# Zero-point energy
<!-- MICROSIMGEN:BEGIN v1.7 — generated by g08_place_microsims.py; three.js first (§15); do not hand-edit inside -->
## Microsims — three.js
### Zero-point energy (three.js)
<div class="microsim-player">
<iframe src="https://wikitube-3d-microsims.netlify.app/Zero-point_energy.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Zero-point energy — three.js microsim"></iframe>
</div>
**Open it full-screen:** [Zero-point_energy.html](https://wikitube-3d-microsims.netlify.app/Zero-point_energy.html) · library `threejs` · route `microsim/threejs/`
### Related microsims
Live sims on neighbouring articles — 2 of them inside this article's own Wikipedia link tree:
- [[Boson]] *(in tree)*
- [[Superconductivity]] *(in tree)*
- [[Dilution_refrigerator]]
- [[Superfluid_helium-4]]
*Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*
<!-- MICROSIMGEN:END -->
## Microsim
### Live player
<div class="microsim-player">
<iframe src="https://editor.p5js.org/sciencenibber/full/npXlwW_az" width="100%" height="620" frameborder="0" sandbox="allow-scripts allow-same-origin"></iframe>
</div>
<div class="microsim-fallback">
<img src="Microsims/thumbs/Zero-point_energy.png" alt="Zero-point_energy 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/npXlwW_az">open sketch in the p5.js editor</a></em></p>
</div>
**Editor URL:** https://editor.p5js.org/sciencenibber/sketches/npXlwW_az
**Description (100 words):**
A quantum harmonic oscillator -- the canonical model in which zero-point energy first appears. The reader sees a structural grey parabolic potential well V(x) = (1/2) m omega^2 x^2 with eight horizontal energy levels stacked at E_n = (n + 1/2) hbar*omega. The n = 0 level glows yellow at exactly (1/2) hbar*omega, the irreducible zero-point energy, and below it a yellow bar marks the ground-state Gaussian width sigma_0 = sqrt(hbar / (m*omega)). A blue |psi_n(x)|^2 lobe sits on whatever level the n slider selects. Three sliders -- omega, mass m, and level n -- let the reader watch the ladder, the wavefunction, and the zero-point width respond. An orange callout reminds: sigma_0/d_NN ~ 0.30 in helium, which is why helium refuses to solidify at T = 0.
```js
// =====================================================================
// Zero-point_energy.js -- Wikitube microsim
// Article: Zero-point energy en.wikitube.io/wiki/Zero-point_energy
// Room: Helium Pattern: D (parametric curves)
// ---------------------------------------------------------------------
// Idea: an interactive quantum harmonic oscillator -- the canonical
// model in which zero-point energy first appears. The reader sees:
//
// 1. A parabolic potential well V(x) = (1/2) k x^2 in the main
// plot, drawn in the structural grey of every other Helium-room
// Pattern-D sketch.
// 2. Horizontal energy levels at E_n = (n + 1/2) hbar*omega,
// starting from n = 0. The n = 0 level -- the zero-point energy
// (1/2) hbar*omega -- is drawn in the yellow TRAJ accent and
// labelled as "ground state -- zero-point".
// 3. The ground-state probability density |psi_0(x)|^2 , a Gaussian
// of width sigma = sqrt(hbar / (m*omega)) , drawn as a filled
// blue glow on top of the potential. This is the irreducible
// "smearing" of the particle that the uncertainty principle
// forbids you to remove, even at T = 0.
// 4. A selected n-level (0 by default) gets its full |psi_n(x)|^2
// overlaid in COLD blue so the reader can sweep n upward and
// watch the wavefunction interpolate toward the classical
// parabolic-bowl probability distribution.
//
// Why this matters for the Helium room:
// Helium has the largest de Boer quantum parameter of any element
// (Lambda_star ~ 2.7 for He-4, ~ 3.1 for He-3). At the nearest-
// neighbour distance of liquid He, the zero-point amplitude
// sigma_0 = sqrt(hbar / (m * omega_Debye)) is roughly 30 percent
// of the interatomic spacing. That single number is why helium
// refuses to solidify at any temperature under its own vapour
// pressure -- the zero-point motion shakes the lattice apart.
