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Quantum EngineeringYear 1: Quantum Mechanics CoreMonth 18Day 500

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Year 1·Month 18·Week 4

Day 500: Partial Wave Analysis

Day 500 of 2,016~18 min read

Learning Objectives

  • •**Expand** a plane wave in spherical harmonics and Bessel functions
  • •**Solve** the radial Schrödinger equation for each partial wave
  • •**Define** and calculate phase shifts from asymptotic behavior
  • •**Express** the scattering amplitude in terms of phase shifts
  • •**Construct** the S-matrix and verify unitarity
  • •**Apply** partial wave analysis to simple potentials

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OverviewScheduleLearning Objectives1 Partial Wave ExpansionAngular Momentum ConservationPlane Wave ExpansionAsymptotic Behavior2 Scattering Wave FunctionThe Full Wave FunctionRadial Schrdinger EquationFree Particle Solution3 Phase ShiftsDefinition of Phase ShiftPhysical InterpretationAsymptotic Wave Function4 Scattering Amplitude from Phase ShiftsExtracting fAlternative FormsDifferential Cross SectionTotal Cross SectionUnitarity Bound5 The S-MatrixDefinitionPropertiesRelation to Partial Wave AmplitudePhysical MeaningS-Matrix for Multi-Channel Scattering6 Worked ExamplesExample 1 Hard Sphere ScatteringExample 2 Phase Shift CalculationExample 3 Verifying the Optical Theorem7 Practice ProblemsLevel 1 Direct ApplicationLevel 2 IntermediateLevel 3 Challenging8 Computational Lab Partial Wave Calculations9 SummaryKey ConceptsKey Formulas10 Daily ChecklistConceptual UnderstandingMathematical SkillsComputational Skills11 Preview Day 501References
Day 499Day 500 of 2,016Day 501