A Gaussian wavepacket is the minimum-uncertainty state. In free space (V=0), the Schrödinger equation gives an exact analytic solution:
The packet width grows as:
Group velocity v_g = ℏk₀/m carries the probability peak. Phase velocity v_ph = ℏk₀/2m is half that.
Dispersion arises because different momentum components travel at different speeds (ω = ℏk²/2m is quadratic). A narrower initial wavepacket (smaller σ₀) spreads faster — a consequence of the Heisenberg uncertainty principle: small Δx → large Δk.