# Quantum mechanics
## Microsim
### Live player
<div class="microsim-player">
<iframe src="https://editor.p5js.org/sciencenibber/full/A_73AMciC" width="100%" height="620" frameborder="0" sandbox="allow-scripts allow-same-origin"></iframe>
</div>
<div class="microsim-fallback">
<img src="Microsims/thumbs/Quantum_mechanics.png" alt="Quantum_mechanics microsim poster" style="width:100%;border:1px solid #4445;border-radius:6px;">
<p><em>Live microsim (desktop) · <a href="https://editor.p5js.org/sciencenibber/sketches/A_73AMciC">open sketch in the p5.js editor</a></em></p>
</div>
**Editor URL:** https://editor.p5js.org/sciencenibber/sketches/A_73AMciC
**Description (100 words):**
The microsim shows a live 1D infinite square well — the textbook particle-in-a-box — with the lowest four energy eigenstates available as building blocks. Four sliders set the amplitudes c_1 through c_4; the canvas plots the resulting probability density |psi|^2 as a filled yellow curve that evolves in time according to the Schrödinger equation. A right-hand energy ladder displays each level E_n at its correct height and shows the user's probability weight |c_n|^2 as a colored bar. Toggling "Show parts" overlays the real and imaginary components of psi. Pure eigenstates stand still; superpositions visibly slosh — exactly how superfluid helium, qubits, and NMR resonance work.
```js
// =====================================================================
// Quantum_mechanics.js -- Wikitube microsim
// Article: Quantum_mechanics en.wikitube.io/wiki/Quantum_mechanics
// Room: Helium Pattern: D / E hybrid (parametric +
// evolving probability density)
// ---------------------------------------------------------------------
// Idea: a live, interactive 1D infinite square well (particle in a
// box) showing how a quantum state evolves in time as a coherent
// superposition of energy eigenstates. The reader dials in the
// amplitude of each of the lowest four eigenstates and watches the
// resulting probability density |psi|^2 slosh inside the well -- the
// cleanest possible demonstration of why "stationary" states are
// stationary and why superpositions are not.
//
// Equations (in natural units hbar = 1, m = 1, L = 1):
//
// Eigenfunctions : psi_n(x) = sqrt(2/L) * sin(n*pi*x/L)
// Eigen-energies : E_n = n^2 * pi^2 * hbar^2 / (2 m L^2)
// Evolution : psi(x,t) = sum_n c_n * psi_n(x) * exp(-i E_n t / hbar)
// Probability : |psi|^2 = (Re psi)^2 + (Im psi)^2
// Schrodinger eq. : i hbar dpsi/dt = H psi
//
// Why this works for the Helium room:
// * Helium-4 superfluidity is a macroscopic occupation of the
// ground state -- a "giant wavefunction" exactly like psi_1.
// * Helium-3 fermionic pairing in dilution refrigerators is
// coherent superposition writ large.
// * Superconducting qubits running in dilute-He environments use
// |0> and |1> as Pattern-D eigenstates and read out interference
// between them. The c_1 + c_2 motion shown here IS the qubit.
//
// Helium landmark quantum facts surfaced in the HUD:
// * He-4 has zero net spin (boson) -> Bose-Einstein condensation
// * He-3 has nuclear spin 1/2 (fermion) -> Cooper pairing below mK
// * Both isotopes have zero-point energy large enough that helium
// remains liquid down to 0 K at atmospheric pressure.
//
// Visual layout (720 x 520 canvas):
// * top-left: HUD title + en.wikitube.io/wiki/Quantum_mechanics
// * top-right: pause / play indicator + time t in natural units
// * main panel: |psi|^2 (yellow filled curve), Re psi (cool blue),
// Im psi (warm orange), each on the same x-axis
// inside a framed well. Wall fills (very dim grey)
// indicate V(x) = infinity outside [0, L].
// * right panel: vertical energy ladder. Horizontal bars at heights
// E_n (scaled to fit), with each bar's filled width
// proportional to |c_n|^2 (probability of measuring
// E_n). Color-codes the eigenstates n = 1..4.
// * bottom: four sliders for the (real) amplitudes c_1..c_4,
// a pause/play button, a reset button, a "show
// components" toggle.
// * bottom-right: canonical equation i*hbar dPsi/dt = H*Psi.
