SCIENCE · INVESTIGATE AND EXPLAIN

Infer a spectral line from an energy-level difference

Use differences between stipulated levels to predict line energies and challenge identification from one line.

Learning outcome: Use differences between stipulated levels to predict line energies and challenge identification from one line.

Starting recap

A transition changes matter energy; an emitted photon carries a positive energy equal to the downward gap.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

An invented atom has allowed levels 0,1 and 3 eV. It can emit different photons depending on the selected upper and lower levels.

Open the experiment

Predict

Does the highest numbered level always emit a 3 eV photon?

Your prediction and reason

Experiment

Select all three transition pairs and find the matching photon energy for each. Capture a mismatch and a match; calculate the corresponding wavelengths.

Infer a stipulated line from energy levels

Worked teaching model · synthetic values · Invented three-level atom with levels 0, 1, 3 eV; all shown downward transitions allowed and upper level populated. No actual element identification, line width or probability calculation.

Transition pair: 0=3→0, 1=3→1, 2=1→0: 0 pair Proposed photon energy: 2 eV

Reset keeps your prediction, captures and practice history.

Quantity	Calculated value	Unit
Level gap	3.0000	eV
Proposed photon	2.0000	eV
Gap minus photon	1.0000	eV

Proposed photon does not match the stipulated transition.

Required vacuum wavelength 413.28 nm. Three levels give three distinct positive gaps here; matching one line does not identify a real element.

Model boundary: Invented three-level atom with levels 0, 1, 3 eV; all shown downward transitions allowed and upper level populated. No actual element identification, line width or probability calculation.

All investigations are virtual. Do not obtain radioactive sources, expose eyes to bright beams, or build high-voltage apparatus. Notebook captures preserve selected recipes, not physical measurements.

Proposed photon does not match the stipulated transition.

Required vacuum wavelength 413.28 nm. Three levels give three distinct positive gaps here; matching one line does not identify a real element.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Read differences, not labels

The investigation uses an invented three-level system with energies 0,1 and 3 eV. These are not the measured levels of a named element. Every displayed downward transition is stipulated to be allowed, and its initial upper level is occupied. Those assumptions let us isolate energy conservation from transition probabilities.

For a transition from 3 eV to 1 eV, matter loses 2 eV and radiation gains 2 eV. The photon does not have a negative energy. For the reverse absorption, matter gains 2 eV while the radiation field loses one 2 eV photon. Changing the reference by adding 5 eV to every level leaves all line energies unchanged.

Construct the possible spectrum

List all different pairs with the upper level greater than the lower. The three downward gaps are 3,2 and 1 eV. Use λ=hc/E to calculate approximately 413.28,619.92 and 1239.84 nm. The longer-wavelength line corresponds to the smaller gap. A straight line connecting two drawn levels is a bookkeeping arrow, not a classical electron trajectory through those energies.

Try choosing 3 eV for the 3→1 transition. The numerical residual is−1 eV: the proposed photon carries more energy than the stipulated matter loss. The system could have other processes, but this particular one-photon proposal is incompatible with the supplied levels. It is not repaired by making the beam brighter.

Distinguish a direct transition from a cascade

A 3→0 transition produces one 3 eV photon. A cascade 3→1→0 produces a 2 eV photon followed by a 1 eV photon. Both conserve 3 eV overall, but their line patterns and photon counts differ. Adding the wavelengths of the two cascade photons does not give the direct-transition wavelength, because wavelength is inversely related to energy.

Transfer: shifted levels

For levels 4,6 and 9 eV, the gaps are 2,3 and 5 eV. The 9→4 line has wavelength about 247.97 nm. A 9→6→4 cascade gives 3 and 2 eV photons, not one 5 eV photon. A common shift of all levels would change none of these results.

Evidence that would strengthen identification

A real spectrum has line widths, intensities and possible overlapping species. Agreement with one line is weaker evidence than agreement with several well-resolved lines under calibrated conditions. Population, selection rules and the detector’s response matter. This lesson predicts allowed energy differences only; it does not reconstruct a real atom or establish that every energetically possible transition has an appreciable rate.

Source and authorship

Reference for the physical relationship. Source checked 30 October 2026. The explanations, numerical cases, questions and diagrams here are original; reference material is not copied. Exact h,c,e follow NIST SI definitions.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Distinguish absolute level energy from gap energy and explain why a single matching line is not a unique fingerprint.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Calculate photon energy without confusing it with power Next: Test a photoelectric threshold

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.