Emission and Absorption Spectra

Bound-state energies are negative because zero is defined at the ionized state, so $-13.6$ eV is hydrogen's lowest level and the levels stack upward toward zero. A transition emits or absorbs a photon whose energy is the difference between two levels, so hydrogen falling from $n = 3$ to $n = 2$ emits $1.89$ eV at $656$ nm. Absorption takes energy in and moves the atom up a level while emission gives it out and drops the atom down, which is why an element shows the same wavelengths as bright lines or as dark gaps. Ionization is the jump to $E = 0$, costing $13.6$ eV from hydrogen's ground state, and any surplus becomes kinetic energy of the freed electron.

Four errors dominate. Ranking levels by the size of the number, so $-13.6$ eV reads as the highest level and every arrow on the diagram reverses. Setting a photon's energy equal to a single level rather than to the difference between two. Having the atom emit to climb and absorb to drop, or reading the dark lines in a stellar spectrum as light the gas produced. And treating ionization as a transition to the highest level drawn rather than to zero binding energy.