An identity, a probability, a line of light: sources & model

Change particle counts, pin the differences, and discover what names an element. Then prepare one-electron hydrogen states, collect possible position outcomes and match energy gaps to light.

What supports the explanation?

What this model assumes

1. The counting desk is an inventory representation, not a nucleus geometry, ion-binding calculation or nuclear decay simulation.
2. The quantum bench always concerns one isolated hydrogen-1 atom with one electron. It does not derive oxygen, carbon, molecules or materials from a hydrogen cloud.
3. The Coulomb model omits spin, relativity, radiative corrections, fine/hyperfine structure, proton finite size, fields and measurement dynamics.
4. Every dot represents a separate simulated outcome. Accumulation order is not a particle trajectory, detector film or non-invasive tracking of one electron.
5. The 30a radial display limit is declared; probability outside it is not renormalized away. Dot size and the proton-location marker are enlarged for visibility.
6. The analytic slice uses a stated logarithmic display mapping shared by states. Screen brightness and dot opacity are not physical emission or charge density.
7. The radius probe is a chosen question boundary, not an atom’s hard edge. Changing it leaves the quantum state fixed.
8. Vacuum model wavelengths are kept distinct from NIST air entries. Screen colors and line widths do not predict calibrated spectral color, intensity or lifetime.
9. The paper activity samples a rounded distribution. It requires no chemical, flame, laser, ultraviolet lamp or electrical apparatus.
10. Exact counting, limited nuclear reference facts: Atomic number Z is proton count. Mass number A = Z + N counts nucleons; charge number is Z − Ne. Charge Q = (Z − Ne)e, with e = 1.602176634 × 10⁻¹⁹ C exactly. The eight curated ground-state entries are from IAEA LiveChart. Unlisted configurations receive no inferred stability label. Charge arithmetic does not determine which isolated ions can bind all selected electrons.
11. Mass number is not a measured mass: Carbon-13 has A = 13, while NIST lists its neutral-atom relative mass near 13.00335483507. Carbon’s standard atomic-weight interval is [12.0096, 12.0116]. Those are distinct quantities. Electrons have nonzero mass, but electron count does not enter the integer mass number.
12. A declared hydrogen approximation: We use a nonrelativistic, spin-independent Coulomb model of hydrogen-1 with reduced electron–proton mass. The relative-coordinate scale a = a₀(1 + me/mp) is about 52.9465 pm. CODATA 2022 constants give RH = R∞/(1 + me/mp), binding scale B = hcRH ≈ 13.5983 eV, and En = −B/n² relative to separated particles at rest.
13. Density and wavefunction: With x = r/a, the normalized spatial amplitudes are ψ1s = exp(−x)/√(πa³), ψ2s = (2−x)exp(−x/2)/√(32πa³), and ψ2pz = x cosθ exp(−x/2)/√(32πa³). Probability density is |ψ|², in pm⁻³ here. Probability in a region is its volume integral. A continuous density assigns zero probability to an exact point.
14. A radial shell adds its geometry: Integrating over directions gives radial densities per pm: 4x²exp(−2x)/a for 1s, x²(2−x)²exp(−x)/(8a) for 2s, and x⁴exp(−x)/(24a) for 2p. The 1s spatial density peaks at the origin, while its radial-shell density peaks at r = a. The shell’s growing volume explains the difference; it does not define an orbit.
15. Sampling a state without a trajectory: The seeded sampler draws the correct radial and angular distributions. For 2p_z, cosθ has density 3cos²θ/2; it is not uniform. Samples come from independent identical preparations. A stationary energy state has a time-independent position density in this model. Presentation time only reveals additional trials.
16. A node is a zero, not a wall: For 2s, density vanishes on r = 2a, but approximately 5.2653% of probability lies inside that sphere. For the chosen real 2p_z state, density vanishes in the plane z = 0. The analytic slice samples density at a plane; the projected dots are different data.
17. Spectral scope and evidence: Only 2p → 1s, 3p → 2s and 4p → 2s are offered as electric-dipole emission examples. The 3p/4p cards use energies; no unsupported clouds are drawn for them. A matching energy is not a complete absorption-probability rule. The 2s → 1s route is not an ordinary allowed one-photon electric-dipole transition.
18. A useful approximation can miss precision evidence: The 2011 Parthey et al. experiment measured a 1S–2S hyperfine-centroid transition frequency of 2,466,061,413,187,035(10) Hz using two-photon spectroscopy. The simple model predicts about 2,466,038,423,686,301 Hz. The discrepancy is much greater than the experimental uncertainty. This is the total transition frequency, not each probe photon’s frequency.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Original calculated projection of independently sampled hydrogen 2p_z position outcomes. The distribution is analytical and source-linked; the points are separate preparations, not many electrons in one atom or a trajectory. The lesson separately credits an unchanged USGS graphite photograph.