// A "Helium reference" callout in the lower-left shows this number.
//
// Canonical equations on the chart:
// E_n = (n + 1/2) * hbar * omega (energy ladder)
// E_0 = (1/2) hbar * omega (zero-point energy)
// sigma = sqrt(hbar / (m * omega)) (ground-state width)
// dx dp >= hbar / 2 (uncertainty bound)
//
// Visual layout (720 x 520 canvas):
// * top-left: HUD title + en.wikitube.io/wiki/Zero-point_energy
// * top-right: control hint summary
// * center: potential well + energy ladder + |psi_n|^2 overlay
// * lower-left: He-4 zero-point amplitude callout (orange dim)
// * bottom: 3 sliders (omega, mass m, level n) + canonical eq
//
// Conventions (Wikitube Betterfire Standard v0):
// * single ARTICLE constant at top, single quotes
// * p5.disableFriendlyErrors = true (clean console)
// * pixelDensity(2); textFont('system-ui')
// * every createSlider has .position(x,y).size(w)
// * non-ASCII (psi, omega, sigma, lambda) lives in COMMENTS ONLY;
// every text() string literal is ASCII
// * Energy-room palette: BG=18, FG=240, HOT, COLD, STRUCT, TRAJ
// * HUD factored into drawHUD()
// =====================================================================
const ARTICLE = 'Zero-point_energy';
const TITLE = ARTICLE.replace(/_/g, ' ');
p5.disableFriendlyErrors = true;
// ----- Energy room palette (P5_JS_EDITOR section 4) ------------------
const BG = 18;
const FG = 240;
const DIM = [240, 240, 240, 140];
const HOT = [220, 110, 60]; // warm accents (He callout)
const COLD = [60, 130, 220]; // cool: |psi_n|^2 overlay
const COLDER = [40, 80, 180, 160]; // |psi_0|^2 fill
const STRUCT = [120, 130, 150]; // structural grey: potential V(x)
const TRAJ = [240, 220, 80]; // accent: ground-state level
const SCRATCH = [120, 120, 120, 90]; // grid / scratch lines
const LEVEL = [200, 200, 210, 200]; // excited-state levels
// ----- Physical constants (natural units; not real SI) ---------------
// We work in a dimensionless system where hbar = 1 and the plot's x
// axis runs over -6 .. +6 in oscillator units. omega and m are
// user-adjustable; the wavefunction width is sigma = 1/sqrt(m*omega).
// E_n = (n + 1/2) * omega in these units.
const HBAR = 1.0;
const X_RANGE = 6.0; // half-width of x window
const E_MAX = 12.0; // y-axis ceiling for energy ladder
const N_MAX = 8; // highest level we draw (above n we stop)
// ----- Plot rectangle in canvas pixels (set in setup) ----------------
let plotX, plotY, plotW, plotH;
// ----- Sliders --------------------------------------------------------
let omegaSlider, massSlider, nSlider;
let omegaVal = 1.0; // angular frequency (natural units)
let massVal = 1.0; // mass (natural units)
let nLevel = 0; // selected energy level
function setup() {
createCanvas(720, 520);
pixelDensity(2);
textFont('system-ui');
// Plot area: leaves room for HUD top + sliders bottom + axis margin.
plotX = 70;
plotY = 70;
plotW = width - 100;
plotH = height - 200;
// omega slider -- range 0.3 .. 3.0 in natural units
omegaSlider = createSlider(0.3, 3.0, 1.0, 0.05);
omegaSlider.position(90, height - 95);
omegaSlider.size(160);
// mass slider -- range 0.3 .. 3.0 (small mass = wider sigma, He-like)
massSlider = createSlider(0.3, 3.0, 1.0, 0.05);
massSlider.position(310, height - 95);
massSlider.size(160);
// n slider -- range 0 .. N_MAX
nSlider = createSlider(0, N_MAX, 0, 1);
nSlider.position(530, height - 95);
nSlider.size(160);
}
function draw() {
background(BG);
// Pull latest control values once per frame.
