Transition Direction Three-Row Scan - Atomic Spectra: Energy Levels and Line Series Step by Step

A transition direction scan checks whether flipping a transition's direction label changes its computed energy gap. Ground to upper reads 9 joules under emission and 9 joules under absorption; middle to upper -- chapter one's own case -- reads 5 joules under absorption exactly for the stated toy level ledger. Chapter one showed this once, for one pair; this scan generalizes the same fact to a second pair, confirming the direction label is bookkeeping about which way a photon travels, not a term that enters the subtraction. Real transition energies also depend on selection rules and finite level linewidths that this toy model does not capture.

Chapter one showed direction doesn't move the gap for one pair; this scan generalizes the fact to a second pair. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A new pair sets a new baseline gap

Ground to upper is a pair chapter one never used. Its emission gap is 9 joules.

E_{\text{upper}}-E_{\text{ground}}=9\ \mathrm{J},\quad direction=emission

Flipping direction never changes the gap, on any pair

Chapter one showed this once, for the middle-to-upper pair. Flipping ground-to-upper from emission to absorption leaves the gap exactly where it was; only the direction label moves.

\begin{array}{c|c|c}\text{pair}&\text{direction}&gap\\g\to u&\text{emission}&9\\g\to u&\text{absorption}&9\\m\to u&\text{absorption}&5\\\end{array}

The middle-to-upper pair is chapter one's own case

This third row is the exact pair and direction chapter one already checked: gap 5 joules, direction absorption. The new ground-to-upper rows above generalize the same fact to a pair chapter one never tried.

E_{\text{upper}}-E_{\text{middle}}=5\ \mathrm{J}