# Schrödinger equation
## Microsim
### Live player
<div class="microsim-player">
<iframe src="https://editor.p5js.org/sciencenibber/full/zMq3oG9Pf" width="100%" height="620" frameborder="0" sandbox="allow-scripts allow-same-origin"></iframe>
</div>
<div class="microsim-fallback">
<img src="Microsims/thumbs/Schrödinger_equation.png" alt="Schrödinger_equation 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/zMq3oG9Pf">open sketch in the p5.js editor</a></em></p>
</div>
**Editor URL:** https://editor.p5js.org/sciencenibber/sketches/zMq3oG9Pf
**Description (100 words):**
This sketch renders a quantum particle in a 1D infinite square well and lets the reader compose its state from up to six eigenstates. Three sliders control the primary quantum number n, a partner state m, and a mixing amplitude c that interpolates from a pure eigenstate to a 50/50 superposition. The left band lights the active rungs of the energy ladder. The right pane shows Re(Psi) in yellow and Im(Psi) in magenta above a filled cyan |Psi|^2 below. Time advances continuously and a pause button freezes it. Setting c above zero makes the probability density slosh visibly inside the well at [[Angular_frequency|angular frequency]] omega_nm = E_m minus E_n.
```js
// =====================================================================
// Schrödinger_equation.js -- Wikitube microsim
// Article: Schrödinger_equation en.wikitube.io/wiki/Schrodinger_equation
// Room: Helium Pattern: E (eigenstate animation)
// ---------------------------------------------------------------------
// Idea: a 1D particle of mass m=1, hbar=1 confined to an infinite
// square well of width L=1. The reader picks two basis eigenstates
// (n, m in 1..6) and a mixing amplitude c in [0, 1] and watches the
// time-dependent Schrodinger equation,
//
// i*hbar*dPsi/dt = H*Psi,
//
// drive the superposition through one full beat at angular frequency
// omega_nm = (E_m - E_n)/hbar. In natural units (hbar=m=L=1) the well
// eigenfunctions and energies are
//
// psi_n(x) = sqrt(2/L) * sin(n*pi*x/L) on 0 < x < L
// E_n = n*n * pi*pi / (2*m*L*L) = n*n * pi*pi / 2
//
// The full state evolves as
//
// Psi(x, t) = sqrt(1-c*c) * psi_n(x) * exp(-i*E_n*t)
// + c * psi_m(x) * exp(-i*E_m*t).
//
// When c = 0 the state is a pure eigenstate: |Psi|^2 is stationary
// and only the real and imaginary parts rotate (the global phase).
// When c > 0 the cross term in |Psi|^2 carries cos((E_m - E_n)*t),
// so the probability density visibly sloshes inside the well -- the
// cleanest demonstration that "stationary states" are stationary in
// modulus only, and that interference between eigenstates is where
// all observable time dependence lives.
//
// Visual layout (720 x 520 canvas):
// * top-left: HUD title + Wikitube subtitle URL
// * top-right: control hints
// * left band: energy ladder E_1..E_6, current n highlighted yellow,
// partner m highlighted magenta when c > 0
// * center: infinite-well plot box, V(x) walls at x=0 and x=L
// - top half: Re(Psi) (yellow) and Im(Psi) (magenta) curves
// - bottom half: |Psi|^2 (cyan filled curve)
// * bottom: sliders for n, m, c, and a play/pause button
// * bottom-right: canonical equation in ASCII
//
// Conventions (Wikitube Betterfire Standard v0):
// * const ARTICLE in single quotes -- the validator regex requires it
// * TITLE and SUBTITLE are ASCII-only so the editor preview pipeline
// does not mangle the umlaut in the slug. The non-ASCII slug only
// appears in the ARTICLE constant (and in this header comment).
// * p5.disableFriendlyErrors = true to keep the editor console clean
// * all sliders have .position(...).size(...)
