Quantum evidence · electric fields

Stark Splitting Spectrometer

How can a split hydrogen line measure an electric field? By the size of the splitting. An electric field splits hydrogen's excited levels by an amount proportional to the field, so one line becomes several polarized components; this is the linear Stark effect. The shifted outer components are polarized along the field. Turn the polarizer across it and only the unshifted middle line remains, which says nothing about the field.

Send excited hydrogen atoms through an electric field and resolve the polarized components of their ultraviolet spectrum. Count photons across the scan, then work out the field strength from the measured splitting.

Hydrogen is modeled as a spinless atom with a fixed nucleus in a pure Coulomb potential, and the Stark shift is computed to first order. The excited states are equally populated and the polarization analyzer is ideal. Fine structure, Lamb shifts, ionization and real discharge kinetics are left out, so the spectrum is not a precision prediction for real hydrogen.

Schematic geometry with enlarged gaps and parts. The colors and moving markers do not show visible light or measured trajectories. Drag to orbit.

Exposure: signal and reference counted separately

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

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Outer-line shift per unit field

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Measured spectrum in 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 with a joint Poisson likelihood

Estimate from both exposures.

Inferred field

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Inferred source rate

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Deviance per degree

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Unsplit alternative

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

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 its settings and seed, for as long as this page stays open.

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

The latest six points are shown. The CSV has every raw count, exposure, setting and unit, plus the seed. Moving the probe never changes data already taken.

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 excited states are treated as spinless, with equal populations maintained externally. Viewed at right angles to the field, the spectrum has two shifted components polarized parallel to the field and a central component polarized perpendicular to it. The ideal analyzer changes their weights but keeps the total flux constant. With the analyzer set perpendicular, the counts carry no information about the field.

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: How can a split hydrogen line measure an electric field?
One-sentence intuition: Fit the signal counts and the reference counts together with one Poisson likelihood, and estimate the source rate from the same data.

Core mathematical model

Stark shift of a degenerate pair

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

The field couples the two states, so diagonalize the pair first and take the line positions from the new energies. To first order, the two transverse orbital states do not shift.

Line weights behind the analyzer

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

These normalized weights assume equal populations in the upper states, viewing at right angles to the field and an ideal analyzer.

Counts in a channel of finite width

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 at half maximum is converted to a Gaussian standard deviation. Integrating over each channel keeps the total probability unchanged.

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]

The background rate is known in advance. The source rate and the spectral parameter are fitted together, and the scatter in the reference counts is included. Counts are drawn directly from the Poisson distribution, not rounded from a Gaussian approximation.

Common difficulties

A good fit does not test the law

Typical misconceptionIf the fit matches the data, that independently confirms the physical law built into the simulator.

Better mental modelThe simulator generates the data from a response it is given. Recovering the parameter shows that this simulated measurement and analysis work together. The deviance and residuals describe how well the curve fits; they are not confidence intervals, significance tests or an outside scientific certification.

Run the experiment

01
Resolve the lines, then estimate the field
Scan the default spectrum. Compare the inferred field with the field you set, then save the run.
What to observe: The two outer lines move in opposite directions. The estimate comes only from the counts and the known instrument response; it never reads the field setting.
02
Rotate the analyzer
Choose the Parallel polarization preset and scan. Then choose No field information and scan again.
What to observe: With the analyzer perpendicular to the field, the shifted lines disappear. The central line does not move with the field, so it cannot tell you the field strength.
03
Lower the resolution
At a small field, increase the instrument FWHM. Compare the new spectrum with the saved run and with the unsplit alternative.
What to observe: When the lines blend, the counts pin down the field to a different degree. A small residual on its own does not prove that only one parameter value fits.

Stark Splitting Spectrometer

Give the spectrum a record.

Which structure survives measurement?

Estimate from both exposures.

Keep the original counts.