Assumptions, results and research status
The curves describe the declared stationary Coulomb wavefunctions; they are not literal electron trajectories or animations of the six generative levels.
What this explorer calculates
A stationary quantum state does not place an electron at one definite orbital radius. This explorer distinguishes three quantities that are often shown under the same label: density at a point, probability in a spherical shell, and total probability inside a chosen radius. Its reference length is the reduced-mass Coulomb radius aμ. [Hydrogen II]
Begin with 1S and x = r/aμ = 1. The point density is largest at the origin, while the radial probability density has its maximum at x = 1. The difference comes from the shell volume, proportional to r². Moving the radius changes the enclosed probability continuously; no sharp atomic surface is implied.
Explore the model
Explore atomic probability and radius
Compare volume density, radial probability, and the probability contained inside a sphere. The initial 1s example places the radius at one reduced-mass Bohr length, x = r/aμ = 1.
- Probability inside selected radius
- 32.3324%
- Probability outside selected radius
- 67.6676%
- Selected radius in physical units
- 52.94654 pm
- Radial probability at selected x
- 0.5413411
- Angular-average density × aμ³
- 0.04307856
1s. The radius 1 aμ contains 32.3324% probability.
The 1s volume density is largest at the origin. Its radial probability is largest at x = 1 because spherical shells have volume proportional to r².
The chart ends at x = 8; 0.001632% of the normalized probability remains beyond that radius.
| x = r/aμ | aμ³ × average density | P(x), per unit x | Probability inside x |
|---|---|---|---|
| 0 | 0.3183099 | 0 | 0% |
| 0.5 | 0.1170997 | 0.3678794 | 8.03014% |
| 1 (selected) | 0.04307856 | 0.5413411 | 32.3324% |
| 2 | 0.005830049 | 0.2930502 | 76.1897% |
| 4 | 0.0001067811 | 0.02146961 | 98.6246% |
| 6 | 0.000001955764 | 0.0008847666 | 99.9478% |
| 8 | 0.00000003582106 | 0.000028809 | 99.9984% |
Several useful meanings of atomic radius
- Mean radius ⟨r⟩
- 1.5 aμ
- Median radius: 50% inside
- 1.33703 aμ
- Radius containing 90%
- 2.66116 aμ
- Root-mean-square radius √⟨r²⟩
- 1.73205 aμ
A radial probability is a density per unit x; its value at one point is not itself a probability for a finite shell. Integrate it over an interval to obtain that probability. The cumulative curve performs that integration from the origin. The most probable radius, the mean radius, and a probability-containing radius answer different questions.
Normalized functions used by the tool
Write x = r/aμ and let P(x) dx be the probability in a spherical shell. For normalized Coulomb states:
2s: P(x) = x²(2 − x)²e−x/8.
2p: P(x) = x⁴e−x/24.
The angular-average volume density is P(x)/(4πx²), with its continuous limiting value used at the origin. The cumulative probability is F(x) = ∫0xP(u)du. The code evaluates the resulting elementary expressions with a small-radius series to avoid cancellation.
For 1s, F(x) = 1 − e−2x(1 + 2x + 2x²). For 2s, F(x) = 1 − e−x(1 + x + x²/2 + x⁴/8). For 2p, F(x) = 1 − e−x(1 + x + x²/2 + x³/6 + x⁴/24).
The 2p radial function is independent of the magnetic quantum number. Its spatial density is not spherically symmetric for a chosen magnetic substate; the density view deliberately shows the average over directions.
The physical length uses aμ = ħ/(μcα), including reduced mass. These are stationary wavefunction distributions in the effective Coulomb model. The chart does not depict a classical electron orbit, an actual sequence of measurement outcomes, or a simulation of the SMM/DTT inner generative levels.
Constants: NIST 2022 CODATA. The calculation uses c = 299792458 m/s, h = 6.62607015 × 10−34 J s, e = 1.602176634 × 10−19 C, α = 7.2973525643 × 10−3, me = 9.1093837139 × 10−31 kg, and mp = 1.67262192595 × 10−27 kg. Displayed digits are rounded model outputs; this tool does not propagate measurement uncertainties.
A worked check you can reproduce
At one characteristic radius, the enclosed probability is 1 − 5e−2, approximately 0.3233. The mean radius is 3aμ/2, which differs from both the radius of the radial maximum and a boundary containing all probability. Verify that the cumulative probability approaches one as the radius increases.
For the excited-state options, distinguish nodes of the radial function from angular nodes. The displayed radial distribution integrates the angular probability; it is not a complete three-dimensional orbital rendering. The state labels and approximations remain visible beside the controls.
Approximation and interpretation
The curves describe the declared stationary Coulomb wavefunctions; they are not literal electron trajectories or animations of the six generative levels.
Use the reset control to restore the reference example. Exported tables should retain their input values and units. The reproducible-calculations guide explains how to distinguish numerical evaluation, uncertainty, model limitations, and independent prediction.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.