SCIENCE · INVESTIGATE AND EXPLAIN

Interpret a radial probability distribution without inventing an orbit

Separate probability per volume, radial shell density and probability over a finite region.

Learning outcome: Compare Z=1 and 2 at the same selected radius, then select their respective radial peaks. Record finite-shell probability and the omitted plot tail. Explain why curve height is not a probability.

Starting recap

Review amount and charge accounting, dimensions and inverse-model assumptions. A fitted number or diagram is not a unique physical identification.

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Try

For hydrogen-like 1 s, radial density peaks at r=a 0/Z while mean radius is 1.5a 0/Z; neither is an orbit radius.

Open the experiment

Predict

Why can local density be greatest at the nucleus while radial shell density is zero there?

Your prediction and reason

Experiment

Compare Z=1 and 2 at the same selected radius, then select their respective radial peaks. Record finite-shell probability and the omitted plot tail. Explain why curve height is not a probability.

Distinguish radial probability from a trajectory

Worked teaching model · synthetic values · Normalized hydrogen-like1s radial density in dimensionless r/a0 coordinate is4Z³r²exp(−2Zr). It is shell probability per unit r/a0, not local volume density or an orbit. Plots stop at6a0 without renormalizing the omitted tail. Selected shell spans radius±.25a0, truncated at0.

Nuclear charge Z for one-electron ion: 1 protons Selected radius in Bohr-radius units: 1 r/a0

Reset keeps your prediction, captures and practice history.

Axes rescale with settings. Discrete stems represent assigned peaks or algebraic fractions, not a continuous spectrum or actual negative populations. Hydrogen bars are energy levels indexed by n, not spatial positions or orbits. Only the radial probability model is drawn as a continuous curve.

Numerical diagram coordinates

Radius in units of a0	Radial probability density
0	0
0.05	0.009048374
0.1	0.03274923
0.15	0.06667364
0.2	0.1072512
0.25	0.1516327
0.3	0.1975722
0.35	0.2433268
0.4	0.2875705
0.45	0.3293214
0.5	0.3678794
0.55	0.402774
0.6	0.4337197
0.65	0.4605787
0.7	0.48333
0.75	0.5020429
0.8	0.5168551
0.85	0.5279554
0.9	0.5355684
0.95	0.5399427
1	0.5413411
1.05	0.5400328
1.1	0.5362873
1.15	0.5303693
1.2	0.5225354
1.25	0.5130312
1.3	0.5020894
1.35	0.4899282
1.4	0.4767509
1.45	0.4627453
1.5	0.4480836
1.55	0.4329228
1.6	0.417405
1.65	0.4016577
1.7	0.385795
1.75	0.3699179
1.8	0.3541154
1.85	0.3384651
1.9	0.3230339
1.95	0.3078795
2	0.2930502
2.05	0.2785867
2.1	0.264522
2.15	0.2508827
2.2	0.2376893
2.25	0.2249572
2.3	0.2126968
2.35	0.2009147
2.4	0.1896134
2.45	0.1787925
2.5	0.1684487
2.55	0.1585764
2.6	0.1491679
2.65	0.1402139
2.7	0.1317035
2.75	0.1236248
2.8	0.115965
2.85	0.1087104
2.9	0.1018469
2.95	0.09536007
3	0.08923508
3.05	0.08345711
3.1	0.07801131
3.15	0.07288294
3.2	0.06805739
3.25	0.06352031
3.3	0.05925763
3.35	0.05525564
3.4	0.05150096
3.45	0.04798066
3.5	0.04468222
3.55	0.04159354
3.6	0.03870301
3.65	0.03599946
3.7	0.0334722
3.75	0.031111
3.8	0.02890607
3.85	0.02684812
3.9	0.02492828
3.95	0.0231381
4	0.02146961
4.05	0.0199152
4.1	0.01846771
4.15	0.01712032
4.2	0.01586664
4.25	0.01470059
4.3	0.01361646
4.35	0.01260888
4.4	0.01167277
4.45	0.01080337
4.5	0.009996194
4.55	0.009247046
4.6	0.008551975
4.65	0.007907282
4.7	0.007309498
4.75	0.006755378
4.8	0.00624188
4.85	0.005766164
4.9	0.005325572
4.95	0.004917621
5	0.004539993
5.05	0.004190525
5.1	0.0038672
5.15	0.003568135
5.2	0.003291577
5.25	0.003035894
5.3	0.002799563
5.35	0.00258117
5.4	0.002379398
5.45	0.002193023
5.5	0.002020906
5.55	0.001861989
5.6	0.001715291
5.65	0.001579899
5.7	0.001454965
5.75	0.001339705
5.8	0.001233389
5.85	0.001135341
5.9	0.001044935
5.95	0.0009615892
6	0.0008847666

