Many interacting worlds

Interactive simulation of the Hall, Deckert and Wiseman model: N classical worlds, each with the particle at a definite position, repel each other and Newton's equations alone reproduce uncertainty, zero-point energy, packet spreading, interference and the collapse of a measurement. It has no dependencies and works offline.

Open it by double-clicking simulations/quantum-mechanics/many-interacting-worlds/index.html (or from any static server); nothing to install. Theory and references: many-interacting-worlds-theory.en.md; Italian version: many-interacting-worlds-simulation-guide.it.md.

Idea

Each world n is a classical particle with position x[n] and velocity v[n]. Neighbouring worlds repel with a force that grows as the gap shrinks (the discrete quantum potential, units ħ = m = 1). With N = 1 the repulsion is zero and plain classical mechanics remains; the quantum behaviour is a property of the ensemble. The formulas and the exact results are in the theory document.

Scenarios

Parameters and query string

Example: index.html?sc=collision&N=40&w=0.5&v0=3.

Language and theme (System, Light, Dark) are set with the gear button at the top right. They are saved in the browser and apply to the other simulations too.

Readings

• Coupled worlds: worlds still interacting with the drawn one / total. It drops at each measurement.
• Δx, Δp: spread of the positions and of the momenta of the coupled ensemble. Δp includes the osmotic term 2U/N, the part of the momentum spread stored in the inter-world repulsion U.
• Δx·Δp next to the exact value (1−1/N)·ħ/2: equal in the stationary state, larger otherwise.
• Total energy next to the expected energy (N−1)·ħω/2 in the trap scenarios. The integrator keeps the energy drift around 10⁻¹⁶ per period in the ground state and below about 10⁻⁴ in the hardest collisions.
• Δx(t)/Δx(0) vs √(1+ω²t²) in the free scenario: the two numbers should match at all times.

After a measurement the kept bin is squeezed to a width of order δ with a large repulsion energy, so Δx first collapses and then grows back fast: that is the momentum kick a sharp position measurement must give.

Graph legend

• Time horizontally, position vertically: one line per world. Worlds never cross.
• The world drawn by a measurement is thick and warm-coloured; discarded bins turn grey from the dashed vertical line marking the measurement time.
• Right strip: histogram of the current world density (grey for decoupled worlds) and, dashed, the quantum density |ψ|² expected in the stationary and free cases.
• Pointer on the graph: tooltip with the world index, position and, on the last instant, velocity.

Structure

simulations/quantum-mechanics/many-interacting-worlds/
├── index.html      page and control panel
├── css/style.css   theme tokens (light, dark, system) and responsive layout
├── js/i18n.js      it/en dictionaries and translation functions
├── js/model.js     forces, ground state, integrator, measurement (also usable from Node)
├── js/graph.js     Canvas renderer: world lines, histogram, tooltip
└── js/app.js       controls, scenarios, query string, animation

The settings button (language and theme) is shared by all simulations and lives in simulations/shared/ (js/settings.js, css/settings.css).

Simplifications

• One particle in one dimension, equally weighted worlds: the toy model of the 2014 paper. No entanglement, no spin (see the theory document, section 5).
• The measurement splits the discarded worlds into bins no wider than δ and each bin keeps evolving as an independent ensemble; the kept ensemble is the one the readings describe.
• Velocity Verlet with adaptive substeps (the repulsion stiffens as 1/g³); a substep that would make worlds cross is halved and retried. The substep budget bounds the work per frame, so extreme parameters can slow the simulated time rather than the browser.

Verification

cd simulations/quantum-mechanics/many-interacting-worlds && node -e 'const M=require("./js/model.js"); const I=require("./js/i18n.js").I18n; const x=M.groundState(11,1); const s=M.statistics(M.createState(x,null,1)); let sq=0; x.forEach(function(p){sq+=p*p}); console.log(sq.toFixed(6), s.energy.toFixed(6), s.product.toFixed(6), I.check().length)'

Expected: 5.000000 5.000000 0.454545 0 (sum rule Σx² = (N−1)/(2ω), energy (N−1)·ω/2, product (1−1/N)/2 for N = 11, and no missing keys in the dictionaries).

cd simulations/quantum-mechanics/many-interacting-worlds && node -e 'const M=require("./js/model.js"); const st=M.createState(M.groundState(21,1),null,0); const d0=M.statistics(st).dx; while(st.t<1-1e-9) M.step(st,0.05,1e6); console.log((M.statistics(st).dx/d0).toFixed(6))'

Expected: 1.414214 (free spreading √(1+t²) = √2 at t = 1).