---
title: "Molecular-Beam Proton Magnetic Moment Interactive Lab | Sandbox Physics"
description: "A molecular beam carries more than proton spin. Interleave field-off and field-on molecular events. Retain upstream flight times and detector positions, then infer an effective proton moment with a declared rotational contribution and the measured velocity distribution."
canonical_url: "https://sandboxphysics.com/labs/molecular-beam-proton-moment/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/molecular-beam-proton-moment/"
last_modified: "2026-10-18"
---

HYDROGEN MOLECULES / MAGNETIC DEFLECTION

# Molecular-Beam Proton Magnetic Moment

Interleave field-off and field-on molecular events. Retain upstream flight times and detector positions, then infer an effective proton moment with a declared rotational contribution and the measured velocity distribution.

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

Prepared ground-vibrational para hydrogen with rotation zero and ortho hydrogen with rotation one. High-field uncoupled nuclear and rotational projections, known ortho fraction and rotational moment, constant-gradient paraxial force, Gaussian reference beam. Molecular shielding, diamagnetism, higher rotation, spin rotation, collisions, fringe fields and calibration uncertainty are omitted. Synthetic events are not Stern’s historical data or a precision free-proton measurement.

Beam travels left to right; positive magnetic moment deflects upward. Transverse displacement is enlarged 56 times relative to the longitudinal scale. Colored paths are source expectations at the mean speed, not recorded trajectories.

SYNTHETIC RECORDS / RETAINED OBSERVATIONS

### Alternate the field. Keep each trigger.

Ready to acquire synthetic records.

Upstream timing triggers alternate field-off / field-on. Detector misses retain a trigger and flight time with an empty position. The known gradient, ortho fraction and rotational moment are calibration inputs; source moment, mean speed and beam width are excluded from inference.

RAW OBSERVATIONS

### A reference beam before a magnetic inference.

$y/\mathrm{mm},\quad N_i$

Green: field off. Brown: field on. Equal detector bins retain zero counts. Gray dashes show saved field-on counts. Missed events remain in the CSV and censored likelihood. No ideal detector image is substituted for these counts.

INFERENCE / DECLARED CALIBRATIONS

### Estimate only what the records resolve.

Inferred effective proton moment

$\text{—}$

Fitted field-off center and width

$\text{—}$

Recorded detector acceptance

$\text{—}$

Second-moment cross-check · full acceptance only

$\text{—}$

Fixed rotational contribution

$\text{—}$

Inspected flight time and position

$\text{—}$

RETAINED EVENTS / PROFILE LIKELIHOOD

### Separate proton and rotational moments.

$\mu_p/\mu_N,\quad 2(\ell_{\max}-\ell)$

The likelihood mixes nine ortho magnetic components and a para reference, using each flight time. Center and width are fixed by field-off records; the profile omits calibration uncertainty and is not a confidence interval. Values above forty are clipped visibly at the top.

CHECK THE RECORD

### Velocity and deflection remain paired.

$t_{\rm flight}/\mathrm{\mu s},\quad y/\mathrm{mm}$

First 600 registered events; all triggers are retained in the export. Slower molecules deflect more. No source mean speed is used in the estimator.

Inspect the last eight raw records

Flight times are in microseconds; detector positions are in millimetres. Missed positions remain blank in CSV.

| Index | Field on | Flight | Position | Detected |
| --- | --- | --- | --- | --- |

Model, apparatus and primary evidence

Prepared ground-vibrational para hydrogen with rotation zero and ortho hydrogen with rotation one. High-field uncoupled nuclear and rotational projections, known ortho fraction and rotational moment, constant-gradient paraxial force, Gaussian reference beam. Molecular shielding, diamagnetism, higher rotation, spin rotation, collisions, fringe fields and calibration uncertainty are omitted. Synthetic events are not Stern’s historical data or a precision free-proton measurement.

$$
\begin{aligned}\frac{\mu_y}{\mu_N}&=2\frac{\mu_p}{\mu_N}m_I+g_Rm_N\\m_I,m_N&\in\{-1,0,1\}\\g_R&=0.88291\end{aligned}
$$

Nine equally populated high-field ortho components; para molecules in rotation zero have no permanent moment in this model.

$$
\begin{aligned}F_y&=\mu_yG\\y-y_0&=\frac{\mu_yGL(D+L/2)}{M v^2}\\K(v)&=\frac{\mu_NGL(D+L/2)}{M v^2}\end{aligned}
$$

The magnet is one tenth of a metre long, followed by half a metre of drift. The downstream detector faces the beam.

$$
v=\frac{\ell}{t_{\rm flight}},\quad \ell=0.10\,\mathrm m
$$

Two upstream timing gates supply the speed of every trigger, including detector misses. A lognormal source distribution is illustrative, not an effusive Maxwell law.

