Particle in a box

Valid inputs immediately show a paused t=0 view. Single state displays one eigenfunction and its density. Compare shows the six normalized states separately, highlighting n₁; it is not their sum. Superposition uses equal weights and initially zero relative phase. Equal levels use one normalized state. Model edits stop playback and rebuild the initial curves. Play/Pause and the time slider select evolution explicitly; Reset returns to t=0 with the same inputs. Options gives the individual energies E₁/E₂. Only exact raw width text, view and level choices are retained, including blank, invalid or comma drafts. Legacy numeric settings migrate. Curves, model time and playback are not saved; reload rebuilds the selected initial view.

The well is 0 < x < L, with ψ = 0 at and beyond both infinite walls. For ħ = m = 1, φₙ(x) = √(2/L) sin(nπx/L), Eₙ = n²π²/(2L²), and ψ(x,t) = [φₙ₁ exp(−iE₁t) + φₙ₂ exp(−iE₂t)]/√2 for distinct levels. The mean energy is (E₁ + E₂)/2 and is constant. Each density integrates to 1. The wave plot shows solid Re ψ and dashed Im ψ during superposition. Neither is a probability: |ψ|²=(Re ψ)²+(Im ψ)². A global phase can change both components without changing the density.

The time slider covers T = 4L²/π, a guaranteed recurrence time of the complete wavefunction; a selected pair may repeat sooner. For distinct levels the density period is 2π/|E₂ − E₁|. Both vertical scales stay fixed throughout a run, so changing peaks are not disguised by automatic scaling. All time and energy values use the stated natural units, not seconds or electronvolts.

Play reveals one full recurrence interval over 12 screen seconds, then stops. This is illustrative model time, not a physical real-time clock. Pause and the slider select intermediate times. Hidden pages stop playback; reduced motion disables animation while keeping manual selection. Width 0.1–20 and levels 1–6 remain supported.

This is the exact ideal infinite-well solution, not a numerical Schrödinger solver, a finite well or a particle trajectory. Source: MIT 8.04, Lecture 6 (the lecture numbers its lowest level 0; this tool starts at 1).