hole for Singlet Hunds First Rule 6 Parahelium and Orthohelium Two Spectroscopic Series Energy Level Diagram The Metastable State Historical Discovery 7 Selection Rules Electric Dipole Selection Rules Consequences for Helium Fine Structure 8 Worked Examples Example 1 Exchange Integral Sign Example 2 Exchange Splitting for 1s2s Example 3 Why No Ground State Triplet 9 Practice Problems Level 1 Direct Application Level 2 Intermediate Level 3 Challenging 10 Computational Lab Exchange Effects in Helium 11 Summary Key Concepts Key Formulas 12 Daily Checklist Conceptual Understanding Mathematical Skills Computational Skills Quantum Computing Connection 13 Preview Day 495 References

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