Calculated values under declared assumptions
Quantity	Value	Unit
Selected radial probability density	0.5413411	per unit r/a0
Selected shell inner radius	0.75	a0
Selected shell outer radius	1.25	a0
Probability in selected finite shell	0.2650337	fraction
Most probable radial coordinate	1	a0
Mean radial coordinate	1.5	a0
Probability beyond plotted6a0	0.0005222581	fraction

Shell probability is an area over a finite radial interval, not the curve height at one radius.

The local1s volume density is largest at the nucleus, but shell volume shrinks to zero there, so radial density vanishes at r=0. No path, shell wall or electron trajectory is drawn. The analytic function is supplied; deriving the wave equation is outside this activity.

Retain predictions, contrasting captures and a changed-case explanation. No graph or numerical match certifies independent reasoning or a performed spectroscopy experiment.

Shell probability is an area over a finite radial interval, not the curve height at one radius.

Observation to attach to a capture

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

Understand

Probability needs a specified region

The modern orbital picture assigns probabilities to measurement outcomes. It does not provide a hidden circular path whose position is merely unknown to us. A local probability density is not itself the probability of finding an electron at one exact mathematical point. Probabilities come from integrating density over a finite region.

Account for spherical shell size

For a hydrogen-like 1s state, local volume density decreases outward from the nucleus. A thin spherical shell, however, has volume proportional to r² times its thickness. Multiplying the local density by shell geometry gives a radial probability density that is zero at r = 0 despite the local density being largest there. Zero shell volume and zero local density are different statements.

Use a supplied normalized function

In the dimensionless coordinate r/a0, the supplied radial density is 4Z³r² exp(−2Zr). The total area from zero to infinity is one. This activity supplies the analytic result; it does not require deriving the Schrödinger equation. The most probable radial coordinate is 1/Z in these units, while the mean is 1.5/Z. Different summaries of the same distribution need not coincide.

Compare distributions rather than trajectories

At Z = 1 the peak lies at 1a0; at Z = 2 it lies at 0.5a0. Both describe one-electron systems, not neutral many-electron helium. Increasing nuclear attraction contracts this Coulomb distribution. The visualization plots a distribution over possible measured radii, not an electron moving along the curve or occupying a shell surface.

Read finite-shell probabilities

The selected shell extends 0.25a0 on either side of the chosen radius, truncated at zero. Its probability is the area over that interval, computed from the supplied cumulative function. A tall narrow density peak can have a finite probability below one; density height may exceed one because its unit is inverse radial coordinate. The plot stops at 6a0 and reports its omitted tail without renormalizing visible values.

Transfer to a careful explanation

Compare a shell centered on the peak with a wider region containing it. The wider region cannot have less probability, even if its midpoint has a smaller density. Explain why a mean radius is not a guaranteed observation and why repeated detections would be needed to test a distribution. This abstract calculation is not an instruction to perform atomic imaging or radiation experiments.

Sources and model boundary

Conceptual references checked 30 October 2026: OpenStax quantum theory. All lesson prose, numerical packets and diagrams are original; no textbook figures or exercises are copied. Synthetic parameter choices are not measured atomic spectra. The activity does not request radiation sources, vacuum equipment, particle beams or chemical manipulation. Saved calculations remain public practice, not independent mastery or physical laboratory evidence.

Additional primary teaching reference checked on the same date: MIT radial probability notes. No source figures or prose are reproduced.

Apply

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

Distinguish supplied quantum wavefunction results from classical trajectories and from a measurement of one electron.

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: Construct allowed quantum labels and node counts Next: Build an orbital occupancy ledger with charge and spin constraints

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