// * non-ASCII (Greek letters, math symbols, arrows) lives ONLY in
// this comment block -- every text(...) string literal is ASCII
// * Energy-room palette: dark BG, hot/cold accents, structural grey,
// bright trajectory yellow for the wavefunction
// =====================================================================
const ARTICLE = 'Schrödinger_equation';
const TITLE = 'Schrodinger equation'; // ASCII override; the editor mangles non-ASCII in text()
const URL_SLUG = 'Schrodinger_equation'; // ASCII URL form
const SUBTITLE = 'Wikitube microsim . en.wikitube.io/wiki/' + URL_SLUG;
const EQN_TEXT = 'i*hbar*dPsi/dt = H*Psi E_n = n^2 * pi^2 / 2';
p5.disableFriendlyErrors = true;
// ----- Energy room palette (P5_JS_EDITOR section 4) ------------------
const BG = 18;
const FG = 240;
const DIM = [240, 240, 240, 150];
const HOT = [220, 110, 60]; // probability density baseline tint
const COLD = [60, 130, 220]; // |Psi|^2 fill
const STRUCT = [120, 130, 150]; // well walls, axis grid
const TRAJ = [240, 220, 80]; // Re(Psi) and the current eigenstate
const ACCENT = [220, 100, 200]; // Im(Psi) and the partner eigenstate
const GAUGE = [120, 220, 140]; // energy-level annotations
// ----- physics constants (natural units: hbar = m = L = 1) -----------
const L_WELL = 1.0; // well width
const NMAX = 6; // maximum quantum number on the ladder
const ENERGY = new Array(NMAX + 1); // ENERGY[n] = n^2 * pi^2 / 2
for (let n = 1; n <= NMAX; n++) {
ENERGY[n] = (n * n * Math.PI * Math.PI) * 0.5;
}
// ----- layout rectangles (computed in setup) -------------------------
let plotX, plotY, plotW, plotH; // wavefunction plot box
let ladderX, ladderY, ladderW, ladderH;// energy ladder box
let controlsBaseY; // y-coordinate of the slider row
// ----- DOM controls --------------------------------------------------
let nSlider, mSlider, cSlider, playBtn, resetBtn;
// ----- simulation state ----------------------------------------------
let simTime = 0; // accumulated time in natural units
let playing = true; // play/pause toggle for time evolution
// ====================================================================
// setup() -- canvas, palette, layout, and DOM controls
// ====================================================================
function setup() {
createCanvas(720, 520);
pixelDensity(2);
textFont('system-ui');
// layout: split the canvas into a left ladder, a center plot, and a
// controls strip across the bottom. Numbers chosen so the plot fits
// 0 <= x <= L cleanly and the energy ladder reads from E_1 (bottom)
// up to E_NMAX (top).
ladderX = 22;
ladderY = 70;
ladderW = 60;
ladderH = 320;
plotX = 110;
plotY = 70;
plotW = 580;
plotH = 320;
controlsBaseY = 420;
// controls row (all sliders explicitly positioned per Betterfire FES2)
nSlider = createSlider(1, NMAX, 1, 1).position(110, controlsBaseY).size(180);
mSlider = createSlider(1, NMAX, 2, 1).position(110, controlsBaseY + 28).size(180);
cSlider = createSlider(0, 1, 0, 0.01).position(110, controlsBaseY + 56).size(180);
playBtn = createButton('pause').position(310, controlsBaseY);
playBtn.mousePressed(() => {
playing = !playing;
playBtn.html(playing ? 'pause' : 'play');
});
resetBtn = createButton('reset t').position(370, controlsBaseY);
resetBtn.mousePressed(() => { simTime = 0; });
}
// ====================================================================
// helpers -- well eigenfunction, energy, full superposition
// ====================================================================
// psi_n(x) = sqrt(2/L) * sin(n*pi*x/L), real-valued on the well
function psi_n(n, x) {
return Math.sqrt(2.0 / L_WELL) * Math.sin(n * Math.PI * x / L_WELL);
}
// Full time-dependent superposition Psi(x, t) returned as {re, im}.
// Psi(x, t) = sqrt(1-c^2) psi_n(x) e^{-i E_n t} + c psi_m(x) e^{-i E_m t}
function psi_super(n, m, c, x, t) {
const an = Math.sqrt(1.0 - c * c);
const am = c;
const En = ENERGY[n];
const Em = ENERGY[m];
const re = an * psi_n(n, x) * Math.cos(En * t)
+ am * psi_n(m, x) * Math.cos(Em * t);
// e^{-i E t} = cos(E t) - i sin(E t), so Im(Psi) carries -sin(E t)
const im = -an * psi_n(n, x) * Math.sin(En * t)
- am * psi_n(m, x) * Math.sin(Em * t);
return { re: re, im: im };
}
// ====================================================================
// draw() -- frame loop
// 1. integrate simulation time
// 2. paint background
// 3. draw energy ladder (left)
// 4. draw wavefunction plot (center)
// 5. draw controls labels (bottom)
// 6. drawHUD() (top-left title + bottom-right equation)
// ====================================================================
function draw() {
background(BG);
// 1) time integration ------------------------------------------------
if (playing) {
// dt in natural units; capped to keep beat smooth on slow frames
const dt = Math.min(deltaTime / 1000, 0.05);
simTime += dt;
}
// read controls once into named locals so the physics block reads
// as physics rather than as UI plumbing
const n = nSlider.value();
const mPart = mSlider.value();
const cMix = cSlider.value();
// 2-5) regions -------------------------------------------------------
drawEnergyLadder(n, mPart, cMix);
drawWavePlot(n, mPart, cMix, simTime);
drawControlsRow(n, mPart, cMix);
drawHUD();
}
// ====================================================================
// Energy ladder (left band)
// Six horizontal rules at heights proportional to E_n / E_NMAX, so
// the n=1 line sits near the bottom and n=6 near the top. The active
// eigenstate (n) is drawn yellow; the superposition partner (m) is
// drawn magenta with reduced alpha when the mixing amplitude is zero.
