---
title: "Partial-Wave Scattering Interactive Lab | Sandbox Physics"
description: "Why does scattering off a single round target show so many minima? Because the partial waves interfere."
canonical_url: "https://sandboxphysics.com/labs/partial-wave-scattering/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/partial-wave-scattering/"
last_modified: "2026-10-27"
---

Q016 · Phase shifts, angles and cross sections

# Partial-Wave Scattering

**Why does scattering off a single round target show so many minima?** Because the partial waves interfere. Each one scatters with its own phase shift, and their complex amplitudes add before the sum is squared, so they reinforce at some angles and cancel at others. The total cross section is a separate quantity: one number for all angles together.

Send a beam of a single energy at a central potential. Move a ring-shaped detector, count the scattered particles, and tell interference in the angular pattern apart from a change in the total amount of scattering.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Particles with the electron mass scatter elastically and nonrelativistically from an attractive spherical square well or a hard sphere. Each exposure is an independent Poisson count, the incident fluence is uniform, and the ring detector is ideal. The model has no Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

Simulated detector exposures

## Count what the detector actually accepts.

$\theta/\mathrm{deg},\quad \overline{d\sigma/d\Omega}\,/\,(\mathrm{nm^2/sr})$ · Blue: current scan. Amber: saved comparison. Drag to choose the angle.

Show the model prediction

$\text{—}$

Counts and uncertainty

Each cell is one simulated exposure. Darker cells contain more counts. The amber curve uses the most recent saved sample other than the active scan. Whiskers are approximate 95% Poisson score intervals; the detector reports a polar-band average.

A conserved probability current

## Every partial wave sits on the same circle.

$\operatorname{Re}t_l,\quad\operatorname{Im}t_l$ · Gold: largest partial cross section. Cyan: other waves.

Change the energy and follow the gold point around the circle.

$1$

Why probability flow gives a circle

$$
\begin{gathered}S_l=e^{2i\delta_l}=1+2it_l\\t_l=e^{i\delta_l}\sin\delta_l\\(\operatorname{Re}t_l)^2+\left(\operatorname{Im}t_l-\tfrac12\right)^2=\tfrac14\end{gathered}
$$

A real potential returns all incoming flux to the elastic channel. Changing the energy moves each amplitude around the circle. The top of the circle gives the largest possible partial cross section at that energy.

This is a nonrelativistic potential model. The dimensionless circle relation also occurs in relativistic partial waves once their phase-space normalization is included.

[Continue from probability flow to scattering amplitudes](https://sandboxphysics.com/stories/scattering-amplitudes/)

Partial cross sections, angular interference and saved samples

03 / Connect the shape to the result

## Different partial waves carry different phases.

$l,\quad \sigma_l/\mathrm{nm^2}$ · This decomposition comes from the model. It is not a measurement of individual partial waves.

$\text{—}$

The last three partial waves show how much the tail contributes. The scientific tests also check convergence by adding eight more waves.

Model view, separate from the counts

## Interference changes the angular pattern.

$\theta/\mathrm{deg},\quad |f(\theta)|^2/\mathrm{nm^2}$ · Solid: coherent sum. Dashed: the squared components added without their cross terms. The dashed curve is only a diagnostic; it is not a second physical scattering model.

$\text{—}$

$r/\mathrm{nm},\quad E/\mathrm{eV}$

04 / Save the evidence

## Save the settings with the records.

You can keep up to eight frozen samples. Restore one to see its original settings. Acquiring after a save starts a new sample. The CSV contains every zero outcome as well as every successful detection.

Model limits, units & sources

A real, finite-range central potential and elastic electron-mass scattering. The incident plane wave travels along positive x. The annular detector accepts every azimuth and a polar interval excluding two-degree beam ports at each pole. Radius and depth define a hypothetical target; no specific material is implied. Target and detector use schematic scales; the guide lines indicate directions rather than trajectories. The hard-sphere option replaces the interior by a zero-wavefunction boundary.

The partial-wave cutoff increases with incident momentum and radius. Detector integration uses Gaussian quadrature; independent angular exposures follow Poisson counting. The displayed dot set is at most 128 conditionally sampled arrivals from the last exposure. The raw CSV retains each full exposure count and fluence. Target recoil, absorption, inelastic channels, multiple scattering and Coulomb tails are omitted.

