Scattering Worked Problems

These seven calculations turn scattering methods into predictions for specific potentials. Start with an exactly solvable example to check signs and normalization, then use that experience to assess an approximation. Each problem asks about an observable—reflection, an angular distribution, a threshold parameter, or a resonance—and compares it with an independent limit or calculation.

Helpful background. Scattering amplitudes and cross sections define the observables. Partial waves are needed for the central-potential examples; first Born theory is needed for the Gaussian and Yukawa calculations.

Choose a model

Worked problem	Main lesson
Delta-potential scattering	Use exact one-dimensional matching to connect reflection, transmission, phases, and a bound-state pole.
Spherical square-well scattering	Match radial waves to find exact phase shifts and a scattering length.
Hard-sphere scattering	Compare a repulsive boundary with low-energy waves and high-energy diffraction.
Gaussian potential in first Born theory	Turn a smooth potential into a calculable angular distribution and test the approximation.
Yukawa potential in first Born theory	Follow how screening affects momentum transfer, forward scattering, and integrated cross sections.
Low-energy s-wave scattering	Compare scattering-length and effective-range predictions with an exact square well.
A square-well resonance	Compare phase motion, real-energy peaks, fitted widths, and an outgoing-wave pole.

Learn through comparisons

The delta potential is a useful first problem because the matching algebra is short. Attractive and repulsive couplings of equal magnitude give the same reflection and transmission probabilities, but different phases and pole structure. Probabilities alone therefore do not tell the whole story.

Next compare the square well with the hard sphere. Both admit partial-wave matching, but an attractive well can be tuned near a bound-state threshold, while a repulsive hard sphere has no bound state. This makes them useful checks on overly broad interpretations of the scattering length.

Then compare the Gaussian and Yukawa Born results with stronger scattering calculations. An analytically correct Fourier transform is only the first step: it must approximate the full distorted-wave problem. The phase-shift computation provides a numerical route for testing a short-range central potential.

Finally study the low-energy and resonance examples together. They show why the peak of a plotted cross section, the motion of a phase shift, and a pole after analytic continuation need not supply identical energies or widths.

Check the quantity you intend to use

For one-dimensional reflection, verify the signed currents before forming nonnegative probabilities. For elastic partial waves, test unit modulus of $S_\ell$ and convergence of the channel sum. For a threshold expansion, vary momentum while keeping the range scales visible. For a pole, check the outgoing boundary condition and the analytic branch.

These checks constrain different errors. A curve can look smooth while using the wrong flux normalization, and a converged numerical solver can faithfully solve the wrong boundary-value problem. The convention dictionary helps when a result disagrees by a sign or a normalization factor.

Exercises

Three hard-sphere areas. For distinguishable particles scattering from a hard sphere of radius $R$, match the limits $4\pi R^2$, $\pi R^2$, and $2\pi R^2$ to low-energy quantum scattering, classical reflection, and high-energy quantum scattering. Why do they differ?

Solution

The low-energy quantum result is $4\pi R^2$, from the isotropic $s$-wave with scattering length $R$. Classical reflection gives the geometric area $\pi R^2$. High-energy quantum scattering approaches $2\pi R^2$, with forward diffraction contributing alongside the classical reflected part. The quantities refer to different limiting wave behavior, not inconsistent normalizations of one limit.

A Born comparison. Two programs agree to ten digits on a first Born amplitude. Does that show that the physical amplitude is accurate to ten digits?

Solution

No. The agreement tests evaluation of the same approximation. Compare with an exact model, a converged radial solution, or controlled higher-order information to assess the omitted scattering processes. Numerical precision and physical approximation accuracy are different questions.

References

• Joachain, C. J. Quantum Collision Theory. 3rd ed., North-Holland (1983). Potential-scattering examples and approximation methods.
• Newton, R. G. Scattering Theory of Waves and Particles. 2nd ed., Dover (2002; reprint of the 1982 edition). Thresholds, diffraction, and resonance structure.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Exact models and partial-wave calculations.