The two timing gates are upstream of the magnet and ten centimetres apart. The detector is half a metre beyond the ten-centimetre magnet and accepts positions within four millimetres of the axis. All event flight times are ideal; timing uncertainty is omitted. Known prepared ortho fraction is not inferred from the profile. The reported positive magnitude cannot determine the sign of the proton moment from a symmetric unpolarized beam.

$$
\begin{aligned}z&=\frac{y-y_0}{K(v)}\\V&=\operatorname{Var}(z)-\left\langle\frac{\sigma_y^2}{K(v)^2}\right\rangle\\V&=f_o\left[\frac83\left(\frac{\mu_p}{\mu_N}\right)^2+\frac23g_R^2\right]\end{aligned}
$$

A separate second-moment cross-check is shown only when every detector event is accepted. Finite acceptance enters the likelihood through Gaussian tail probability for each upstream trigger. No diamagnetic or shielding correction is applied; the inferred parameter is effective within this molecular model.

[Stern · Molecular rays](https://www.nobelprize.org/uploads/2018/06/stern-lecture.pdf) [Barnes, Bray & Ramsey · Rotational moment](https://journals.aps.org/pr/abstract/10.1103/PhysRev.94.893) [NIST · CODATA](https://physics.nist.gov/cuu/Constants/)

[Single-Electron Penning Trap & Spin Readout](https://sandboxphysics.com/labs/penning-electron-spin-readout/) [Quantum evidence collection](https://sandboxphysics.com/stories/quantum-evidence/)

Physics tutorial

## Molecular-Beam Proton Magnetic Moment

**Background** Interleave field-off and field-on molecular events. Retain upstream flight times and detector positions, then infer an effective proton moment with a declared rotational contribution and the measured velocity distribution.

**Why it matters** A constant is inferred through a calibrated response and retained observations.

Start with the essentials

- **Focus question**: A molecular beam carries more than proton spin.
- **One-sentence intuition**: Keep source assumptions separate from estimates made from records.

### Core mathematical model

#### Molecular magnetic moment

$$
\begin{aligned}\frac{\mu_y}{\mu_N}&=2\frac{\mu_p}{\mu_N}m_I+g_Rm_N\\m_I,m_N&\in\{-1,0,1\}\\g_R&=0.88291\end{aligned}
$$

Nine equally populated high-field ortho components; para molecules in rotation zero have no permanent moment in this model.

#### Deflection

$$
\begin{aligned}F_y&=\mu_yG\\y-y_0&=\frac{\mu_yGL(D+L/2)}{M v^2}\\K(v)&=\frac{\mu_NGL(D+L/2)}{M v^2}\end{aligned}
$$

The magnet is one tenth of a metre long, followed by half a metre of drift. The downstream detector faces the beam.

#### Measured velocity

$$
v=\frac{\ell}{t_{\rm flight}},\quad \ell=0.10\,\mathrm m
$$

Two upstream timing gates supply the speed of every trigger, including detector misses. A lognormal source distribution is illustrative, not an effusive Maxwell law.

### Common difficulties

#### A simulation supplies its law

**Typical misconception** Recovering a source parameter independently confirms the law.

**Better mental model** These synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

### Run the experiment

1. 01

   #### Measure the reference

   Acquire the mixed-source preset; compare field-off and field-on positions.

   **What to observe:** Reference records fit the common center and Gaussian width, rather than using their source settings.

2. 02

   #### Keep the velocity distribution

   Use the broad-speed preset and inspect the flight-time scatter.

   **What to observe:** The likelihood uses each measured velocity and includes the rotational contribution and finite detector misses.

3. 03

   #### Remove the nuclear signal

   Compare pure para hydrogen or zero gradient.

   **What to observe:** A narrow reference beam is insufficient to identify a proton moment without force and ortho population.

Physical conclusion

Prepared ground-vibrational para hydrogen with rotation zero and ortho hydrogen with rotation one. High-field uncoupled nuclear and rotational projections, known ortho fraction and rotational moment, constant-gradient paraxial force, Gaussian reference beam. Molecular shielding, diamagnetism, higher rotation, spin rotation, collisions, fringe fields and calibration uncertainty are omitted. Synthetic events are not Stern’s historical data or a precision free-proton measurement.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired 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)