QUANTUM EVIDENCE / ELECTRIC FIELDS

Stark Splitting Spectrometer

Send excited hydrogen through an electric field. Resolve its polarized ultraviolet spectrum, acquire photon counts and infer the field from the recorded splitting.

Spinless, fixed-nucleus hydrogen in the first-order Coulomb model. Equal excited-state populations and an ideal analyzer are assumed. Fine structure, Lamb shifts, ionization and real discharge kinetics are omitted; this is not a precision hydrogen prediction.

Schematic geometry · enlarged gaps and components · colors and moving markers do not represent visible light or measured trajectories. Drag to orbit.

EXPOSURE / TWO INDEPENDENT COUNTS

Give the spectrum a record.

Each coordinate has a signal exposure and a separate reference: whole-line monitor. Exposure time controls Poisson noise; scan playback is accelerated.

Ready. Predict, then acquire.

Recorded points

0

Outer-line separation · source

0

Single outer shift per field

0

ACQUIRED SPECTRUM / COUNTS PER SECOND

Which structure survives measurement?

$\Delta\nu/\mathrm{GHz}$ · $N_i/\tau_i\;(\mathrm{s^{-1}})$

Teal: source expectation. Gold: acquired rates with fitted Poisson standard deviations, once a fit exists. Purple dashes: inference from records. Gray dots: saved run. Background is included.

Click or drag to move the probe; its slider supports keyboard input. The vertical scale follows the data.

INFERENCE / JOINT POISSON LIKELIHOOD

Estimate from both exposures.

Inferred field

\text{—}

Inferred source rate

\text{—}

Deviance per degree

\text{—}

Unsplit alternative

\text{—}

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

Gold: signal residuals. Blue: reference residuals. Known response, splitting/width controls and background; unknown source rate and spectral parameter. At least sixteen distinct probe positions are required. Descriptive residuals and deviance do not establish identifiability, confidence intervals or discovery significance.

Keep the original counts.

Save a full or partial run with settings and seed in this page session.

#	$\Delta\nu/\mathrm{GHz}$	Signal	Whole-line monitor	$\tau/\mathrm{s}$

Latest six points shown. CSV includes all raw counts, exposures, settings, units and seed. Changing the probe never changes acquired records.

Scientific scope, geometry & primary sources

The electric field points vertically; excited atoms cross the plate gap perpendicular to the photon collection axis. The analyzer angle is measured from the field. The ideal ultraviolet selector counts a 2.5 GHz channel after Gaussian broadening.

The four spinless excited states have equal, externally maintained populations. Observation perpendicular to the field gives two parallel-polarized shifted components and a perpendicular central component. An ideal analyzer preserves constant total flux as it changes their weights. At perpendicular polarization the records contain no field information.

This first-order, fixed-nucleus Coulomb calculation deliberately neglects fine structure, Lamb shifts, quadratic corrections, Doppler kinetics and ionization. At low fields it is not the spectrum of real hydrogen; even at high fields it is only a qualitative teaching limit. The reference frequency is the gross Lyman line; the plotted axis is detuning, not a high-accuracy absolute wavelength.

Independent Poisson exposures, fixed calibrated background and stable source. No drift, dead time or calibration uncertainty. The source follows the displayed model; successful recovery validates the simulator, not the historical discovery.

Fitzpatrick · linear Stark effect Cambridge · perturbation theory

Start with the essentials

Focus question: Split the line. Rotate the evidence.
One-sentence intuition: Keep the signal and reference counts. Fit their joint Poisson likelihood while estimating the source rate from the same records.

Core mathematical model

A degenerate pair

H'=eEz,\qquad \langle 2s|z|2p_0\rangle=-3a_0,\qquad \Delta E=\{ -3ea_0E,0,0,3ea_0E\}

Diagonalize the coupled pair before computing line positions. The two transverse orbital states remain unshifted to first order.

Polarization selects evidence

w_-=w_+=\tfrac12\cos^2\alpha,\qquad w_0=\sin^2\alpha,\qquad \Delta\nu_\pm=\pm\frac{3ea_0E}{h}

Equal upper-state populations and observation perpendicular to the field give these normalized weights after an ideal analyzer.

Count a finite channel

g_i=\sum_j w_j\int_{\nu_i-b/2}^{\nu_i+b/2}\frac{e^{-(\nu-\nu_j)^2/(2s^2)}}{\sqrt{2\pi}s}\,\mathrm d\nu,\qquad b=2.5\,\mathrm{GHz}

The instrument full width is converted to Gaussian standard deviation. Channel integration preserves probability.

Two independent exposures

N_i\sim\operatorname{Poisson}\!\left[\tau_i(Sg_i+b)\right],\qquad M_i\sim\operatorname{Poisson}\!\left[\tau_i(S+b)\right]

Background is independently known. Source rate and the spectral parameter are jointly inferred; reference uncertainty is included. Counts are never Gaussian-rounded.

Common difficulties

Recovery is not discovery

Typical misconceptionA good fit independently confirms the law used by the simulator.

Better mental modelThe source response is supplied. Parameter recovery validates this synthetic measurement loop. Descriptive deviance and residuals are not confidence intervals, significance tests or external scientific certification.

Run the experiment

01
Resolve, then infer
Scan the default spectrum. Compare the inferred field with its source setting, then save the run.
What to observe: The outer components move in opposite directions. The estimator reads the counts and known response, not the source field.
02
Rotate the analyzer
Choose parallel polarization, scan, then choose the no-field-information preset and scan again.
What to observe: The shifted components disappear under perpendicular polarization. An unchanged central line cannot identify the field.
03
Lose the resolution
Broaden the instrument at a small field. Compare the recorded spectrum with the saved run and the unsplit alternative.
What to observe: Blending changes how well the inverse problem is constrained. A small residual alone is not proof of a unique parameter.