//
// Conventions (Wikitube Betterfire Standard v0):
// * single ARTICLE constant at the top, single quotes (validator)
// * p5.disableFriendlyErrors = true to keep editor console clean
// * all sliders positioned + sized explicitly, no floating defaults
// * non-ASCII (Greek psi, pi, hbar) lives in COMMENTS ONLY; every
// text() string literal is plain ASCII.
// * Energy-room palette (P5_JS_EDITOR section 4): dark BG, HOT/COLD
// tones, STRUCT grey, TRAJ accent.
// =====================================================================
const ARTICLE = 'Quantum_mechanics';
const TITLE = ARTICLE.replace(/_/g, ' ');
p5.disableFriendlyErrors = true;
// ----- Energy room palette (P5_JS_EDITOR section 4, line 165) --------
const BG = 18;
const FG = 240;
const DIM = [240, 240, 240, 140];
const HOT = [220, 110, 60]; // warm: Im(psi)
const COLD = [60, 130, 220]; // cool: Re(psi)
const STRUCT = [120, 130, 150]; // structural grey
const TRAJ = [240, 220, 80]; // bright accent: |psi|^2
const SCRATCH = [120, 120, 120, 90]; // grid lines
const WALL = [50, 50, 60]; // V = infinity walls
// Per-eigenstate ladder colors for the energy panel
const E_COLORS = [
[240, 220, 80], // n = 1, ground state (yellow)
[120, 220, 140], // n = 2 (green)
[ 60, 130, 220], // n = 3 (blue)
[220, 110, 200] // n = 4 (magenta)
];
// ----- Quantum-mechanical constants (natural units) ------------------
const HBAR = 1.0;
const MASS = 1.0;
const L_BOX = 1.0; // well width (natural units)
const N_MAX = 4; // number of eigenstates shown
const N_X = 240; // spatial-sample resolution
// E_n = n^2 * pi^2 * hbar^2 / (2 m L^2). Precompute angular freqs.
const E_N = []; // filled in setup()
const OMEGA_N = []; // omega_n = E_n / hbar
// ----- UI / sim state -----------------------------------------------
let cSliders = []; // user-set amplitudes c_1..c_4
let pauseBtn, resetBtn, compToggle;
let showComponents = false;
let paused = false;
let tSim = 0; // simulation time (natural units)
const DT_MAX = 0.02; // cap dt for stable visuals
// ----- Panel rectangles (set in setup) ------------------------------
let mainX, mainY, mainW, mainH; // wavefunction panel
let ladderX, ladderY, ladderW, ladderH; // energy ladder
let controlsY; // slider row baseline
function setup() {
createCanvas(720, 520);
pixelDensity(2);
textFont('system-ui');
// Precompute eigen-energies E_n = n^2 * pi^2 * hbar^2 / (2 m L^2).
// Note: in natural units this is simply n^2 * pi^2 / 2.
for (let n = 1; n <= N_MAX; n++) {
const e = (n * n * PI * PI * HBAR * HBAR) / (2 * MASS * L_BOX * L_BOX);
E_N.push(e);
OMEGA_N.push(e / HBAR);
}
// ---------- Panel layout ----------
// Main wavefunction panel takes the left ~70% of the canvas.
mainX = 60;
mainY = 56;
mainW = 440;
mainH = 320;
// Energy ladder on the right.
ladderX = mainX + mainW + 30;
ladderY = mainY;
ladderW = 160;
ladderH = mainH;
// ---------- Controls ----------
// Four amplitude sliders for c_1..c_4, defaulting to a 50/50
// superposition of n = 1 and n = 2 (the textbook "sloshing" state).
controlsY = mainY + mainH + 30;
const defaults = [1.0, 1.0, 0.0, 0.0];
for (let n = 1; n <= N_MAX; n++) {
const s = createSlider(-1.5, 1.5, defaults[n - 1], 0.05)
.position(mainX + (n - 1) * 110, controlsY)
.size(90);
cSliders.push(s);
}
// Pause / Reset / Components-toggle.
pauseBtn = createButton('Pause')
.position(ladderX, controlsY)
.size(70, 22);
pauseBtn.mousePressed(() => {
paused = !paused;
pauseBtn.html(paused ? 'Play' : 'Pause');
});
resetBtn = createButton('Reset t')
.position(ladderX + 80, controlsY)
.size(70, 22);
resetBtn.mousePressed(() => { tSim = 0; });
compToggle = createButton('Show parts')
.position(ladderX, controlsY + 30)
.size(150, 22);
compToggle.mousePressed(() => {
showComponents = !showComponents;
compToggle.html(showComponents ? 'Hide parts' : 'Show parts');
});
}
function draw() {
background(BG);
// ---------- Time integration (purely analytic; no ODE step) ----
// Each eigenstate's phase rotates at omega_n = E_n / hbar. We do not
// integrate numerically -- we just advance tSim and resample.
