Klein-Gordon Wave Packet: Mass, Dispersion and the Light Cone

An interactive Klein-Gordon wave packet (natural units $c = \hbar = 1$; Peskin and Schroeder; Greiner, Relativistic Quantum Mechanics). A Gaussian packet $\psi(x,0) = \exp(-x^2/2 \sigma_0^2) e^{i k_0 x}$ is Fourier-synthesised with the relativistic dispersion $\omega(k) = \sqrt{k^2 + m^2}$: the phase velocity $\omega/k$ exceeds $c$ (no signal), the group velocity $v_g = k/\omega = p/E$ is strictly sub-luminal for $m \gt 0$ and exactly $c$ for $m = 0$, and $v_g v_p = 1$. A massive packet disperses (its width grows because $\omega$ is nonlinear in $k$) while its centroid moves at $v_g$ inside the light cone; a massless packet is dispersion-free ($\omega = |k|$, constant width) and rides $x = t$. The packet panel shows $|\psi|^2$ against the light cone with the initial packet for comparison; the dispersion panel shows $\omega(k)$, the light line and the $v_g$ tangent; the track panel shows the centroid against the light cone and the RMS width. The group and phase velocities satisfy $v_g v_p = 1$ with $v_g \lt c$ for a massive packet and exactly $c$ for a massless one, the massless packet is dispersion-free, and a heavier packet moves more slowly while its centroid stays inside the light cone.