Relation Cross-Scans

Weighted Energy Contribution Cross-Scan

A weighted-energy-contribution cross-scan varies the state energy across rows while holding the weights fixed, showing how each state's p_i * E_i contribution to the average moves row by row. The arithmetic is exact only for the stated energies, weights, and rows in the ideal ensemble model. Real systems add preparation bias, finite samples, degeneracy/model choices, thermalization limits, detector/counting calibration, finite precision, and measurement uncertainty.

The average-energy numerator is built from three visible weighted terms. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Energy contribution row 1

Energy 0 times weight 4 contributes 0 joules to the numerator.

1⋅0⋅4=01\cdot 0\cdot 4=0
Weighted energy row 1Each row is one term of the checked average-energy numerator.E=0, weight=4, contribution=0

Energy contribution row 2

Energy 2 times weight 2 contributes 4 joules to the numerator.

1⋅2⋅2=41\cdot 2\cdot 2=4
Weighted energy row 2Each row is one term of the checked average-energy numerator.E=2, weight=2, contribution=4

Energy contribution row 3

Energy 4 times weight 1 contributes 4 joules to the numerator.

1⋅4⋅1=41\cdot 4\cdot 1=4
Weighted energy row 3Each row is one term of the checked average-energy numerator.E=4, weight=1, contribution=4

All weighted terms make the average-energy numerator

The three contributions sum to eight joules. Dividing by the same partition total seven gives the average energy eight sevenths of a joule.

⟨E⟩=0+4+47=87\langle E\rangle=\frac{0+4+4}{7}=\frac{8}{7}
Weighted energy contribution cross-scanThe numerator is shown as three exact source-bound terms.0 + 4 + 4 over Z=7 gives average energy 8/7