// ====================================================================
function drawEnergyLadder(n, mPart, cMix) {
noFill();
stroke(STRUCT[0], STRUCT[1], STRUCT[2], 180);
strokeWeight(1);
rect(ladderX, ladderY, ladderW, ladderH);
// gridlines and labels for E_1..E_NMAX
textAlign(RIGHT, CENTER);
textSize(10);
const Emax = ENERGY[NMAX];
for (let k = 1; k <= NMAX; k++) {
const yk = ladderY + ladderH - (ENERGY[k] / Emax) * ladderH;
const lit = (k === n) || (k === mPart && cMix > 0.001);
let col = STRUCT;
if (k === n) col = TRAJ;
if (k === mPart && cMix > 0.001) col = ACCENT;
const a = lit ? 240 : 110;
stroke(col[0], col[1], col[2], a);
strokeWeight(lit ? 2 : 1);
line(ladderX + 6, yk, ladderX + ladderW - 22, yk);
noStroke();
fill(col[0], col[1], col[2], a);
text('E' + k, ladderX + ladderW - 4, yk);
}
// header label for the ladder
noStroke();
fill(DIM[0], DIM[1], DIM[2], DIM[3]);
textAlign(LEFT, TOP);
textSize(11);
text('energy ladder', ladderX, ladderY - 14);
}
// ====================================================================
// Wavefunction plot (center)
// Top half: Re(Psi) (yellow) and Im(Psi) (magenta) sampled across
// the well at ~2 px per sample.
// Bottom half: |Psi|^2 = Re^2 + Im^2, drawn as a filled cyan curve
// above the x-axis.
// The walls of the infinite well are drawn as solid grey rectangles
// outside [0, L] and the inside is left dark.
// ====================================================================
function drawWavePlot(n, mPart, cMix, t) {
// axes: split the box into a top half for Psi and a bottom half for |Psi|^2
const midY = plotY + plotH * 0.5;
const halfH = plotH * 0.5;
// background panels (the well floor V=0 is drawn dark; walls at x=0
// and x=L are conceptual infinite barriers drawn as thicker grey rules)
noStroke();
fill(28);
rect(plotX, plotY, plotW, plotH);
// sample the wavefunction across the well at ~1.5 px steps
const samples = 256;
let maxAbs2 = 0;
const reArr = new Array(samples + 1);
const imArr = new Array(samples + 1);
const xArr = new Array(samples + 1);
for (let i = 0; i <= samples; i++) {
const x = (i / samples) * L_WELL;
const psi = psi_super(n, mPart, cMix, x, t);
reArr[i] = psi.re;
imArr[i] = psi.im;
xArr[i] = x;
const a2 = psi.re * psi.re + psi.im * psi.im;
if (a2 > maxAbs2) maxAbs2 = a2;
}
// amplitude axis scaling: pin to sqrt(2/L)*1.2 so the curves do not
// jitter when sliders move. The well's maximum eigenstate amplitude
// is exactly sqrt(2/L) so 1.2x leaves a clean margin.