omegaVal = omegaSlider.value();
massVal = massSlider.value();
nLevel = nSlider.value();
// Layer order: grid -> potential -> psi_n overlay -> energy ladder
// -> ground-state highlight -> labels -> HUD/sliders.
drawAxes();
drawPotentialWell();
drawPsiOverlay(); // |psi_n|^2 for selected n
drawEnergyLadder();
drawGroundStateHighlight();
drawLandmarks();
drawHeliumCallout();
drawHUD();
drawSliderLabels();
}
// =====================================================================
// Coordinate transforms: (x [units], E [units]) <-> (px, py) in pixels
// =====================================================================
function xToPx(x) {
return map(x, -X_RANGE, X_RANGE, plotX, plotX + plotW);
}
function eToPy(E) {
// Energy 0 at bottom of plot, E_MAX at top.
return map(E, 0, E_MAX, plotY + plotH, plotY);
}
// =====================================================================
// Physical functions: V(x), E_n, sigma_0, psi_n^2
// =====================================================================
// Parabolic potential in natural units: V(x) = (1/2) m omega^2 x^2
function potential(x) {
return 0.5 * massVal * omegaVal * omegaVal * x * x;
}
// Energy ladder: E_n = (n + 1/2) * hbar * omega
function energyOfLevel(n) {
return (n + 0.5) * HBAR * omegaVal;
}
// Ground-state Gaussian width: sigma_0 = sqrt(hbar / (m omega))
function sigmaZero() {
return Math.sqrt(HBAR / (massVal * omegaVal));
}
// Hermite polynomials by recurrence H_{n+1} = 2 x H_n - 2 n H_{n-1}.
// Used for |psi_n(x)|^2 visualisation only (not for high-precision
// quantum mechanics -- numerics fine up to n = 8).
function hermite(n, x) {
if (n === 0) return 1.0;
if (n === 1) return 2.0 * x;
let h0 = 1.0;
let h1 = 2.0 * x;
for (let k = 1; k < n; k++) {
const h2 = 2.0 * x * h1 - 2.0 * k * h0;
h0 = h1;
h1 = h2;
}
return h1;
}
// |psi_n(x)|^2 -- absolute square of the n-th harmonic oscillator
// eigenfunction. The full prefactor 1/(2^n n! sqrt(pi) sigma) is
// absorbed by an empirical max-normalisation at plot time, so the
// peak of each |psi_n|^2 lands at a consistent visual height.
function psiSquared(n, x) {
const sig = sigmaZero();
const xi = x / sig;
const Hn = hermite(n, xi);
const gauss = Math.exp(-xi * xi);
return Hn * Hn * gauss;
}
// Find the max of |psi_n|^2 on [-X_RANGE, X_RANGE] for normalisation.
function psiSquaredMax(n) {
let m = 0;
const STEPS = 200;
for (let i = 0; i <= STEPS; i++) {
const x = -X_RANGE + 2 * X_RANGE * (i / STEPS);
const v = psiSquared(n, x);
if (v > m) m = v;
}
return m || 1;
}
// =====================================================================
// Rendering: axes, potential, levels, wavefunction overlay, labels
// =====================================================================
function drawAxes() {
push();
// Plot bounding box
noFill();
stroke(...SCRATCH);
strokeWeight(1);
rect(plotX, plotY, plotW, plotH);
// x-axis ticks at integer values
textSize(10);
noStroke();
fill(...DIM);
textAlign(CENTER, TOP);
for (let xv = -X_RANGE + 1; xv <= X_RANGE - 1; xv++) {
if (xv === 0) continue;
const px = xToPx(xv);
stroke(...SCRATCH);
line(px, plotY + plotH, px, plotY + plotH - 4);
noStroke();
text(xv.toFixed(0), px, plotY + plotH + 4);
}
// y-axis ticks at integer energies
textAlign(RIGHT, CENTER);
for (let E = 2; E <= E_MAX - 2; E += 2) {
const py = eToPy(E);
stroke(...SCRATCH);
line(plotX, py, plotX + 4, py);
noStroke();
text(E.toFixed(0), plotX - 6, py);
}
// Axis labels
fill(...DIM);
textAlign(CENTER, BOTTOM);
textSize(11);
text('position x (oscillator units)', plotX + plotW / 2, plotY + plotH + 28);
push();
translate(plotX - 38, plotY + plotH / 2);
rotate(-HALF_PI);
textAlign(CENTER, TOP);
text('energy E (hbar * omega units)', 0, 0);
pop();
pop();
}
function drawPotentialWell() {
push();
noFill();
stroke(...STRUCT);
strokeWeight(2);
beginShape();
const STEPS = 200;
for (let i = 0; i <= STEPS; i++) {
const x = -X_RANGE + 2 * X_RANGE * (i / STEPS);
const V = potential(x);
if (V > E_MAX) {
// Clip the parabola at the plot ceiling so it doesn't escape.
