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Quantum EngineeringYear 1: Quantum Mechanics CoreMonth 18Day 492

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Year 1·Month 18·Week 3

Day 492: Perturbation Approach to Helium

Day 492 of 2,016~17 min read

Learning Objectives

  • •**Set up** the first-order perturbation calculation for helium
  • •**Evaluate** the electron-electron repulsion integral ⟨1/r₁₂⟩
  • •**Derive** the result $E^{(1)} = \frac{5}{4}Z$ Hartree
  • •**Calculate** the total first-order energy and compare with experiment
  • •**Analyze** why perturbation theory improves on zeroth order
  • •**Understand** the limitations of first-order perturbation theory

Today's Schedule (7 hours)

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OverviewScheduleLearning Objectives1 Review First-Order Perturbation TheoryThe Perturbation FrameworkApplication to HeliumWhat We Need to Calculate2 Setting Up the IntegralExplicit FormChoosing CoordinatesThe Expansion of 1rThe Addition TheoremSimplification for s-Orbitals3 Evaluating the Radial IntegralSplitting the RegionThe Inner IntegralThe Outer IntegralStandard IntegralsFinal EvaluationThe Final Result4 Total Energy and ComparisonFirst-Order Total EnergyComparison TableAnalysis5 Physical InterpretationWhat Does 1r RepresentAverage Electron SeparationWhy Repulsion is So LargeScreening Picture6 Worked ExamplesExample 1 First-Order Energy for He-like IonsExample 2 Scaling with ZExample 3 Second-Order Energy Setup7 Practice ProblemsLevel 1 Direct ApplicationLevel 2 IntermediateLevel 3 Challenging8 Computational Lab First-Order Perturbation9 SummaryKey ConceptsKey Formulas10 Daily ChecklistConceptual UnderstandingMathematical SkillsComputational SkillsQuantum Computing Connection11 Preview Day 493References
Day 491Day 492 of 2,016Day 493