Scattering

University:
Massachusetts Institute of Technology
Course:
8.322 | Quantum Theory II
Academic year:

2021
Views:

77

Pages:

30
Author:

Alayna Ibarra

B. Scattering 8.1 General discussion Many physics experiments: [Diagram: A beam of particles labeled "A" is incident upon a target labeled "B." Scattered particles are detected by detectors positioned around the target.] possible outcomes: elastic scattering: particle A bounces off target without changing internal structure of A or B (ex. e-e scattering) inelastic scattering: internal structure of A or B changes; kinetic energy absorbed in collision (ex. e- scattering of H atom, collision raises energy of H) rearrangement collisions: outgoing particles are not A & B (ex. e+e- → μ+μ-) We will focus on elastic scattering. Simplifying assumptions particles A, B spinless assume A, B have no internal structure (elastic) assume interaction from potential V(r₁ - r₂) in COM frame, scattering of a particle of mass μ by potential V(r) (potential scattering) General discussion [Diagram: A beam of particles is incident on a target, and scattered particles are detected at an angle (θ, φ).] F = flux number of particles per unit angle: dσ/dΩ dN/dt = F (dσ/dΩ) dΩ dσ/dΩ : differential scattering cross-section σ = ∫ dσ(θ, φ) dΩ : total scattering cross-section units: area (1 barn = 10⁻²⁴ cm²) σ = cross-sectional area of target Looking for solution to (-ħ²/2m ∇² + V(x)) ψ(x) = E ψ(x) (-ħ²/2m ∇² + V(x)) ψ(x) = ħ²k²/2m ψ(x) Next: Partial wave approach to scattering Recall for decay of excited states (k) basis unnatural for photons w/ particles A.M. → want vector spherical harmonics Similar for scattering - use spherical waves 8.5 Spherical waves Want to find (continuum) states of free particle w/ fixed A.M. (eigenvalues) L², L_z. i.e., H = E = ħ²k²/2m, L² = ħ²ℓ(ℓ+1), L_z = ħm Ψ(x) = = C Reℓ(r) Yℓm(x) Free Schrödinger eqn: Rℓ'' + 2/r Rℓ' + (k² - ℓ(ℓ+1)/r²) Rℓ = 0 use p = kr Rℓ''(p) + 2/p Rℓ'(p) + (1 - ℓ(ℓ+1)/p²) Rℓ(p) = 0 Solution: spherical Bessel functions jℓ(p) = (π/2p)^(1/2) Jℓ+1/2(p) nℓ(p) = -(π/2p)^(1/2) Nℓ+1/2(p) jℓ(p) = Bessel function (regular @p=0) nℓ(p) = Neumann function (singular @p=0)

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