vertex(xToPx(x), eToPy(E_MAX));
} else {
vertex(xToPx(x), eToPy(V));
}
}
endShape();
// "V(x)" annotation up on the right wall of the parabola
noStroke();
fill(...STRUCT);
textSize(11);
textAlign(LEFT, BOTTOM);
text('V(x) = (1/2) m omega^2 x^2', xToPx(2.6), eToPy(8.5));
pop();
}
function drawEnergyLadder() {
push();
textSize(10);
// Draw E_n for n = 1 .. N_MAX (n = 0 gets the special highlight).
for (let n = 1; n <= N_MAX; n++) {
const E = energyOfLevel(n);
if (E > E_MAX) break;
// Find the classical turning points x_n = sqrt(2 E_n / (m omega^2))
// so the level only stretches inside the parabolic well.
const xTurn = Math.sqrt(2 * E / (massVal * omegaVal * omegaVal));
const xLo = Math.max(-X_RANGE, -xTurn);
const xHi = Math.min(X_RANGE, xTurn);
stroke(...LEVEL);
strokeWeight(n === nLevel ? 2.5 : 1);
line(xToPx(xLo), eToPy(E), xToPx(xHi), eToPy(E));
noStroke();
fill(...LEVEL);
textAlign(LEFT, CENTER);
text('n=' + n, xToPx(xHi) + 6, eToPy(E));
}
pop();
}
function drawGroundStateHighlight() {
push();
const E0 = energyOfLevel(0);
const xTurn = Math.sqrt(2 * E0 / (massVal * omegaVal * omegaVal));
const xLo = Math.max(-X_RANGE, -xTurn);
const xHi = Math.min(X_RANGE, xTurn);
// The zero-point level itself -- thick yellow line.
stroke(...TRAJ);
strokeWeight(nLevel === 0 ? 3.5 : 2.5);
line(xToPx(xLo), eToPy(E0), xToPx(xHi), eToPy(E0));
// Label
noStroke();
fill(...TRAJ);
textSize(11);
textAlign(LEFT, BOTTOM);
text('n=0 zero-point E_0 = (1/2) hbar*omega = ' + nf(E0, 0, 3),
xToPx(xHi) + 6, eToPy(E0) - 2);
pop();
}
function drawPsiOverlay() {
push();
const n = nLevel;
const E_n = energyOfLevel(n);
const yBase = eToPy(E_n);
// Vertical scale: the wavefunction lobe should sit ON the level
// line and rise by at most ~1 inter-level gap so neighbouring
// levels don't visually collide.