const ampScale = Math.sqrt(2.0 / L_WELL) * 1.2;
// |Psi|^2 axis: pin to (2/L)*1.2 for the same reason
const prob2Scale = (2.0 / L_WELL) * 1.2;
// top half: Re and Im traces
// axis line at midY (Psi = 0)
stroke(STRUCT[0], STRUCT[1], STRUCT[2], 120);
strokeWeight(1);
line(plotX, plotY + halfH * 0.5, plotX + plotW, plotY + halfH * 0.5);
// Im(Psi) drawn first (under) in magenta
noFill();
stroke(ACCENT[0], ACCENT[1], ACCENT[2], 200);
strokeWeight(1.6);
beginShape();
for (let i = 0; i <= samples; i++) {
const px = plotX + (xArr[i] / L_WELL) * plotW;
const py = plotY + halfH * 0.5 - (imArr[i] / ampScale) * (halfH * 0.45);
vertex(px, py);
}
endShape();
// Re(Psi) drawn on top in trajectory yellow
stroke(TRAJ[0], TRAJ[1], TRAJ[2], 240);
strokeWeight(1.8);
beginShape();
for (let i = 0; i <= samples; i++) {
const px = plotX + (xArr[i] / L_WELL) * plotW;
const py = plotY + halfH * 0.5 - (reArr[i] / ampScale) * (halfH * 0.45);
vertex(px, py);
}
endShape();
// bottom half: |Psi|^2 filled cyan curve
// baseline at the bottom of the plot box
const baseY = plotY + plotH - 6;
stroke(STRUCT[0], STRUCT[1], STRUCT[2], 120);
strokeWeight(1);
line(plotX, baseY, plotX + plotW, baseY);
noStroke();
fill(COLD[0], COLD[1], COLD[2], 120);
beginShape();
vertex(plotX, baseY);
for (let i = 0; i <= samples; i++) {
const px = plotX + (xArr[i] / L_WELL) * plotW;
const p2 = reArr[i] * reArr[i] + imArr[i] * imArr[i];
const py = baseY - (p2 / prob2Scale) * (halfH - 8);
vertex(px, py);
}
vertex(plotX + plotW, baseY);
endShape(CLOSE);
noFill();
stroke(COLD[0], COLD[1], COLD[2], 220);
strokeWeight(1.6);
beginShape();
for (let i = 0; i <= samples; i++) {
const px = plotX + (xArr[i] / L_WELL) * plotW;
const p2 = reArr[i] * reArr[i] + imArr[i] * imArr[i];
const py = baseY - (p2 / prob2Scale) * (halfH - 8);
vertex(px, py);
}
endShape();
// walls of the infinite square well: thick grey rules at the plot
// edges with little "infinity" markers above. The well floor V=0 is
// the dark panel between them.
stroke(STRUCT[0], STRUCT[1], STRUCT[2], 220);
strokeWeight(3);
line(plotX, plotY + 4, plotX, plotY + plotH - 4);
line(plotX + plotW, plotY + 4, plotX + plotW, plotY + plotH - 4);
// mid-panel divider between Re/Im and |Psi|^2
stroke(STRUCT[0], STRUCT[1], STRUCT[2], 80);
strokeWeight(1);
line(plotX, midY, plotX + plotW, midY);
// panel labels
noStroke();
textSize(11);
textAlign(LEFT, TOP);
fill(TRAJ[0], TRAJ[1], TRAJ[2], 240);
text('Re Psi', plotX + 6, plotY + 4);
fill(ACCENT[0], ACCENT[1], ACCENT[2], 240);
text('Im Psi', plotX + 56, plotY + 4);
fill(COLD[0], COLD[1], COLD[2], 240);
text('|Psi|^2', plotX + 6, midY + 4);
// wall labels x=0, x=L
fill(DIM[0], DIM[1], DIM[2], DIM[3]);
textAlign(LEFT, BOTTOM);
text('x = 0', plotX + 6, plotY + plotH - 8);
textAlign(RIGHT, BOTTOM);
text('x = L', plotX + plotW - 6, plotY + plotH - 8);
// live readout: simulation time and beat frequency
textAlign(LEFT, BOTTOM);
fill(GAUGE[0], GAUGE[1], GAUGE[2], 220);
const omega = Math.abs(ENERGY[mPart] - ENERGY[n]);
const tStr = 't = ' + t.toFixed(2);
const wStr = 'omega_nm = ' + omega.toFixed(2);
text(tStr + ' ' + wStr, plotX + 80, plotY + plotH - 8);
}
// ====================================================================
// Controls row (bottom strip)
// Each slider gets a label and a current-value readout. The labels
// align with the slider .position() x-coordinate so the layout is
// a clean three-line block.
