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.

Recorded trials $\text{—}$

Transmission at probe $\text{—}$

01 / Change the boundary and see what gets through

Raise the ceiling and count the neutrons that get through.

Scan the gap and save. Repeat with only the ground state. Then speed up the neutrons: the energy levels stay put, but the measured edge moves.

Neutrons fly horizontally above a hard mirror. The absorber covers the whole mirror, the vertical gap is greatly enlarged, and the detector sits after the mirror.

Absorber height

h/\mathrm{\mu m}

20

Ready. Each scan setting uses 200 independent incident particles.

Try a comparison

Gravity

g/(\mathrm{m/s^2})

9.81

Horizontal speed

v_x/(\mathrm{m/s})

5

Overlap loss rate

\Gamma/\mathrm{s^{-1}}

160

Vertical preparation

Sampling settings

Sampling seed

s_0

1729

Moving the probe preserves readings. Changing the device or preparation starts a fresh sample; save first. Nothing runs by itself.

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 · Voronin et al. · absorbing waveguide and limits of a simple loss model

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 misconceptionNeutrons sit only on discrete height shelves.

Better mental modelThe 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 misconceptionAgreement here certifies a real device.

Better mental modelUltracold 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

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.
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.
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.