Motivation and learning objective
How can the exact scattering amplitude stay finite while its Born series diverges?
An exactly solvable one-dimensional delta potential reveals how the outgoing Green function repeatedly returns a wave to the same scatterer. Compare the exact unitary reflection amplitude against successive Born-series partial sums, and see why their radius of convergence matters.
Recommended prerequisites: the relevant mathematical formulation in the English lecture notes, complex amplitudes, expectation values, and a basic understanding of quantum operators. Read the derivation below before interpreting the plots as physical observables.
Predictions and benchmarks
Live model outputs
Scattering probabilities versus coupling
The exact scattering result respects flux conservation. The first Born approximation is a perturbative prediction and can exceed unity when the potential becomes strong.
Exact reflection and first Born reflection as a function of potential coupling
At fixed k, R=u²/(1+u²) is bounded by one, while Rfirst≈u² only for |u|≪1. Born values above 1.15 are intentionally omitted from this scale because truncated perturbative probabilities need not be normalized.
Successive scattering orders and their error
Each extra return to the scatterer contributes another factor of −iu. The resulting error decreases only when |u|<1.
Error of Nth-order reflected amplitude versus Born iteration count
For the geometric series, |rN−rExact|=|rExact||u|^N. At |u|=1 the partial sums fail to converge even though the exact amplitude exists.
Trajectory in the complex amplitude plane
Partial sums of the reflection amplitude trace a complex path. In the convergent regime they approach the exact complex reflection coefficient.
Real versus imaginary parts of successive Born approximations for reflection
The exact reflection is displayed as the orange point; the blue curve connects orders N=1,…,12 in the complex plane.
Derivation and assumptions
Reconstruct the physical calculation
Set ℏ²/(2m)=1 and consider a one-dimensional scattering state with incident kinetic energy E=k² (k>0) and a short-range potential V(x)=gδ(x). The free outgoing resolvent is G₀⁺(x,x′;k)=−i exp[ik|x−x′|]/(2k), giving G₀⁺(0,0;k)=−i/(2k). This is the outgoing Green function specified by the radiation boundary condition.
The exact outgoing wave is ψ(x)=exp(ikx)+r exp(−ikx) for x<0 and ψ(x)=t exp(ikx) for x>0, where u=g/(2k), r=−iu/(1+iu), t=1/(1+iu). Reflection and transmission probabilities are R=u²/(1+u²), T=1/(1+u²), satisfying the exact current-conservation identity R+T=1.
Iterating Lippmann–Schwinger algebraically yields the Born series T₀₀,N=g Σj=0N−1(−iu)j, convergent only for |u|<1. The order-N reflected amplitude is rN=−iT₀₀,N/(2k), and the exact finite-order error is rN−rExact=−rExact(−iu)N, hence |rN−rExact|=|rExact||u|N.
A well-defined nonperturbative scattering amplitude may therefore coexist with a divergent perturbation series. The condition |u|<1 is the radius of convergence for this explicit geometric Born expansion, not a universal criterion for every scattering potential or three-dimensional partial-wave calculation. At negative g the model supports a bound state, but the displayed energy is positive and the on-shell scattering still satisfies unitarity.
Check exact and singular limits
At g→0, R≈g²/(4k²) and the first Born approximation becomes reliable. At |g|≫2k, exact R approaches one while the first Born probability grows without bound and higher geometric partial sums diverge. At g=0, r=0 and t=1. At |u|=1, the series lies on its convergence boundary and the exact amplitude is nevertheless finite.
Numerical and physical validity conditions
Ideal one-dimensional delta potential, continuous line, positive incident energy and an outgoing Green-function boundary condition. The delta interaction is used to supply an analytically exact multiple-scattering benchmark. No three-dimensional angular differential cross section is inferred from this one-dimensional model.
Active understanding
Three independent investigations
- Recover the exact solution. Integrate the Schrödinger equation across x=0 to obtain the derivative jump ψ′(0+)−ψ′(0−)=gψ(0), then rederive r and t.
- Test the Born convergence circle. Increase |u| through unity and add terms. Separate the behavior of the exact amplitude from that of its series.
- Distinguish amplitude from probability. Square the truncated rN and compare against the exact R. Explain why a finite Born approximation need not satisfy exact scattering unitarity.
Self-check: For the delta scatterer, what condition guarantees convergence of the geometric Born T-series?
Extension: reproduce one model with a separate implementation or an AI-assisted notebook. Check numerical convergence, normalizations, phase conventions, or independent limiting-case results before trusting the answer.
Physical conclusions
Interpret the invariant or approximation carefully
This model links the outgoing Green function, T-matrix resummation, perturbative convergence and unitary scattering in one independently solvable experiment. It explicitly demonstrates why a strong-scattering calculation may require resummation even though each perturbative step is algebraically simple.
Lecture notes: 03_lippmann_schwinger.tex ↗.