Read bound energy levels and ionization thresholds
In the simple hydrogen approximation, bound energies are En=−13.6/n² eV relative to an electron and proton separated at rest. The negative sign encodes binding relative to this chosen zero; it is not a negative photon energy. A transition from a higher level to a lower level releases the positive difference Eupper−Elower. Reverse absorption requires the same gap when that transition is allowed. Ionization from level n instead requires reaching the continuum zero, an energy 13.6/n² eV.
Learning outcome: Compare upper levels 3,4,6 at fixed lower2, then switch the lower level to 1. Predict gap and wavelength direction before calculating. Compare a bound transition with ionization and identify which intensity or orbital claims remain unsupported.
Starting recap
Separate per-event energy, expectation rate and measurement indication. Check particle identities, conservation, threshold branches and calibration roles before making a source or detector claim.
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For nupper=4 and nlower=2, the energies are−0.85 and−3.4 eV. Emission releases2.55 eV with wavelength1240/2.55≈486.27451 nm. Ionizing from n=2 requires3.4 eV, a different question from this bound-to-bound transition.
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With the continuum zero convention, ionization from hydrogen n=2 needs:
Experiment
Compare upper levels 3,4,6 at fixed lower2, then switch the lower level to 1. Predict gap and wavelength direction before calculating. Compare a bound transition with ionization and identify which intensity or orbital claims remain unsupported.
Use energy differences rather than orbit pictures as spectral evidence
Worked teaching model · synthetic values · Hydrogen-like one-electron hydrogen approximation E_n=−13.6/n² eV with continuum zero. Supplied hc=1240 eV nm. Fine structure, recoil and selection probabilities omitted. Discrete horizontal energies are not a spatial orbit or a guarantee that every transition has equal intensity.
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Energy levels and the selected emission gap
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: level n=1 (solid).2: level n=2 (dashed).3: level n=3 (dotted).4: level n=4 (dotted).5: level n=5 (dotted).6: level n=6 (dotted).7: selected transition endpoints (dotted). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Energy levels and the selected emission gap
level n=1
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0 | -13.6 |
| 1 | -13.6 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
level n=2
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0 | -3.4 |
| 1 | -3.4 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
level n=3
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0 | -1.51111 |
| 1 | -1.51111 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
level n=4
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0 | -0.85 |
| 1 | -0.85 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
level n=5
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0 | -0.544 |
| 1 | -0.544 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
level n=6
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0 | -0.377778 |
| 1 | -0.377778 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
selected transition endpoints
| schematic horizontal coordinate (not distance) | energy relative to free electron (eV) |
|---|---|
| 0.5 | -0.85 |
| 0.5 | -3.4 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Upper bound-state energy | -0.85 | eV |
| Lower bound-state energy | -3.4 | eV |
| Emitted photon energy | 2.55 | eV |
| Photon wavelength | 486.275 | nm |
| Ionization threshold from lower level | 3.4 | eV |
| Atom energy change during emission | -2.55 | eV |
A spectral photon carries a difference of allowed energies, not either level energy by itself.
The same positive gap is needed for absorption from lower to upper and released in emission from upper to lower, when the transition is allowed. Negative bound-state energies refer to a chosen zero at separated electron/proton; ionization needs the positive distance to zero. The simple hydrogen formula omits fine structure, recoil and transition strengths. Horizontal lines are energy labels, not literal circular trajectories; modern orbitals describe probabilities. A thermal or electron-driven source can produce continuum as well as discrete emission, so one line alone is not a complete source model.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A spectral photon carries a difference of allowed energies, not either level energy by itself.
The same positive gap is needed for absorption from lower to upper and released in emission from upper to lower, when the transition is allowed. Negative bound-state energies refer to a chosen zero at separated electron/proton; ionization needs the positive distance to zero. The simple hydrogen formula omits fine structure, recoil and transition strengths. Horizontal lines are energy labels, not literal circular trajectories; modern orbitals describe probabilities. A thermal or electron-driven source can produce continuum as well as discrete emission, so one line alone is not a complete source model.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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