Compare two spectral fits with equal data budgets and a new observation

Define the state, source and evidence

A pair of level indications can be fitted by more than one family. Compare a quadratic model y=A n²+B with an affine-in-n model y=C n+D. Both have two unknown parameters. Give both only the level labels n=1,2 and their indicated energies y1,y2. The generating coefficient and offset are revealed teaching controls, not independent inputs to either inverse fit.

Derive the quantitative relation

The quadratic fit has A=(y2−y1)/3 and B=y1−A, so its frozen n=3 prediction is (8y2−5y1)/3. The affine fit predicts 2y2−y1. Reveal the third record only after those predictions are recorded. Renaming one fitted record as a test does not restore independence.

Check normalization, limits and inference

If the two calibration indications have independent hard errors bounded epsilon, direct prediction bounds are 13epsilon/3 for the quadratic family and 3epsilon for the affine family. Their fitted parameter intervals are correlated because they reuse the same readings. Add a fresh-record error separately before assessing a residual. These are conditional bounds, not confidence levels or an unrestricted proof of the potential.

Compare a changed premise

An energy-zero offset changes indicated absolute levels without changing energy gaps. Even three matching levels cannot uniquely identify every possible potential. A separate spatial test adds different information: a uniform position density puts probability 1/4 in the left quarter, while a normalized infinite-well ground state gives 1/4−1/(2π). Observing that interval would test a spatial prediction rather than another copy of the calibration spectrum.

Worked original case

For quadratic coefficient 1 eV and offset 0.5 eV, the first two records are 1.5 and 4.5 eV. The quadratic family predicts 9.5 eV at n=3; the affine family predicts 7.5 eV. With epsilon=0.01 eV their calibration-only prediction bounds are about 0.0433333 and 0.03 eV. The fresh third record is 9.5 eV within this supplied generator, not independent physical confirmation.

Investigation

Fit both hypotheses with exactly the same two records and freeze their third predictions. Propagate the original reading bounds with signed weights. Then design a separate left-quarter probability test and list one calibration or potential alternative that the available data still cannot exclude.

Changed-case practice

Try the changed case before feedback

For y1=2 eV and y2=5 eV, find the quadratic family prediction at n=3 in eV.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions, numerical packets and diagrams are original. These models do not establish actual beam, radiation, vacuum, optical or electrical laboratory performance. Public feedback is practice, not authenticated independent mastery. Preserve first reasoning, criterion critique, revision and a genuinely new assigned packet in the course evidence workspace.