// ====================================================================
function drawControlsRow(n, mPart, cMix) {
noStroke();
textSize(11);
textAlign(LEFT, CENTER);
fill(DIM[0], DIM[1], DIM[2], DIM[3]);
text('n (state) = ' + n, 300, controlsBaseY + 8);
text('m (partner) = ' + mPart, 300, controlsBaseY + 36);
text('c (mix amp) = ' + cMix.toFixed(2), 300, controlsBaseY + 64);
// hint about superposition
fill(GAUGE[0], GAUGE[1], GAUGE[2], 200);
let hint;
if (cMix < 0.005) {
hint = 'pure eigenstate -- |Psi|^2 is stationary';
} else if (cMix > 0.995) {
hint = 'pure partner eigenstate -- |Psi|^2 is stationary';
} else {
hint = 'superposition -- |Psi|^2 oscillates at omega_nm';
}
text(hint, 440, controlsBaseY + 64);
}
// ====================================================================
// HUD (top-left title block + bottom-right canonical equation)
// Required by Betterfire Standard: title + Wikitube subtitle URL up
// top, equation text in the bottom-right with an "=" so the BF4 check
// finds it. All text() literals are ASCII per the room standard.
// ====================================================================
function drawHUD() {
// top-left: title block
noStroke();
fill(0, 0, 0, 140);
rect(8, 6, 360, 36, 4);
fill(FG);
textAlign(LEFT, TOP);
textSize(20);
text(TITLE, 14, 10);
fill(DIM[0], DIM[1], DIM[2], DIM[3]);
textSize(11);
text(SUBTITLE, 14, 30);
// top-right: control hints
fill(DIM[0], DIM[1], DIM[2], DIM[3]);
textAlign(RIGHT, TOP);
textSize(11);
text('drag n, m, c sliders below', width - 12, 10);
text('press pause to freeze t', width - 12, 24);
// bottom-right: canonical equation
noStroke();
fill(0, 0, 0, 140);
rect(width - 360, height - 28, 352, 22, 4);
fill(FG);
textAlign(RIGHT, BOTTOM);
textSize(12);
text(EQN_TEXT, width - 14, height - 10);
}
```
## Links (Wikipedia order)
<!-- injected from _registry/childlinks/Schrödinger_equation.json (2026-07-30T02:09:12Z) -->
`(2+1)-dimensional_topological_gravity` · `4D_N_=_1_global_supersymmetry` · `4D_N_=_1_supergravity` · `6D_(2,0)_superconformal_field_theory` · `ABJM_superconformal_field_theory` · `Abdus_Salam` · `Action_(physics)` · `AdS/CFT_correspondence` · `Adiabatic_quantum_computation` · `Adrian_Kent` · `Albert_Einstein` · `Algebraic_quantum_field_theory` · `Algorithmic_cooling` · `American_Journal_of_Physics` · `American_Mathematical_Society` · `Amplitude_amplification` · `Annalen_der_Physik` · `Anomalous_magnetic_dipole_moment` · `Antiparticle` · `Anton_Zeilinger` · `Applied_physics` · `Arnold_Sommerfeld` · `Arthur_Compton` · `Ashcroft_and_Mermin` · `Asher_Peres` · `Astrophysics` · `Atomic,_molecular,_and_optical_physics` · `Atomic_orbital` · `Atomic_physics` · `Axiomatic_quantum_field_theory` · `Azimuthal_quantum_number` · `BB84` · `BF_model` · `BHT_algorithm` · `BQP` · `Bacon–Shor_code` · `Barton_Zwiebach` · `Bas_van_Fraassen` · `Batalin–Vilkovisky_formalism` · `Bekenstein_bound` · `Bell's_theorem` · `Bell_test` · `Bernstein–Vazirani_algorithm` · `Bhabha_scattering` · `Biophysics` · `Black-body_radiation` · `Black_hole` · `Black_hole_complementarity` · `Black_hole_information_paradox` · `Black_hole_thermodynamics` · `Bloch's_theorem` · [[Bohr_model]] · `Bohr_radius` · `Born_rule` · `Born–Infeld_model` · `Boson_sampling` · `Bosonic_string_theory` · `Bousso's_holographic_bound` · `Bra–ket_notation` · `Breit–Wheeler_process` · `Bremsstrahlung` · `Brillouin_zone` · `Bullough–Dodd_model` · `Bunch–Davies_vacuum` · `C._V._Raman` · `CA-duality` · `CGHS_model` · `CSS_code` · [[Calculus]] · `Cambridge_University_Press` · `Canonical_commutation_relation` · `Canonical_quantization` · `Canonical_quantum_gravity` · `Casimir_effect` · `Causal_dynamical_triangulation` · `Causal_patch` · `Causal_sets` · `Cavity_quantum_electrodynamics` · `Chandralekha_Singh` · `Charge_qubit` · `Chern–Simons_theory` · `Chiral_model` · `Circuit_quantum_electrodynamics` · `Cirq` · `Classical_capacity` · `Classical_mechanics` · `Claude_Cohen-Tannoudji` · `Cloud-based_quantum_computing` · `Cluster_state` · `Commutator` · `Complementarity_(physics)` · `Complex_number` · [[Complex_system]] · `Compton_scattering` · `Condensed_matter_physics` · `Conformal_field_theory` · `Consciousness_causes_collapse` · `Consistent_histories` · `Continuity_equation` · `Continuous-variable_quantum_information` · `Convex_set` · `Copenhagen_interpretation` · `Corpuscular_theory_of_light` · `Cosmic_censorship_hypothesis` · `Cosmic_string` · `Cosmology` · [[Coulomb's_law]] · `Cross-entropy_benchmarking` · `Crystal_momentum` · `David_Hilbert` · `David_J._Griffiths` · `Davisson–Germer_experiment` · `De_Broglie–Bohm_theory` · `Decoy_state` · `Degenerate_energy_levels` · `Delayed-choice_quantum_eraser` · `Delbrück_scattering` · `Density_matrix` · `Derivative` · `Deutsch–Jozsa_algorithm` · `DiVincenzo's_criteria` · [[Differential_equation]] · `Dirac_delta_function` · `Dirac_equation` · 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## From the Real GENERATIVE library
