Bundle Comparison: Waves

Starting from Gaussian wave packet at ( t=0 ): [ \psi(x,0) = \left( \frac12\pi\sigma_0^2 \right)^1/4 e^-x^2/(4\sigma_0^2) e^ik_0x ] Fourier transform gives ( A(k) \propto e^-\sigma_0^2 (k-k_0)^2 ). Using ( \omega = \hbar k^2/(2m) ), integrate to get [ |\psi(x,t)|^2 = \frac1\sqrt2\pi , \sigma(t) e^-(x - v_g t)^2/(2\sigma(t)^2), \quad \sigma(t) = \sigma_0 \sqrt1 + \left( \frac\hbar t2m\sigma_0^2 \right)^2 ] Hence width grows unbounded as ( t \to \infty ). ∎

A nondispersive medium (( \omega \propto k )) preserves shape. A dispersive medium (any curvature in ( \omega(k) )) causes spreading. Quantum free space is inherently dispersive; vacuum EM is not. We have compared wave bundles across three fundamental domains. All are described by Fourier superpositions, but their evolution depends entirely on the dispersion relation. Mechanical strings and vacuum EM allow distortion-free propagation; quantum free particles invariably spread. This comparison clarifies why laser pulses can travel across the universe without broadening (in vacuum), while an electron’s position certainty decays rapidly. waves bundle comparison

If ( \omega(k) ) is linear in ( k ), the bundle propagates without distortion. If nonlinear, the envelope spreads over time. Governing equation: 1D wave equation [ \frac\partial^2 y\partial t^2 = v^2 \frac\partial^2 y\partial x^2, \quad v = \sqrtT/\mu ] where ( T ) = tension, ( \mu ) = linear density. Starting from Gaussian wave packet at ( t=0

[ \omega = c|k| \quad \text(linear, nondispersive) ] A dispersive medium (any curvature in ( \omega(k)