const gap = HBAR * omegaVal;
const lobeH = Math.min(gap * 0.85, 1.1);
const pxLobeH = (gap > 0)
? (eToPy(E_n) - eToPy(E_n + lobeH))
: 30;
const norm = psiSquaredMax(n);
const STEPS = 240;
// Filled lobe (cold blue glow)
noStroke();
fill(...COLDER);
beginShape();
vertex(xToPx(-X_RANGE), yBase);
for (let i = 0; i <= STEPS; i++) {
const x = -X_RANGE + 2 * X_RANGE * (i / STEPS);
const v = psiSquared(n, x) / norm;
vertex(xToPx(x), yBase - v * pxLobeH);
}
vertex(xToPx(X_RANGE), yBase);
endShape(CLOSE);
// Outline for crispness
noFill();
stroke(...COLD);
strokeWeight(1.2);
beginShape();
for (let i = 0; i <= STEPS; i++) {
const x = -X_RANGE + 2 * X_RANGE * (i / STEPS);
const v = psiSquared(n, x) / norm;
vertex(xToPx(x), yBase - v * pxLobeH);
}
endShape();
// Label: "|psi_n(x)|^2 for selected n"
noStroke();
fill(...COLD);
textSize(10);
textAlign(LEFT, BOTTOM);
text('|psi_' + n + '(x)|^2', xToPx(-X_RANGE) + 6, yBase - pxLobeH - 4);
pop();
}
function drawLandmarks() {
push();
// Mark the ground-state width sigma_0 with a small horizontal bar
// below the n=0 level, between -sigma_0 and +sigma_0.
const sig = sigmaZero();
const yBar = eToPy(0) - 12;
stroke(...TRAJ);
strokeWeight(2);
line(xToPx(-sig), yBar, xToPx(sig), yBar);
// Tick marks at the ends
line(xToPx(-sig), yBar - 4, xToPx(-sig), yBar + 4);
line(xToPx( sig), yBar - 4, xToPx( sig), yBar + 4);
noStroke();
fill(...TRAJ);
textSize(10);
textAlign(CENTER, BOTTOM);
text('+/- sigma_0 = +/- ' + nf(sig, 0, 3), xToPx(0), yBar - 4);
pop();
}
function drawHeliumCallout() {
// Lower-left mini box: liquid-helium reference number.
// sigma_4He / d_NN ~ 0.30 at the He-4 Debye frequency.
push();
const boxX = 14;
const boxY = height - 158;
const boxW = 240;
const boxH = 52;
// Border
noFill();
stroke(...HOT);
strokeWeight(1);
rect(boxX, boxY, boxW, boxH, 4);
noStroke();
fill(...HOT);
textSize(11);
textAlign(LEFT, TOP);
text('Helium reference (de Boer)', boxX + 8, boxY + 4);
fill(...DIM);
textSize(10);
text('Lambda_star(He-4) ~ 2.7 Lambda_star(He-3) ~ 3.1', boxX + 8, boxY + 20);
text('sigma_0 / d_NN ~ 0.30 -> no solid at T=0', boxX + 8, boxY + 34);
pop();
}
// =====================================================================
// HUD + slider labels
// =====================================================================
function drawHUD() {
// Top-left: title + Wikitube URL (Betterfire Standard rule 2)
noStroke();
fill(FG);
textAlign(LEFT, TOP);
textSize(20);
text(TITLE, 14, 12);
fill(...DIM);
textSize(12);
text('Wikitube microsim . en.wikitube.io/wiki/Zero-point_energy', 14, 36);
// Top-right: control hints (Betterfire Standard rule 3)
textAlign(RIGHT, TOP);
textSize(10);
text('omega -- oscillator frequency', width - 14, 12);
text('mass m -- particle mass', width - 14, 24);
text('n -- selected energy level', width - 14, 36);
// Bottom-right: canonical equation (Betterfire Standard rule 4)
textAlign(RIGHT, BOTTOM);
fill(FG);
textSize(13);
text('E_n = (n + 1/2) hbar * omega', width - 14, height - 6);
// Just above that: ground-state energy live readout
textSize(11);
fill(...DIM);
text('E_0 = ' + nf(0.5 * omegaVal, 0, 3) +
' sigma_0 = ' + nf(sigmaZero(), 0, 3),
width - 14, height - 24);
}
function drawSliderLabels() {
push();
noStroke();
fill(...DIM);
textSize(10);
textAlign(LEFT, BOTTOM);
// Above each slider: a tiny label with the current value.
