A Free Particle Moves at the Group Velocity and Spreads
An unconfined free particle has no boxed-up energy rungs. Instead any wavenumber k is allowed, and each plane wave oscillates with the dispersion relation ω(k) = ℏk²/2m. Something curious follows. The crests of the wave drift slowly at the phase velocity vp, yet the speed at which the particle actually travels is twice that, the group velocity vg. And as time passes a once-tight wave packet steadily spreads. We will watch crests, packets, the dispersion curve, and spreading one figure at a time.
Start with a single plane wave. Re(ψ) = cos(kx − ωt) is an endlessly repeating ripple. The k slider changes how tight the ripple is, and the time slider marches the crests forward. Here k = 2π/λ is the wavenumber. The speed at which a crest drifts reads out as the phase velocity vp = ℏk/2m. Raise k and the crests race faster.
Now sum several cosines of nearby k into a single packet. Let time run and the two speeds split apart. One marked crest creeps along at the phase velocity vp, while the peak of the whole packet runs ahead at the group velocity vg, twice as fast. You can watch the peak overtake the crests. For matter waves vg = 2vp always holds, and the group velocity is the speed at which the particle truly moves.
The identity of the two speeds is all inside the dispersion curve ω(k) = ℏk²/2m. This ω(k) is the dispersion frequency tying frequency to wavenumber; it is not the oscillator ω, nor a beat frequency. Drag the point along the curve. The slope of the tangent there, dω/dk, is the group velocity vg, and the slope of the chord from the origin to that point, ω/k, is the phase velocity vp. Because it is a parabola the tangent is always twice as steep as the chord, so the readout confirms vg = 2vp.
A free particle spreads with time even when left alone. As you push the time slider, |ψ(x,t)|² grows wider and flatter. This happens because the dispersion curve is curved, that is, d²ω/dk² = ℏ/m is not zero. The narrower you make the initial packet, the faster it spreads. That ties exactly to uncertainty: squeeze position and momentum widens.
Finally, place the free particle and the box side by side. The free particle allows any k, so E(k) = ℏ²k²/2m is an unbroken continuous curve. The confined box, by contrast, allows only a sparse ladder with E ∝ n². Toggle between them and the contrast is clear: free is a smooth continuum, the box is a discrete staircase. Confinement quantizes; release makes it continuous.