Gaussian Wavepacket: Position-Momentum Uncertainty

[solas99]

how can the gaussian wavepacket presents a physical picture of the origin of position-momentum uncertainty?

 

[jtbell]

I would prefer to say that the Gaussian wave packet is an example of position-momentum uncertainty.

All wave packets, no matter what shape, have a position-momentum uncertainty relationship ΔxΔp = k, where k depends on the shape of the packet. The Gaussian wave packet is special because it can be shown from Fourier analysis theory that it has the smallest k, namely $\hbar / 2$. All other shapes of packets have larger k's. Therefore for any wave packet,

$$\Delta x \Delta p \ge \frac{\hbar}{2}$$

(the Heisenberg uncertainty principle)