This sim builds a localised wave packet on a ring out of plane waves with whole-number k, then lets the clock run, in units where ħ = 2m = 1. It shows how the packet's shape comes from the amplitudes and phases of its waves, and so how a free quantum particle's wave function changes with time.

What's on screen

• Cards across the top, one per wave drawn: its k, its amplitude |c|, an arrow for its phase, and the wave itself.
• Fourier Sum |ψ|² (white): the probability density of the waves you draw, with the real and imaginary parts of ψ in lime and rose.
• Target Gaussian |ψ|² (dashed slate): the packet at t = 0 built from all 80 waves. It never changes.
• Statistics: the clock t, Total Norm, Plotted Norm and Captured. The ⟳ FULL REVIVAL badge in the top bar lights near t = 2π.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: ħ = 2m = 1; the ring is −π ≤ x ≤ π; the 80 waves are k = −40 … 39; the packet has no net drift (its mean k is 0); the clock wraps at 4π.

On the ring only plane waves with whole-number k fit. Any packet can be written as a sum of them, with one complex number c_k per wave: |c_k| is the wave's amplitude and the phase of c_k sets where along x it starts.

The sim draws only the N waves with the largest |c_k| and adds them. Captured is the share of the packet's probability, Σ|c_k|², that those N waves carry.

• |ψ|² is a probability density. Over the whole ring it integrates to 1 (Total Norm).
• The dashed target is the t = 0 packet built from all 80 waves. At σ = 0.25, N = 3 captures about 55% of it and N = 13 about 99.9%.

The sim's packet is a Gaussian, ψ(x, 0) ∝ exp[−(x − x₀)²/4σ²]. Its amplitudes are a Gaussian too, |c_k| ∝ e^−k²σ², so the two widths are tied together:

• Δx = σ, the rms width of |ψ|².
• Δk = 1/2σ, the rms width of |c_k|².
• Δx·Δk = ½, the smallest product the uncertainty relation allows (Δp = ħΔk).

Halving σ doubles Δk, so a narrower packet needs waves out to larger |k|. A wider packet needs fewer: at σ = 1.0 the three waves k = 0, ±1 carry over 99%, against about 55% at σ = 0.25.

At the widest settings the ring cuts the Gaussian's tails and the sim wraps them round, so σ is then the Gaussian's parameter rather than a measured width.

Each wave k has energy E = ħ²k²/2m, which is k² with ħ = 2m = 1, so its phase turns at the angular frequency ω_k = k²:

• Every |c_k| stays fixed. Only the phases turn: k = 0 stands still, k = ±1 turn together, k = ±2 turn four times faster, k = ±3 nine times faster.
• |ψ|² changes because the waves drift out of step. Re ψ and Im ψ hold the phase information that |ψ|² hides: at t = 0 the packet is real, so Im ψ starts flat.
• The amber dashed line vg marks x₀ + v_gt, where v_g = dω/dk = 2k at the packet's mean k. That mean is 0 here, so the line stays at x₀.

Waves with large |k| turn fastest, so they fall out of step with the slow ones. For a free particle on a line the Gaussian's width then grows as

• At t = τ = σ₀² the width is √2 larger and the peak of |ψ|² is √2 lower. Total probability stays 1; it is spread over more of x.
• A narrower packet has a larger Δk, so it spreads sooner: halve σ₀ and τ is four times shorter.
• On the ring this law holds only until the packet meets itself. At σ₀ = 0.25 that is about t = 0.45, when the law's width reaches π/√3 ≈ 1.81, the width of probability spread evenly round the ring.

On the ring every k is a whole number, so every turning rate k² is too. After t = 2π, wave k has turned exactly k² times and all the phases are back together. At t = π the factor e^−ik²π equals (−1)^k, which is what shifting x by π does to e^ikx.

• t = 2π: the packet is exactly as it began.
• t = π: the same packet, on the opposite side of the ring.
• t = π/2: two copies, at x₀ and x₀ + π, each carrying half the probability.

The ⟳ badge lights for t within 0.3 of a multiple of 2π, so it also lights just after a restart at t = 0. That is the start, not a revival.

• A free electron spreads almost at once. In real units τ = 2mσ₀²/ħ. For σ₀ = 1 nm that is about 1.7 × 10⁻¹⁴ s, and the width has doubled by about 3 × 10⁻¹⁴ s.
• A heavy object barely spreads at all. A 1 g bead with σ₀ = 1 µm has τ of about 2 × 10¹⁹ s, roughly forty times the age of the universe.
• Spreading and revival are seen in experiments. Wave packets of highly excited (Rydberg) atoms spread and later come back, fully or in fractions, approximately.

Key points

• ψ = Σ c_k e^ikx; |ψ|² is a density that integrates to 1.
• Δx = σ, Δk = 1/2σ, Δx·Δk = ½: narrow in x means wide in k.
• Each |c_k| stays fixed; wave k's phase turns at ω = k² (ħ = 2m = 1).
• The width grows by √2 at τ = σ₀². On the ring the packet returns at 2π, shifted at π, in two halves at π/2.

Common mix-ups

• "The packet spreads because each wave spreads." Each wave already fills the ring and its amplitude never changes. The phases drift out of step.
• "The packet loses probability as it spreads." Total Norm stays 1; the same probability covers more of x.
• "Captured says how accurate the physics is." It says how much of the packet the N chosen waves draw. Those waves evolve exactly.

Things to try on your own

• At σ = 0.25, find the smallest N with Captured above 99%. Repeat at σ = 0.5.
• Set x₀ = −2, pause near t/2π = 0.5, and read off where the packet sits.
• Hide Re ψ, keep Im ψ, and describe it at t = 0 and a little later.