---
title: "Gravitational Quantum Bouncer Interactive Lab | Sandbox Physics"
description: "How tall must a slit be for a neutron to get through? Taller than the lowest gravitational quantum state, which reaches about 14 micrometres above the mirror."
canonical_url: "https://sandboxphysics.com/labs/gravitational-quantum-bouncer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gravitational-quantum-bouncer/"
last_modified: "2026-10-27"
---

Q017 · Mirror / absorber / neutron counts

# Gravitational Quantum Bouncer

**How tall must a slit be for a neutron to get through?** Taller than the lowest gravitational quantum state, which reaches about 14 micrometres above the mirror. As the slit opens, transmission rises in steps tied to these states. Each state is spread continuously in height, though, so the edge of a step is not itself an energy level.

Raise an absorber above a horizontal mirror and count the neutrons that get through at each height. Compare a mixture of four states with the ground state alone. Then change the flight time to see why the edge of the transmission curve is not an energy level.

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

Ultracold neutrons fly above an ideal hard mirror in uniform gravity. They are prepared in an incoherent mixture of four vertical states, and an assumed overlap-loss model removes neutrons during the horizontal flight. Distortion of the states by the absorber, scattering from roughness and mixing between modes are left out. The density varies continuously with height; the classical turning heights are not shelves where neutrons sit.

02 / Counts first, then the curve

## The gap opens gradually.

Model reference

$h/\mathrm{\mu m}$ · Blue: current records. Amber: last saved other sample. Dashed: optional model. Drag to inspect; the slider supports keyboard input.

At this probe setting

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No recorded particles yet.

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95% pointwise Wilson interval. It is not a simultaneous band and carries no guarantee under optional stopping.

Latest 60 raw outcomes

Blue: transmitted. Gray: reflected or absorbed. No hidden trajectory is recorded.

03 / What can these records tell us?

## The measured edge is not the energy level itself.

Model only: each bar shows its contribution before and after the absorber. Absorption changes populations, not the energy markers.

Scan interval where half transmission is first seen

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Independent model comparison

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Model only; nothing new is recorded

## Discrete energies. Continuous height.

$z/\mathrm{\mu m}$ · Drag the absorber. Gray: incoming density. Blue: surviving unnormalized density per incident neutron. Orange markers: classical turning heights.

$E_n/\mathrm{peV}$ · Numerically computed eigenstates above a hard mirror. The incoming mixture contains only the first four.

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04 / Save the evidence

## Save before changing the experiment.

Eight frozen samples. Restoring inspects the original records; acquiring again starts a new run. CSV contains individual outcomes, scan membership and original settings.

Physical model, units and sources

$$
\ell=0.12\,\mathrm m,\quad\psi(0)=0,\quad V(z)=m_ngz
$$

The ground-state scale uses the neutron mass and SI constants. A 360-point finite-difference solution extends to eighteen gravity lengths. The absorber is a diagonal phenomenological loss channel proportional to each unperturbed mode’s probability above the gap; it does not impose a second hard wall. Coherent mixing, distorted eigenstates, real roughness, energy-dependent material absorption, loading efficiency and neutron decay are omitted. Preparation is ideal and the detector has unit efficiency. Geometry magnification is anisotropic; vertical position is a probability distribution, never a hidden ballistic track.

[Nesvizhevsky et al. · mirror, absorber and transmitted neutrons](https://arxiv.org/abs/hep-ph/0306198) · [Voronin et al. · absorbing waveguide and limits of a simple loss model](https://arxiv.org/abs/quant-ph/0512129)

Physics tutorial

## Gravitational Quantum Bouncer

**Background** Raise an absorber above a horizontal mirror and count the neutrons that get through at each height. Compare a mixture of four states with the ground state alone. Then change the flight time to see why the edge of the transmission curve is not an energy level.

**Why it matters** A boundary changes which waves are allowed, and a detector only ever records a finite sample.

Start with the essentials

- **Focus question**: How tall must a slit be for a neutron to get through?
- **One-sentence intuition**: Even when the energies are discrete, the position distribution is continuous.

### Core mathematical model

#### Gravity length

$$
z_0=\left(\frac{\hbar^2}{2m_n^2g}\right)^{1/3}
$$

About 5.87 micrometers at Earth gravity.

#### States above a hard mirror

$$
\psi_n(z)\propto\operatorname{Ai}(z/z_0-a_n),\quad E_n=m_ngz_0a_n
$$

The magnitudes of the Airy zeros set the energies. The probability density reaches above the classical turning height.

#### A simple absorber model

$$
S(h)=\sum_n w_n\exp\!\left[-\Gamma\frac{\ell}{v_x}\int_h^\infty|\psi_n(z)|^2\,dz\right]
$$

This is a phenomenological loss that acts on each state separately, not a solution for a rough absorbing waveguide. The mirror is twelve centimeters long.

### Common difficulties

#### What was measured?

**Typical misconception** Neutrons sit only on discrete height shelves.

**Better mental model** The energies are discrete, but each state spreads continuously over height. The detector only records whether a neutron survives past the absorber.

#### An idealized model

**Typical misconception** Agreement here certifies a real device.

**Better mental model** Ultracold neutrons fly above an ideal hard mirror in uniform gravity. They are prepared in an incoherent mixture of four vertical states, and an assumed overlap-loss model removes neutrons during the horizontal flight. Distortion of the states by the absorber, scattering from roughness and mixing between modes are left out. The density varies continuously with height; the classical turning heights are not shelves where neutrons sit.

### Run the experiment

1. 01

   #### Predict

   Choose the four-state mixture. Predict whether a wider slit can lower the transmission.

   **What to observe:** Keep the reference curve hidden until you have recorded data.

2. 02

   #### Scan and save

   Scan heights and save. Prepare only the ground state and repeat.

   **What to observe:** A saved sample keeps the settings that produced it.

3. 03

   #### Find the limits

   Increase horizontal speed at the same height. The state energies remain fixed, but less flight time means less loss.

   **What to observe:** The height at which half the neutrons get through summarizes the data; it is not a measurement of energy.

Physical conclusion

Ultracold neutrons fly above an ideal hard mirror in uniform gravity. They are prepared in an incoherent mixture of four vertical states, and an assumed overlap-loss model removes neutrons during the horizontal flight. Distortion of the states by the absorber, scattering from roughness and mixing between modes are left out. The density varies continuously with height; the classical turning heights are not shelves where neutrons sit.

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)