Wave Packet

Describe particles with well defined position and momentum

Take the Wave function ψ(x)\psi(x) A wave packet is a real envelope ϕ(x)\phi(x) modulating a plane wave

ψ(x)=ϕ(x) eikx,ϕ(x)∈R\psi(x) = \phi(x)\, e^{ikx}, \quad \phi(x) \in \mathbb{R} ψ∗(x)=ϕ(x)e−ikx\psi^*(x) = \phi(x)e^{-ikx}

The envelope represents the probability that the particle will appear at exactly xx. ∣ψ(x)∣2=ϕ2(x)|\psi(x)|^2=\phi^2(x) is the probability density for the position.

P(a≤x≤b)=∫ab∣ψ(x)∣2 dxP(a \le x \le b) = \int_a^b |\psi(x)|^2\,dx