text('omega = ' + nf(omegaVal, 0, 2), 90, height - 100);
text('mass m = ' + nf(massVal, 0, 2), 310, height - 100);
text('level n = ' + nLevel, 530, height - 100);
// Just below each slider: short caption.
fill(...DIM);
textSize(9);
textAlign(LEFT, TOP);
text('oscillator frequency', 90, height - 78);
text('particle mass', 310, height - 78);
text('selected eigenstate', 530, height - 78);
pop();
}
// =====================================================================
// End of Zero-point_energy.js -- Wikitube microsim, Helium room,
// Pattern D (parametric curves, energy ladder, |psi_n|^2 overlay).
// =====================================================================
```
## Links (Wikipedia order)
<!-- injected from _registry/childlinks/Zero-point_energy.json (2026-07-30T02:09:12Z) -->
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## From the Real GENERATIVE library
*Zero-point energy — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Nuclear room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:2_Helium.png).*
*Animated: Zero-point energy — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Nuclear room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:QHO-groundstate-animation-color.gif).*
> Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty principle.[1] Therefore, even at absolute zero, atoms and molecules retain some vibrational motion. ([Wikipedia](https://en.wikipedia.org/wiki/Zero-point_energy))
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## Media (PD/CC)
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!Gif Library/Zero-point energy/QHO-groundstate-animation-color.gif
*QHO-groundstate-animation-color.gif · Geek3 · CC BY 3.0 · [source](https://commons.wikimedia.org/wiki/File:QHO-groundstate-animation-color.gif)*
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**Semiotic universals** (the notations and alphabet letters this article speaks — each opens its canonical card): energy · oscillation · harmonic · distribution · frequency. Index: the glyph gallery · SEMIOTICS PORTAL.
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> **Room:** [[Helium]] · **Status:** ✅ shipped
## Overview
**Zero-point energy** is the lowest possible [[Energy|energy]] that a quantum mechanical [[System|system]] retains in its ground state — the irreducible residual motion that survives even at absolute zero. For the quantum harmonic oscillator, the canonical model of every chemical bond, lattice mode and electromagnetic cavity field, the ground-state energy is **E_0 = (1/2)hbar*omega**, a direct consequence of the **Heisenberg uncertainty principle** dx*dp >= hbar/2, which forbids a particle from simultaneously possessing zero kinetic energy and a definite position. The idea was introduced by **Planck (1911)** in his second theory of black-body radiation, formalised by **Einstein and Stern (1913)** for the heat capacity of hydrogen, and made fully rigorous when Heisenberg, Born and Jordan derived matrix mechanics in 1925-26. In condensed matter the relevant landmark is the **de Boer quantum parameter** Lambda_star = h / (sigma * sqrt(m*epsilon)) of the inter-atomic Lennard-Jones well: for helium-4 (Lambda_star ~ 2.7) and helium-3 (Lambda_star ~ 3.1) the zero-point amplitude exceeds thirty percent of the nearest-neighbour distance, so helium **refuses to solidify** at any temperature under its own vapour pressure and remains liquid down to absolute zero, the only element with this property. Zero-point energy underlies the **Casimir [[Force|force]]** between conducting plates, the **Lamb shift** of atomic hydrogen, vacuum fluctuations in quantum electrodynamics, the residual phonon noise that ultimately limits gravitational-[[Wave|wave]] interferometers, and the zero-point pressure that keeps He-3 / He-4 mixtures from fully phase-separating in a **[[Dilution_refrigerator|dilution refrigerator]]** below 0.87 K, making it indispensable to [[Cryogenics|cryogenics]], quantum optics and [[Quantum_computing|quantum computing]] alike.
## 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 142 of the Helium sheet on 2026-05-14T16:51:27Z.*
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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/Zero-point_energy) : [Wikitube](https://en.wikitube.io/wiki/Zero-point_energy)
## Previous hub tags
Tree parents: [[Helium]] · [[Helium-3]].
Legacy hubs: none.
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*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*