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Quantum Engineering Year 1: Quantum Mechanics Core Month 18Day 501

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Year 1·Month 18·Week 4

Day 501: Low-Energy Scattering

Day 501 of 2,016~18 min read

Learning Objectives

•**Explain** why s-wave dominates at low energies
•**Define** and calculate the scattering length a
•**Apply** the effective range expansion
•**Connect** scattering length to bound state properties
•**Distinguish** positive and negative scattering lengths
•**Apply** these concepts to ultracold atomic physics

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Overview Schedule Learning Objectives 1 s-Wave Dominance Why Only s-Wave at Low Energy Quantitative Criterion Phase Shift Behavior 2 The Scattering Length Definition Physical Interpretation Sign of Scattering Length Low-Energy Cross Section 3 Effective Range Expansion Beyond the Scattering Length Physical Meaning Partial Wave Amplitude Zero-Energy Limit Low-Energy Cross Section with Range Correction 4 Scattering and Bound States Connection Between Scattering and Binding Derivation Three Scenarios Levinsons Theorem 5 Examples and Applications Hard Sphere Square Well Neutron-Proton Scattering Ultracold Atoms Feshbach Resonances 6 Worked Examples Example 1 Extracting Scattering Length Example 2 Bound State Energy Example 3 Effective Range Correction 7 Practice Problems Level 1 Direct Application Level 2 Intermediate Level 3 Challenging 8 Computational Lab Low-Energy Scattering 9 Summary Key Concepts Key Formulas 10 Daily Checklist Conceptual Understanding Mathematical Skills Computational Skills 11 Preview Day 502 References

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