Enumerate allowed labels before making a degeneracy claim
For a one-electron central Coulomb problem, a principal quantum number n is a positive integer. Orbital angular label l takes values 0 through n−1. For each l, magnetic projection label m takes integer values −l through +l. A proposed label outside this set is rejected rather than rounded or clipped into a different state.
Learning outcome: Create a full label table for two shells and sum the magnetic labels rather than memorizing the total. Test an invalid l and identify which selected-state quantities must be withheld. Compare angular magnitude and projection without drawing a classical trajectory.
Starting recap
Review finite sums, exponential and logarithmic functions, radial integration measures, first-order coupled equations and mass-energy units. Keep state counts, occupations, calibration records and unused tests distinct.
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Try
For n=3, the allowed l values 0,1,2 contribute 1+3+5=9 spatial states and 18 spin-orbitals. For l=2, the five m labels run −2 through +2, magnitude is sqrt(6)hbar and maximum absolute projection is 2hbar. The proposed pair n=2,l=2 is invalid.
Open the experimentPredict
For n=2, which proposed orbital label is invalid?
Experiment
Create a full label table for two shells and sum the magnetic labels rather than memorizing the total. Test an invalid l and identify which selected-state quantities must be withheld. Compare angular magnitude and projection without drawing a classical trajectory.
Count labels without drawing a classical orbit
Worked teaching model · synthetic values · One-electron orbital labels have integer l from zero through n−1 and m from −l through l. Spin doubles one-electron state count. A proposed invalid l is rejected, not clipped. Degeneracy here assumes the ideal spin-independent Coulomb Hamiltonian.
Reset keeps your prediction, captures and practice history.
Allowed projections for one orbital label
Graph axes rescale independently; read units and values, not screen angles. 1: m = -2 (open-circle marked location).2: m = -1 (open-circle marked location).3: m = 0 (open-circle marked location).4: m = 1 (open-circle marked location).5: m = 2 (open-circle marked location). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Allowed projections for one orbital label
m = -2
| m label | Lz divided by hbar |
|---|---|
| -2 | -2 |
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.
m = -1
| m label | Lz divided by hbar |
|---|---|
| -1 | -1 |
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.
m = 0
| m label | Lz divided by hbar |
|---|---|
| 0 | 0 |
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.
m = 1
| m label | Lz divided by hbar |
|---|---|
| 1 | 1 |
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.
m = 2
| m label | Lz divided by hbar |
|---|---|
| 2 | 2 |
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 |
|---|---|---|
| Proposed orbital label valid | 1 | flag |
| Orbital states in entire principal shell | 9 | integer |
| One-electron states including spin | 18 | integer |
| Magnetic labels for selected orbital label | 5 | integer |
| Orbital angular momentum magnitude divided by hbar | 2.44949 | ratio |
| Maximum absolute projection divided by hbar | 2 | ratio |
Allowed projections count distinct eigenstates, not classical paths.
For each l=0,..., n−1 there are 2l+1 magnetic labels. Summing consecutive odd integers gives n² spatial states; two electron spin projections give2n² spin-orbitals. Orbital magnitude sqrt[l(l+1)]hbar differs from maximum projection l hbar. The ideal degeneracy and labels do not assert classical simultaneous angular components or multi-electron energy equality.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Allowed projections count distinct eigenstates, not classical paths.
For each l=0,..., n−1 there are 2l+1 magnetic labels. Summing consecutive odd integers gives n² spatial states; two electron spin projections give2n² spin-orbitals. Orbital magnitude sqrt[l(l+1)]hbar differs from maximum projection l hbar. The ideal degeneracy and labels do not assert classical simultaneous angular components or multi-electron energy equality.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which independent calibration, state-counting or physical-domain premise is needed, and which fresh observation could challenge the inference?
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