[Fitzpatrick · Partial waves](https://farside.ph.utexas.edu/teaching/qm/Quantum/node107.html) · [Phase shifts](https://farside.ph.utexas.edu/teaching/qm/Quantum/node109.html) · [Resonant scattering](https://farside.ph.utexas.edu/teaching/qm/Quantum/node112.html)

Physics tutorial

## Partial-Wave Scattering

**Background** Send a beam of a single energy at a central potential. Move a ring-shaped detector, count the scattered particles, and tell interference in the angular pattern apart from a change in the total amount of scattering.

**Why it matters** Symmetry restricts the amplitude. Measure what that restriction does, then change one part of the preparation.

Start with the essentials

- **Focus question**: Why does scattering off a single round target show so many minima?
- **One-sentence intuition**: Add the complex partial-wave amplitudes first and square the magnitude afterward. That is why each angle keeps its own interference terms.

### Core mathematical model

#### Scattering amplitude as a sum of partial waves

$$
f(\theta)=\frac1k\sum_{l=0}^{l_{\max}}(2l+1)e^{i\delta_l}\sin\delta_l P_l(\cos\theta)
$$

The polar angle is measured from the incoming direction, positive x, which is also the beam axis in the drawing.

#### Counts in a detector of finite size

$$
\langle N\rangle=\mathcal F\int_{\Delta\Omega}|f(\theta)|^2\,d\Omega
$$

The ring accepts every azimuth over a finite band of polar angles. Dividing the counts by exposure and solid angle estimates the average over that band, not the value at a single angle.

#### Total cross section, computed independently

$$
\sigma=\frac{4\pi}{k^2}\sum_l(2l+1)\sin^2\delta_l=\frac{4\pi}{k}\operatorname{Im} f(0)
$$

A real potential with only elastic scattering conserves flux, so integrating over angle and using the optical theorem must both agree with the partial-wave sum.

### Common difficulties

#### A picture is not a trajectory

**Typical misconception** The outgoing guide line is a computed classical path.

**Better mental model** The line only connects the scatterer to the angle the detector accepts. The counts come from complex wave amplitudes, not from classical paths.

#### What the model leaves out

**Typical misconception** If the numerical checks pass, a real apparatus has been reproduced.

**Better mental model** Particles with the electron mass scatter elastically and nonrelativistically from an attractive spherical square well or a hard sphere. Each exposure is an independent Poisson count, the incident fluence is uniform, and the ring detector is ideal. The model has no Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

### Run the experiment

1. 01

   #### Predict before counting

   Predict whether a larger round target can produce minima in the angular pattern, then record an angular scan.

   **What to observe:** The page begins with a seeded simulated angular scan. Your first acquisition starts a fresh sample. Each point counts arrivals in a finite detector band.

2. 02

   #### Change one physical cause

   Compare the low-energy and multi-wave presets. Then widen the detector and repeat at the same angle.

   **What to observe:** A wide aperture washes out narrow minima. The cross section and the conditional angular distribution are different observables.

3. 03

   #### Look at a case where the signal vanishes

   Remove the potential. Exposures with zero counts are still recorded. Then look at an impenetrable sphere on its own.

   **What to observe:** Export the raw records, zero outcomes included, and you can recompute the displayed estimate yourself.

Physical conclusion

Particles with the electron mass scatter elastically and nonrelativistically from an attractive spherical square well or a hard sphere. Each exposure is an independent Poisson count, the incident fluence is uniform, and the ring detector is ideal. The model has no Coulomb tail, target recoil, absorption, multiple scattering or calibrated material potential.

## Related experiments and articles

Scattering amplitudes · Reading path

### From probability flow to scattering amplitudes and bootstrap

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a scattering amplitude becomes a detector count**](https://sandboxphysics.com/stories/scattering-amplitudes/flux-and-amplitude/)
2. [Article **How a circle limits the strength of scattering**](https://sandboxphysics.com/stories/scattering-amplitudes/partial-wave-unitarity/)

Quantum evidence · Reading path

### Probability and exclusion

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Born: how a scattering direction became a probability**](https://sandboxphysics.com/stories/quantum-evidence/born-scattering-probability/)
2. [Lab **Quantum State & Born Measurement** Choose how to measure a qubit and predict the outcome probabilities](https://sandboxphysics.com/labs/qubit-state/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)