*Schrödinger equation — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Energy room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Grave_Schroedinger_%28detail%29.png).*
*Animated: Schrödinger equation — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (Energy room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Wavepacket-a2k4-en.gif).*
> The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system.[1]: 1–2 Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work ([Wikipedia](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation))
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## Media (PD/CC)
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!Gif Library/Schrödinger equation/Wavepacket-a2k4-en.gif
*Wavepacket-a2k4-en.gif · Xcodexif · CC BY-SA 4.0 · [source](https://commons.wikimedia.org/wiki/File:Wavepacket-a2k4-en.gif)*
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**Semiotic universals** (the notations and alphabet letters this article speaks — each opens its canonical card): greek in science · energy · sampling · superposition · amplitude. Index: the glyph gallery · SEMIOTICS PORTAL.
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> **Room:** [[Helium]] · **Status:** ✅ shipped
## Overview
The Schrödinger equation is the linear [[Partial_differential_equation|partial differential equation]] that governs the time [[Evolution|evolution]] of the quantum state of a non-relativistic [[Physical_system|physical system]]. Formulated by Erwin Schrödinger in 1926 — building on Louis de Broglie's matter-[[Wave|wave]] hypothesis and Werner Heisenberg's matrix mechanics — it is the wave-mechanical foundation of quantum theory, on the same footing in [[Physics|physics]] as Newton's second law in classical mechanics. The time-dependent form states that the [[Energy|energy]] operator (the Hamiltonian H) generates translations of the wave function in time, so that i ℏ ∂Ψ/∂t = H Ψ. For a single non-relativistic particle of mass m in a potential V, the Hamiltonian is H = −(ℏ²/2m) ∇² + V, and stationary states satisfy the eigenvalue form H ψ = E ψ. The squared magnitude |Ψ|² is interpreted, following Max Born, as a probability [[Density|density]] over configuration space, so the equation predicts not trajectories but the statistics of measurement outcomes.
Closed-form solutions exist for a small canonical set — the free particle, the infinite and finite square wells, the harmonic oscillator, the hydrogen atom — and these solutions undergird the quantum numbers, selection rules, and tunneling probabilities used throughout atomic, molecular, condensed-matter, and nuclear physics. For helium, the two-[[Electron|electron]] Schrödinger equation has no analytic solution and motivated the variational and perturbative methods that became standard tools in quantum [[Chemistry|chemistry]]. The same equation governs the qubit evolution that quantum-computing platforms implement on superconducting and trapped-ion hardware, and it sets the energy gap [[Structure|structure]] exploited by helium-cooled detectors, masers, and spectrometers.
## 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 159 of the Helium sheet on 2026-05-14T21:13:43Z.*
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*Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*
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**Microsim — three.js (Wikitube framework):** *Schrödinger equation*
<div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Schrödinger_equation.html" data-title="Schrödinger equation"></div>
*Built from `MICROSIM_GUIDE/specs/sims/Schrödinger_equation.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).*
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## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Schrödinger_equation) : [Wikitube](https://en.wikitube.io/wiki/Schrödinger_equation)
## Previous hub tags
Tree parent: [[Hydrogen]].
Legacy hubs: none.
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*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*