const dt = min(deltaTime / 1000, DT_MAX);
if (!paused) tSim += dt;
// ---------- Read amplitudes once per frame --------------------
const cRaw = cSliders.map(s => s.value());
const norm = sqrt(cRaw.reduce((a, b) => a + b * b, 0)) || 1.0;
const c = cRaw.map(v => v / norm); // normalized amplitudes
// ---------- Draw the three panels ----------------------------
drawWell(mainX, mainY, mainW, mainH, c);
drawLadder(ladderX, ladderY, ladderW, ladderH, c);
drawHUD();
drawFooter();
}
// =====================================================================
// Region: WELL PANEL
// Draws V(x) walls, axis frame, and the live psi(x, t) decomposition.
// Inputs: panel rect (px), normalized amplitude vector c[n-1].
// =====================================================================
function drawWell(x0, y0, w, h, c) {
// Background frame + walls.
noStroke(); fill(...WALL);
// Left wall.
rect(x0 - 28, y0 - 4, 28, h + 8);
// Right wall.
rect(x0 + w, y0 - 4, 28, h + 8);
// Inner well background (subtle).
fill(28); rect(x0, y0, w, h);
// Grid + zero line.
stroke(...SCRATCH); strokeWeight(1);
for (let i = 1; i < 4; i++) {
const yy = y0 + (h * i) / 4;
line(x0, yy, x0 + w, yy);
}
// Strong zero line (psi = 0 axis).
stroke(...STRUCT, 160); strokeWeight(1.5);
const yZero = y0 + h * 0.5;
line(x0, yZero, x0 + w, yZero);
// Sample psi(x, t).
const psiRe = new Array(N_X);
const psiIm = new Array(N_X);
for (let i = 0; i < N_X; i++) {
const x = (i / (N_X - 1)) * L_BOX;
let re = 0, im = 0;
for (let n = 1; n <= N_MAX; n++) {
const amp = c[n - 1] * sqrt(2 / L_BOX) * sin(n * PI * x / L_BOX);
const phase = -OMEGA_N[n - 1] * tSim;
re += amp * cos(phase);
im += amp * sin(phase);
}
psiRe[i] = re;
psiIm[i] = im;
}
// Vertical scale: amplitude window [-2, 2] (normalized eigenfns peak
// near sqrt(2/L) ~ 1.41, superpositions a touch higher).
const yMin = -2.2, yMax = 2.2;
const yPx = v => map(v, yMin, yMax, y0 + h, y0);
// ---------- |psi|^2 filled curve (TRAJ accent) ----------
noStroke(); fill(...TRAJ, 90);
beginShape();
vertex(x0, yPx(0));
for (let i = 0; i < N_X; i++) {
const px = x0 + (i / (N_X - 1)) * w;
const p2 = psiRe[i] * psiRe[i] + psiIm[i] * psiIm[i];
vertex(px, yPx(p2));
}
vertex(x0 + w, yPx(0));
endShape(CLOSE);
// Outline on top of the fill.
noFill(); stroke(...TRAJ); strokeWeight(2);
beginShape();
for (let i = 0; i < N_X; i++) {
const px = x0 + (i / (N_X - 1)) * w;
const p2 = psiRe[i] * psiRe[i] + psiIm[i] * psiIm[i];
vertex(px, yPx(p2));
}
endShape();
// ---------- Re psi (cool) and Im psi (warm) overlays ----------
if (showComponents) {
// Re psi
stroke(...COLD); strokeWeight(1.5); noFill();
beginShape();
for (let i = 0; i < N_X; i++) {
const px = x0 + (i / (N_X - 1)) * w;
vertex(px, yPx(psiRe[i]));
}
endShape();
// Im psi
stroke(...HOT); strokeWeight(1.5); noFill();
beginShape();
for (let i = 0; i < N_X; i++) {
const px = x0 + (i / (N_X - 1)) * w;
vertex(px, yPx(psiIm[i]));
}
endShape();
}
// ---------- Axis labels ----------
noStroke(); fill(...STRUCT); textSize(11); textAlign(LEFT, TOP);
text('x = 0', x0 - 4, y0 + h + 4);
textAlign(RIGHT, TOP);
text('x = L', x0 + w + 4, y0 + h + 4);
textAlign(LEFT, CENTER);
text('psi', x0 - 36, yZero - 6);
textAlign(LEFT, TOP);
text('|psi|^2', x0 + 8, y0 + 4);
if (showComponents) {
fill(...COLD); text('Re psi', x0 + 8, y0 + 20);
fill(...HOT); text('Im psi', x0 + 8, y0 + 36);
}
textAlign(LEFT, BASELINE);
}
// =====================================================================
// Region: ENERGY LADDER
// Stacks horizontal bars at heights ~ E_n. Bar fill width = |c_n|^2.
