Two-State Populations

Population Ratio Uses the Same Weights

The population ratio between the two states is just the ratio of their weights, n_high / n_low = w_high / w_low, with the partition total canceling out. That cancellation is exact only for stated weights, the two listed states, fixed temperature/model, and equilibrium sampling. Real systems add preparation bias, finite sampling, calibration, degeneracy, and uncertainty.

The population ratio is another view of the same weight ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The ratio uses the same weights

Upper over lower is one over three.

NupperNlower=13\frac{N_{\rm upper}}{N_{\rm lower}}=\frac{1}{3}
Population ratioThe ratio is checked from the same weights.energies=lower=0J|upper=2Jweights=lower=3|upper=1lowerFraction=3/4upperFraction=1/4upperLowerRatio=1/3weightOrigin=explicit integer weights

No new population source is invented

The ratio and fractions share one two-state check.

Nupper:Nlower=1:3N_{\rm upper}:N_{\rm lower}=1:3
Shared sourceA copied ratio cannot replace the weight ledger.energies=lower=0J|upper=2Jweights=lower=3|upper=1lowerFraction=3/4upperFraction=1/4upperLowerRatio=1/3weightOrigin=explicit integer weights