Gaussian wavepacket dispersion (free particle)

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