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Wavepacket Dispersion in 1D

What you are seeing: a Gaussian wavepacket evolving under the free-particle Schrödinger equation. The orange curve is the real part of ψ(x,t)\psi(x, t); the cyan curve is ∣ψ∣2|\psi|^2. The packet center drifts at v=ℏk0/mv = \hbar k_0 / m, while the width broadens as σ(t)=σ01+(ℏt/2mσ02)2\sigma(t) = \sigma_0 \sqrt{1 + (\hbar t / 2 m \sigma_0^2)^2}.

Figure 1. Free-particle Gaussian wavepacket.
σ_0 (width)1.00
k_0 (momentum)3.0

WHAT TO TRY

  • Watch the packet glide right and flatten: the bottom plot draws its centre as a straight worldline (slope = group velocity v_g = hbar k0/m) inside a band that fans out, the spreading sigma(t).
  • Make sigma0 small (a sharply localized packet): it disperses fastest, because a narrow packet needs a broad spread of momenta. A wide packet barely changes width over the same time.
  • Raise k0: the de Broglie carrier inside the envelope oscillates faster and the whole packet drifts quicker, but the spreading rate is set by sigma0, not k0.