Wave Packet Derivation -

Then (ignoring dispersion):

This is a Gaussian envelope moving at (v_g) — a localized pulse. If (\omega'' \neq 0), the (\kappa^2) term broadens the packet over time: [ \text{Width}(t) = \sqrt{\sigma^2 + \left( \frac{\omega'' t}{2\sigma} \right)^2 } ] so the wave packet spreads. wave packet derivation

[ \Psi(x,t) = e^{i(k_0 x - \omega_0 t)} \cdot e^{-\sigma^2 (x - v_g t)^2} \cdot \text{(constant)} ] Then (ignoring dispersion): This is a Gaussian envelope

Here’s a clear, step-by-step derivation of a from the superposition of plane waves, showing how it leads to a localized disturbance. wave packet derivation