A particle scatters off a potential and the measured angular distribution shows strong forward scattering. What does this tell you about the scattering amplitude f(θ,φ)?
Af(θ,φ) is large for small θ, because dσ/dΩ = |f|² and the differential cross section is large near θ = 0
Bf(θ,φ) is small for small θ, because forward scattering means particles are not deflected
CThe total cross section σ must be small, since most particles pass straight through
DThe scattering amplitude cannot be determined from the angular distribution alone
The differential cross section dσ/dΩ = |f(θ,φ)|² directly connects what is measured (the angular distribution of scattered particles) to the scattering amplitude. Strong forward scattering means dσ/dΩ is large at small θ, which means |f|² is large there, so |f| itself is large. The total cross section σ = ∫|f|²dΩ could still be large or small depending on the full angular dependence — forward-peaked scattering often comes with a large total cross section.
Question 2 Multiple Choice
In the asymptotic wavefunction ψ ≈ e^{ikz} + f(θ,φ)e^{ikr}/r, a student argues the scattered amplitude should fall as 1/r² to match how intensity decreases with distance from a point source. Evaluate this reasoning.
ACorrect — both the amplitude and intensity should decrease as 1/r²
BIncorrect — the amplitude must fall as 1/r so that the probability density |ψ|² ∝ 1/r², conserving total probability flux through an expanding spherical shell
CIncorrect — the amplitude falls as 1/r² and the intensity falls as 1/r⁴, which is steeper than a classical point source
DCorrect in reasoning but wrong conclusion — the 1/r dependence comes from angular momentum conservation, not probability conservation
Probability flux must be conserved: the total number of particles crossing a sphere of radius r per unit time must be independent of r. The area of a sphere grows as r², so the probability density |ψ|² must fall as 1/r², which requires the amplitude to fall as 1/r. If the amplitude fell as 1/r², the probability density would fall as 1/r⁴, and the total flux through a large sphere would shrink to zero — a violation of probability conservation.
Question 3 True / False
The scattering amplitude f(θ,φ) has dimensions of length.
TTrue
FFalse
Answer: True
The differential cross section dσ/dΩ = |f(θ,φ)|² must have units of area per steradian. Since steradians are dimensionless, dσ/dΩ has units of area (e.g., m², barn). Therefore |f|² has units of area and f itself has units of length. This dimensional analysis also explains why the Born approximation, which integrates the potential over a volume, produces a result with the right dimensions for f.
Question 4 True / False
Measuring the complete angular distribution |f(θ,φ)|² at a fixed energy fully determines the scattering amplitude f(θ,φ), including its phase.
TTrue
FFalse
Answer: False
The angular distribution measurement gives dσ/dΩ = |f(θ,φ)|², which reveals only the magnitude |f| at each angle — the phase of f is not directly measurable in a single-beam scattering experiment. Recovering the phase requires interference measurements (e.g., Coulomb-nuclear interference) or use of partial wave analysis to exploit unitarity constraints. This 'phase problem' in scattering is analogous to the phase problem in X-ray crystallography.
Question 5 Short Answer
Why is the scattering amplitude f(θ,φ) described as encoding 'all the physics of the interaction in the far field'? What information does it capture, and what does it not capture?
Think about your answer, then reveal below.
Model answer: f(θ,φ) encodes everything needed to predict experimental observables: the angular distribution of scattered particles (via dσ/dΩ = |f|²) and the total scattering rate (via σ = ∫|f|²dΩ). Different potentials V(r) produce different f, so measuring f characterizes the interaction. However, f does not capture the wavefunction inside or near the potential region — it is strictly an asymptotic (large-r) quantity. Also, f is a complex function, but experiments typically measure only |f|², so the phase information is inaccessible without special interference setups.
The scattering amplitude is the bridge between the quantum mechanical description (wavefunction solving Schrödinger's equation) and experimental measurements (particle counts at each angle). Its power is that two very different potentials that produce the same f(θ,φ) are experimentally indistinguishable — scattering experiments constrain but do not uniquely determine the underlying interaction, a fact with deep implications for nuclear and particle physics.