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.

PREPARE / DEFLECT / INFER

A molecular beam carries more than proton spin.

Calibrated transverse field gradient

G/(\mathrm{T}\,\mathrm{m}^{-1})

2200

Source mean speed · excluded from fit

\bar v/(\mathrm{m}\,\mathrm{s}^{-1})

1000

Lognormal speed spread · excluded from fit

s_{\ln v}

0.18

Source transverse standard deviation · fitted from reference

\sigma_y/\mathrm{mm}

0.1

Calibrated ortho fraction · rotation one

f_o

0.75

Source effective proton moment · excluded from fit

\mu_p/\mu_N

2.793

Triggered molecules per field setting

N_{\rm trig}

1200

Inspect retained event · no acquisition change

i

0

Inspection keeps every record. Acquisition changes clear current data; a saved comparison remains in memory.

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

\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 Barnes, Bray & Ramsey · Rotational moment NIST · CODATA

Single-Electron Penning Trap & Spin Readout Quantum evidence collection

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 misconceptionRecovering a source parameter independently confirms the law.

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

Run the experiment

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.
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.
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.