SCIENCE · INVESTIGATE AND EXPLAIN

Integrate shell probability rather than reading a spatial-density peak

A normalized hydrogenic 1s density per physical volume is rho(r)=exp(−2r/a)/(pi a³). It is largest at r=0. That does not mean a thin shell at the origin contains the most probability: a spherical shell has volume 4pi r²dr. Multiplying gives radial probability density P(r)=4r²exp(−2r/a)/a³, with units of inverse length.

Learning outcome: Normalize with the spherical volume measure and independently derive the cumulative law. Compare the two density modes with the mean, then change the scale at fixed physical radius and at fixed radius-to-scale ratio. Explain why these interventions differ.

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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For a=0.1 nm, the radial mode is 0.1 nm and mean radius is 0.15 nm, while the spatial density peaks at zero. Probability inside r=a is 1−5 exp(−2)≈0.323323584. Doubling a while also doubling the enclosing radius leaves this dimensionless probability unchanged.

Open the experiment

Predict

Which factor converts a spherical spatial density into radial shell probability density?

Your prediction and reason

Experiment

Normalize with the spherical volume measure and independently derive the cumulative law. Compare the two density modes with the mean, then change the scale at fixed physical radius and at fixed radius-to-scale ratio. Explain why these interventions differ.

Integrate a spherical probability measure

Worked teaching model · synthetic values · Normalized hydrogenic 1s spatial density exp(−2r/a)/(pi a³). The radial shell density is 4r² exp(−2r/a)/a³; no classical orbit is assigned. Effective scale is supplied rather than inferred from a real atom.

Effective radial scale: 0.1 nm Upper radius divided by scale: 1 ratio

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Shell probability includes the volume measure

Graph axes rescale independently; read units and values, not screen angles. 1: 4 u squared exp(−2u) (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Shell probability includes the volume measure

4 u squared exp(−2u)

r divided by scale	probability density per dimensionless radius
0	0
0.08333333	0.02351338
0.1666667	0.07961459
0.25	0.1516327
0.3333333	0.2281854
0.4166667	0.3018043
0.5	0.3678794
0.5833333	0.4238544
0.6666667	0.4686171
0.75	0.5020429
0.8333333	0.5246545
0.9166667	0.5373736
1	0.5413411
1.083333	0.5377901
1.166667	0.5279585
1.25	0.5130312
1.333333	0.4941045
1.416667	0.4721656
1.5	0.4480836
1.583333	0.4226091
1.666667	0.3963777
1.75	0.3699179
1.833333	0.3436606
1.916667	0.3179491
2	0.2930502
2.083333	0.2691641
2.166667	0.2464345
2.25	0.2249572
2.333333	0.2047887
2.416667	0.1859531
2.5	0.1684487
2.583333	0.1522531
2.666667	0.1373284
2.75	0.1236248
2.833333	0.1110845
2.916667	0.09964353
3	0.08923508
3.083333	0.07979058
3.166667	0.07124149
3.25	0.06352031
3.333333	0.0565615
3.416667	0.05030212
3.5	0.04468222
3.583333	0.0396452
3.666667	0.03513797
3.75	0.031111
3.833333	0.02751833
3.916667	0.02431755
4	0.02146961
4.083333	0.01893875
4.166667	0.01669232
4.25	0.01470059
4.333333	0.01293656
4.416667	0.01137578
4.5	0.009996194
4.583333	0.00877789
4.666667	0.007702973
4.75	0.006755378
4.833333	0.005920706
4.916667	0.005186079
5	0.004539993

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.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Probability inside selected radius	0.3233236	ratio
Radial shell mode	0.1	nm
Spatial density mode radius	0	nm
Mean radius	0.15	nm
Second radial moment	0.03	nm²
Radial variance	0.0075	nm²

The spatial density peaks at the origin while the radial shell density vanishes there and peaks at one scale length.

Integrate4u²exp(−2u)du with u=r/a. Repeated integration by parts gives cumulative probability 1−exp(−2u)(1+2u+2u²), mean radius3a/2 and second moment3a². A shell has volume4pi r²dr; dropping that factor confuses density per volume with probability per radial length. Neither mode is a particle orbit.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

The spatial density peaks at the origin while the radial shell density vanishes there and peaks at one scale length.

Observation to attach to a capture

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

Understand

Define the state and evidence

Derive the quantitative relation

Use u=r/a and dr=a du. Normalization becomes integral from zero to infinity of 4u²exp(−2u)du=1. Repeated integration by parts gives the cumulative probability F(u)=1−exp(−2u)(1+2u+2u²). At zero the probability is exactly zero; it approaches one as the enclosing sphere grows. A density evaluated at one radius is not this integrated probability.

Test a changed model or boundary

Differentiate u²exp(−2u) to place its nonzero maximum at u=1. Integrating u and u² against the normalized radial law gives mean radius 3a/2 and second moment 3a², hence radial variance 3a²/4. The spatial density mode, radial shell mode and mean radius are therefore three different quantities with different interpretations.

Limit the inference

This is a spherical quantum probability law, not a circular electron orbit or an actual reconstructed atomic image. The supplied effective scale can represent a controlled comparison, but additional Hamiltonian and calibration assumptions would be needed to identify a real atom. No probability is obtained by integrating the spatial density over dr without the volume factor.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

Find probability inside r=2a for this 1s density.

Numerical result in the requested units

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Sources and boundaries

Original derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions and numerical packets are original. Supplied theoretical distributions and synthetic records are not actual laboratory measurements. No nuclear, electrical, heating, beam or material-handling procedures are requested. Retain first reasoning, actual criterion critique, revision and a separately assigned unfamiliar task in the course evidence workspace; public keys do not authenticate mastery.

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Which independent calibration, state-counting or physical-domain premise is needed, and which fresh observation could challenge the inference?

Your explanation and limitations

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