Wave Packet Derivation -

We’ll start with the simplest 1D case. A single plane wave [ \psi_k(x,t) = e^{i(kx - \omega(k) t)} ] has definite momentum ( \hbar k ) but extends infinitely in space. To get a localized wave, we superpose many plane waves with different (k) values. 2. Wave packet definition Consider a continuous superposition:

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) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} A(k) , e^{i(kx - \omega(k) t)} , dk ] We’ll start with the simplest 1D case

Here’s a clear, step-by-step derivation of a from the superposition of plane waves, showing how it leads to a localized disturbance. t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} A(k)