A quantum system can have separated allowed energies instead of a continuous ladder. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Allowed energies are separated

The diagram shows allowed energy levels, not every height between them. The labels are training levels, not real atom constants.

allowed levels are separated\text{allowed levels are separated}
Discrete energy levelsThree exact ratio levels are shown.groundmiddleupper3/45/36

A shifted level table shows the gaps

Taking the ground level as the zero reference, the same three allowed levels become a small exact table. The spacing is not uniform.

levelE00134289\begin{array}{c|c}\text{level}&E\\0&0\\1&\frac{3}{4}\\2&\frac{8}{9}\end{array}
Discrete energy levelsThe table is the shifted version of the drawn levels.groundmiddleupper3/45/36

A level gap has an exact size

The middle to ground gap is 3/4 of the chosen energy unit.

ΔEmiddle to ground=34\Delta E_{\text{middle to ground}} = \frac{3}{4}
Discrete energy levelsThe transition arrow is the computed level gap.groundmiddleupper3/45/36