Reserve fresh temperatures after equal-budget heat-capacity calibration

Define the state and evidence

For a quantum oscillator with levels En=(n+1/2)hbar omega, the geometric partition sum gives mean energy U=hbar omega[1/2+1/(exp(Theta/T)−1)], where Theta=hbar omega/kB. Differentiate at fixed oscillator frequency to obtain C/kB=F(Theta/T), with F(x)=x²exp(x)/(exp(x)−1)². The temperature-independent zero-point energy contributes no heat capacity.

Derive the quantitative relation

The classical oscillator has C/kB=1. The quantum factor approaches one at high temperature and falls toward zero when temperature is small compared with Theta. This single-frequency independent-oscillator model does not reproduce every real solid, especially the full low-temperature phonon spectrum. Its controlled comparison is a model test, not an identity claim.

Test a changed model or boundary

Here Theta is independently stipulated, but response amplitude A is unknown to the analyst. Both quantum response A F(Theta/T) and a classical constant response fit one amplitude using only the200 K indication. Freeze predictions at100 and400 K before revealing those records. Giving one model extra fitted parameters or a withheld record would invalidate the intended equal-budget comparison.

Limit the inference

With calibration hard error epsilon, the quantum prediction bound at T is epsilon F(Theta/T)/F(Theta/200), while the classical constant has bound epsilon. Add the independent fresh-reading bound before comparing a residual. A temperature-dependent response gain could mimic suppression and requires an independent calibration test; a successful finite comparison does not prove a unique oscillator spectrum.

Worked original case

Take Theta=300 K and generating amplitude2. The200 K indication is2F(1.5). The quantum fitted amplitude is that reading divided byF(1.5), then predicts2F(3) at100 K and2F(.75) at400 K. The classical family predicts the same calibrated200 K indication at both new temperatures. Both used exactly one calibration datum.

Investigation

Derive the energy and heat-capacity law from the geometric sum. Fit both amplitudes at the same temperature, retain two frozen predictions and propagate the direct calibration ratio. Then propose a temperature-dependent gain alternative and the independent measurement needed to test it.

Changed-case practice

Try the changed case before feedback

If quantum prediction factor Ffresh/Fcal is0.4 and calibration hard error is0.02 response units, find the prediction error bound before adding fresh error.

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 and numerical packets are original. Supplied theoretical distributions and synthetic records are not actual laboratory measurements. No nuclear, electrical, heating, beam or material-handling procedures are requested. Retain first reasoning, actual criterion critique, revision and a separately assigned unfamiliar task in the course evidence workspace; public keys do not authenticate mastery.