Hydrogen_atom - Wikiparency

# Hydrogen atom  ## Microsims — three.js ### Hydrogen atom (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Hydrogen_atom.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Hydrogen atom — three.js microsim"></iframe> </div> **Open it full-screen:** [Hydrogen_atom.html](https://wikitube-3d-microsims.netlify.app/Hydrogen_atom.html) · library `threejs` · route `microsim/threejs/` ### Related microsims Live sims on neighbouring articles: - [[Diatomic_molecule]] - [[Ammonia]] - [[Metallic_hydrogen]] - [[Hydrogen_line]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  ## Overview A hydrogen atom is one proton and one electron, and it is the only atom the Schrodinger equation solves exactly in closed form. Add a second electron and the repulsion between them makes the equation non-separable; from helium onward every result comes from variational methods, perturbation theory or numerics. That exactness makes hydrogen the reference system for all of atomic physics -- every orbital drawn for any other element is a hydrogenic orbital with the labels borrowed -- and a test bench for quantum electrodynamics, since the prediction is limited by calculable corrections rather than by approximation error. Measure a hydrogen frequency to a part in 10^15 and you are testing QED. ## The physics Separating variables in a central potential gives psi_nlm(r,theta,phi) = R_nl(r) Y_lm(theta,phi), a radial function times a spherical harmonic. Three integers label the solution: n = 1, 2, 3, ... sets the energy and the size, l = 0 ... n-1 is the orbital angular momentum, m = -l ... +l its projection on z. Those constraints follow from requiring a solution finite at the origin and normalizable at infinity. The same two factors fix the nodes: R_nl vanishes at exactly n-l-1 finite radii, nested spheres buried invisibly inside the cloud; Y_lm supplies l angular nodes, cones about z plus planes through it for the real chemist's combinations. Total n-1, always. The energies are E_n = -R_inf hc / n^2, with R_inf hc = 13.605 693 122 990(15) eV and R_inf = 10 973 731.568 157(12) m^-1 (CODATA 2022). That Rydberg energy belongs to an idealised one-electron atom with an infinitely heavy point nucleus and no relativistic or QED corrections; real hydrogen in its ground state is bound by 13.598 434 599 702(12) eV. The 7.3 meV gap is mostly the finite proton mass, and it dwarfs the uncertainty on either figure: -13.6057 eV and -13.5984 eV are different numbers about different objects. The sim's HUD prints the first. Because the energy depends on n alone, all n^2 orbital states at a given n -- 2n^2 with spin -- share E_n. The m-degeneracy at fixed l is ordinary rotational symmetry, SO(3), and holds for any central potential. The l-degeneracy does not: it belongs to the 1/r potential alone. The Coulomb problem conserves an extra quantity, the Laplace-Runge-Lenz vector, constant only for an exact inverse-square force; with L it closes an so(4) algebra, and each degenerate n-shell is one irreducible representation of SO(4). The old label, accidental degeneracy, is wrong: it is dynamical, from a real if non-geometric symmetry (Pauli 1926, Fock 1935). Nor is it general to atoms -- in any many-electron atom the inner electrons screen the nucleus, the potential stops being pure 1/r, SO(4) breaks, and 3s, 3p and 3d separate in energy. The shape of the periodic table is that broken symmetry. Bohr's 1913 model returns exactly this E_n, and every mechanical picture in it is wrong. Bohr's n = 1 state carries angular momentum L = hbar; the true 1s state has l = 0 and none at all. Bohr's electron runs a circle of radius a_0 = 5.291 772 105 44(82) x 10^-11 m; the 1s state is a spherically symmetric cloud in which a_0 is only the most probable radial distance, expectation value 1.5 a_0. Right numbers, wrong picture -- stranger than a rough first approximation. The model omits real physics, not rounding. No electron spin, hence no fine structure (order alpha^2 E_n, about 10^-4 eV). No Lamb shift, the QED effect that lifts 2s(1/2) above 2p(1/2) by roughly 1057.8 MHz although the Dirac equation makes them degenerate -- states of the same n and j. No hyperfine structure from the electron-proton spin coupling, whose ground-state splitting is the 21 cm line of [[Hydrogen_line]]. The proton-radius puzzle from that precision programme is now closed: a February 2026 measurement of the 2S-6P transition at 0.66 parts per trillion gives r_p = 0.8406(15) fm, matching the muonic-hydrogen value and CODATA 2022's recommended 0.840 75(64) fm. ## Controls -> what each maps to | Control | Maps to | Range / values | Physical meaning | |---|---|---|---| | n | principal quantum number | 1 to 5 | Sets E_n = -13.6057/n^2 eV and the overall size; watch the ruler ticks grow like n^2 | | l | orbital angular momentum | 0 to n-1, relisted live | Sets the angular shape, s through g; the option list enforces the constraint | | m | magnetic quantum number | -l to +l, relisted live | Orientation about the quantization axis z | | form | real vs complex harmonics | real (p_x, d_xy) / complex (L_z) | Real gives the chemist's oriented lobes with no definite L_z; complex gives L_z = m hbar and phase winding instead of nodal planes | | points | Monte Carlo sample size | 3000 to 36000 | More dots, less shot noise; the sampling error on the measured mean radius shrinks with it | | cutaway | clipping plane position | -100 to +100 | Slices the cloud open so the n-l-1 nested radial shells can be counted | | nodal surfaces | -- | on / off | Draws the nodes themselves: white spheres for radial, amber cones and planes for angular | | running | -- | on / off | Slow rotation about z plus the resampling shimmer; starts off under prefers-reduced-motion | | Reset | -- | button | Returns the camera, the cutaway and the sample to their initial state | ## Learning objective After playing, a learner can name the three quantum numbers and their constraints, predict any orbital's node structure from n and l before drawing it, and explain why every l shares one energy in hydrogen but in no other atom. ## Limits and connections The cloud is a Monte Carlo sample of |psi_nlm|^2 drawn by rejection sampling: each dot is one draw from the probability density, not an electron. These are stationary states: nothing here is really moving. The shimmer is resampling noise, the rotation is the camera's. Colour encodes the phase of psi -- its sign, for the real form -- so the hue flips across every node; it is not density, and a sparse region can still be bright. The cloud is rescaled to fill the same screen volume at every n, so growth like n^2 shows in the ruler ticks rather than the picture. One electron in a fixed 1/r well is all this is: add a second proton and the closed-form solution is gone, which is where [[Diatomic_molecule]] begins; give the proton its spin and the ground state splits in two, which is [[Hydrogen_line]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Hydrogen_atom) : [Wikitube](https://en.wikitube.io/wiki/Hydrogen_atom) ## Previous hub tags Tree parent: [[Hydrogen]]. Legacy hubs: `HYDROGEN`. --- *Created 2026-08-05 - append-only - authored to WIKI_REPOPULATION_PROTOCOL v1.0 section 5 (PORTAL_Hydrogen batch 1) - 0 deletions*