// Color-codes each eigenstate so the well-panel curves can be cross-
// referenced even when Show-parts is off.
// =====================================================================
function drawLadder(x0, y0, w, h, c) {
// Frame.
noFill(); stroke(...STRUCT, 120); strokeWeight(1);
rect(x0, y0, w, h);
// Map E_n into the ladder height. Top of panel = E_max plus a margin.
const eMax = E_N[N_MAX - 1] * 1.15;
const eToY = e => map(e, 0, eMax, y0 + h, y0);
// Zero-energy baseline.
stroke(...SCRATCH); strokeWeight(1);
line(x0, y0 + h, x0 + w, y0 + h);
// Header label.
noStroke(); fill(...STRUCT); textSize(11);
textAlign(LEFT, TOP);
text('Energy ladder', x0 + 6, y0 - 16);
// Draw each eigenstate.
textSize(11); textAlign(LEFT, CENTER);
for (let n = 1; n <= N_MAX; n++) {
const e = E_N[n - 1];
const y = eToY(e);
const col = E_COLORS[n - 1];
const cn = c[n - 1];
const w2 = cn * cn; // probability weight
// Dim level line spans the panel.
stroke(...col, 130); strokeWeight(1);
line(x0 + 4, y, x0 + w - 4, y);
// Filled probability bar grows rightward from the left margin.
noStroke(); fill(...col, 80);
rect(x0 + 4, y - 6, w - 8, 12);
fill(...col);
rect(x0 + 4, y - 6, (w - 8) * constrain(w2, 0, 1), 12);
// Label: state index and (rounded) E_n.
fill(...col);
text('n=' + n, x0 + 8, y + 18);
fill(...STRUCT);
text('E=' + nf(e, 1, 1), x0 + 50, y + 18);
text('|c|^2=' + nf(w2, 1, 2), x0 + 105, y + 18);
}
textAlign(LEFT, BASELINE);
}
// =====================================================================
// Region: HUD
// Top-left title block + top-right time / pause status.
// All strings ASCII; non-ASCII glyphs live in comments only.
// =====================================================================
function drawHUD() {
// Title bar background.
noStroke(); fill(0, 200);
rect(8, 8, 470, 38);
// Title (bright).
fill(FG); textSize(22); textAlign(LEFT, TOP);
text(TITLE, 14, 12);
// Subtitle (dim).
fill(...DIM); textSize(12);
text('Wikitube microsim . en.wikitube.io/wiki/' + ARTICLE, 14, 32);
// Top-right status (sim time + pause indicator).
fill(...STRUCT); textSize(12); textAlign(RIGHT, TOP);
const status = (paused ? 'PAUSED' : 'PLAYING');
text(status + ' t = ' + nf(tSim, 1, 2), width - 12, 14);
text('infinite square well, hbar=m=L=1', width - 12, 30);
textAlign(LEFT, BASELINE);
}
// =====================================================================
// Region: FOOTER
// Bottom-right canonical equation, bottom-left slider labels.
