Derivation [best]: Wave Packet

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.

[ \Psi(x,0) = \left( \frac2\alpha\pi \right)^1/4 \frac1\sqrt2\pi \int_-\infty^\infty e^-\alpha (k - k_0)^2 + ikx , dk ] wave packet derivation

We define the wave packet as an integral over a range of wavenumbers . Instead of a single , we use a weighting function , typically a Gaussian distribution: A single plane wave [ \psi_k(x,t) = e^i(kx

[ E = \fracp^22m \quad \Rightarrow \quad \hbar \omega = \frac\hbar^2 k^22m \quad \Rightarrow \quad \omega(k) = \frac\hbar k^22m ] Instead of a single , we use a

$$ \omega(k) \approx \omega(k_0) + \left. \fracd\omegadk \right|_k_0 (k-k_0) + \frac12 \left. \frac{d^2

So the integral becomes:

The true power of the wave packet derivation lies in analyzing how it moves. We return to the full time-dependent integral: