QUANTUM EVIDENCE / MAGNETIC MOMENT

Atomic Magnetic-Moment Comparator

Compare two ideal orbital–spin manifolds in the same magnetic field. Record state-selected atom counts, locate both resonances and separate the electron spin factor from a shared field drift.

An illustrative nuclear-spin-free doublet with orbital angular momentum one and spin one half, fixed orbital factor one, linear Zeeman shifts and a calibrated incoherent Lorentz transfer response. No real gallium hyperfine spectrum, coherent pulse, field-gradient trajectory, atomic-structure correction or QED calculation is solved. Inspired by magnetic-factor comparisons; not a refit of Kusch–Foley data.

Ideal state preparation → transverse RF interaction in a shared vertical field → state selection → atom counter. The second state selection is acquired sequentially on the same apparatus. No calculated deflection trajectories are shown.

OBSERVE / KEEP THE RECORD

Compare the two resonances.

At each coordinate: a state-selected count and a separate whole-beam reference for each branch. All four exposures are independent Poisson observations. Scan playback is accelerated.

Ready. Predict, then acquire.

STATE-SELECTED RATES / PER SECOND

A field shift or a factor shift?

Applied RF · first branch

0

Applied RF · second branch

0

$x/\mathrm{kHz},\quad N/\tau\;(\mathrm{s^{-1}})$

Teal: first branch. Gold: second branch. Points: acquired rates. Fine curves: source expectation. Purple dashes: inference from records. Gray: saved observations. Click the plot to move the probe.

INFER / SOURCE PARAMETERS EXCLUDED

Infer field and spin factor together.

Inferred spin factor

\text{—}

Inferred shared field

\text{—}

Spin-factor-two deviance / degree

\text{—}

First resonance · estimate

\text{—}

Second resonance · estimate

\text{—}

Fit deviance / degree

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$r_i=(N_i-\widehat\mu_i)/\sqrt{\widehat\mu_i}$

Each center and its source rate are profiled from the joint signal/reference likelihood. Both centers are searched over the full displayed interval. At least sixteen distinct coordinates are required. Deviance per degree is descriptive; no confidence interval or discovery significance is reported.

COMPARE / RESTORE

Latest six raw observations
#	Offset / kHz	First / ref	Second / ref

CSV includes raw observations, seed, units and source/instrument settings. Source settings are simulator metadata, excluded from inverse estimation. One saved run lasts for this page session.

MODEL / EVIDENCE

What this measurement leaves out.

\begin{aligned}P_J&=\frac{0.4}{1+4(\nu-\nu_J)^2/w^2}\\N_J&\sim\operatorname{Poisson}[\tau(S_JP_J+b)]\end{aligned}

Known transfer contrast 0.4, width and background 8 per second; independently unknown source rates. Linear LS coupling with no nuclear spin, hyperfine mixing, isotope shifts, state lifetime or atomic-structure corrections. The two electronic manifolds and selectors are idealized; this is not calibrated gallium. A small input spin anomaly is supplied, not calculated from QED. The fixed coarse field only labels RF coordinates.

Steck · LS / Zeeman conventions · Kusch–Foley 1948 · author abstract · historical method only; no original data refit.

Read the historical evidence Measure an oil-drop charge

Start with the essentials

Focus question: Two resonances. One unknown field.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

Two magnetic factors

\begin{aligned}g_{1/2}&=\frac{4-g_S}{3}\\g_{3/2}&=\frac{2+g_S}{3}\\\nu_J&=g_J\frac{\mu_B B}{h}\end{aligned}

Positive factor magnitudes, orbital factor one. Electron magnetic moment opposes angular momentum. Nuclear spin is omitted.

A calibrated frequency coordinate

\begin{aligned}\nu_J&=g_J^{(0)}(\nu_c+x)\\\nu_c&=\frac{\mu_B}{h}(0.5\,\mathrm{mT})\\g_J^{(0)}&\in\{2/3,4/3\}\end{aligned}

The scan coordinate is a known frequency mapping; it does not use the unknown true source field or spin factor. Each branch has its own physical RF frequency.

Eliminate the shared field

\begin{aligned}r&=\widehat\nu_{3/2}/\widehat\nu_{1/2}\\\widehat g_S&=\frac{4r-2}{1+r}\\\widehat B&=\frac{h(\widehat\nu_{1/2}+\widehat\nu_{3/2})}{2\mu_B}\end{aligned}

Fit both resonance centers and independent source rates from four Poisson exposures per coordinate; then invert their frequencies.

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
Find the two centers
Scan both state selections through the same coarse frequency interval.
What to observe: The two scaled peaks respond differently to the spin factor.
02
Move the field
Save the run and select the shared-field drift preset.
What to observe: Both centers move, while the inferred spin factor stays consistent within sampling variation.
03
Compare a constrained model
Choose spin factor two and acquire a fresh scan.
What to observe: The alternative fits only the shared field and two rates. Deviance is descriptive, not a discovery significance.