// =====================================================================
function drawFooter() {
// Slider labels.
noStroke(); fill(...STRUCT); textSize(11);
textAlign(LEFT, TOP);
for (let n = 1; n <= N_MAX; n++) {
const lx = mainX + (n - 1) * 110;
fill(...E_COLORS[n - 1]);
text('c_' + n, lx, controlsY - 14);
}
// Canonical equation bottom-right.
fill(...DIM); textSize(12); textAlign(RIGHT, BOTTOM);
text('i*hbar dPsi/dt = H*Psi', width - 12, height - 30);
text('E_n = n^2 pi^2 hbar^2 / (2 m L^2)', width - 12, height - 12);
textAlign(LEFT, BASELINE);
}
```
## Links (Wikipedia order)
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## Images
<figure class="wt-gifplate">
<img src="https://commons.wikimedia.org/wiki/Special:FilePath/Electron_configuration_order.gif" alt="Electron Shells" loading="lazy" decoding="async">
<figcaption><strong>Electron Shells</strong> — Illustrate electron shell filling and quantum numbers.<br>
<span class="wt-credit">Wikimedia Commons · <strong>licence pending verification</strong> (run <code>g17_gif_verify.py</code> on a networked lane) · <a href="https://commons.wikimedia.org/wiki/File:Electron_configuration_order.gif">Details</a></span></figcaption>
</figure>
*Still companion to the 1 live microsim above: the sim is the instrument, the plate is the glance. §15 keeps the player first; this sits in the image slot on [[Quantum_mechanics]].*
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## From the Real GENERATIVE library
*Quantum mechanics — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (STEM and Music room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Hydrogen_Density_Plots.png).*
*Animated: Quantum mechanics — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (STEM and Music room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Guassian_Dispersion.gif).*
> Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms.[2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. ([Wikipedia](https://en.wikipedia.org/wiki/Quantum_mechanics))
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## From the vault media library
!Quantum mechanics thumb.png
*Quantum Mechanics — from the vault's own media holdings, placed 2026-07-09. MTN / Wikitube.io original · CC BY-SA 4.0.*
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## Media (PD/CC)
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!Gif Library/Quantum mechanics/Guassian Dispersion.gif
*Guassian_Dispersion.gif · CC0*
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**Semiotic universals** (the notations and alphabet letters this article speaks — each opens its canonical card): energy · amplitude · probability · resonance · superposition. Index: the glyph gallery · SEMIOTICS PORTAL.
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> **Room:** [[Helium]] · **Status:** ✅ shipped
## Overview
Quantum mechanics is the fundamental theory of [[Physics|physics]] describing the behavior of matter and [[Energy|energy]] at atomic and subatomic scales. Developed in the 1920s by Heisenberg, Schrödinger, Born, Dirac, and Pauli, it replaced classical mechanics for systems where action approaches Planck's constant h ≈ 6.626 × 10⁻³⁴ J·s. Its central object is the complex-valued wavefunction ψ(x,t), whose squared magnitude |ψ|² gives the probability [[Density|density]] of finding a particle at position x. The wavefunction evolves under the time-dependent [[Schrödinger_equation|Schrödinger equation]], iℏ ∂ψ/∂t = Ĥψ, where Ĥ is the Hamiltonian operator. Observables correspond to Hermitian operators whose eigenvalues are the only possible measurement outcomes; non-commuting observables obey the Heisenberg uncertainty principle, Δx · Δp ≥ ℏ/2.
The theory introduced quantization of energy levels, [[Wave|wave]]-particle duality, identical-particle statistics (bosons and fermions), and entanglement. Bound states yield discrete spectra (the hydrogen atom, harmonic oscillator, infinite well), while scattering problems yield continuous spectra. Spin, an intrinsic angular momentum with no classical analog, governs magnetic resonance and the periodic table via the Pauli exclusion principle.
Quantum mechanics is the foundation of nearly every modern technology touching helium: it explains superfluidity in helium-4, Bose-Einstein and Fermi-Dirac condensation, [[Nuclear_magnetic_resonance|nuclear magnetic resonance]] in MRI, electron paramagnetic resonance, superconducting qubits operating in dilution refrigerators, mass spectrometry of helium isotopes, and the [[Zero-point_energy|zero-point energy]] that keeps [[Liquid_helium|liquid helium]] liquid down to absolute zero at ambient pressure.
## See also
- Room hub: [[Helium]]
- p5.js Editor conventions: P5 JS EDITOR
- Wiki root: MAIN
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*Scaffolded by `generative-microsim` from row 137 of the Helium sheet on 2026-05-14T16:50:49Z.*
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*Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*
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## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Quantum_mechanics) : [Wikitube](https://en.wikitube.io/wiki/Quantum_mechanics)
## Previous hub tags
Tree parents: [[Dynamical_system]] · [[Helium]] · [[Helium-3]] · [[Hydrogen]] · [[Phase_space]] · [[Self-organization]].
Legacy hubs: none.
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*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*