\delta x \delta p=m \times {\hbar  \over 2 }\label{Harmonic}.\end{aligned}$$

A wave packet emerges from the matter effects. The wave in the medium is affected by disorders and is described by one function within a finite length, which we define as a coherence length. When this wave of finite coherence is emitted from the medium into the vacuum, it has a finite time length and a finite energy width, because the wave in the matter lives for a finite period of time, of the order of a mean free time. Consequently, this wave packet in the vacuum is not the eigenstate of the Hamiltonian but it is the linear combination of different energies. Its width is determined from the interactions of the particle with scatterers. There are other situations where wave packets are formed. We will study them and estimate wave packet sizes in §2.

Since the wave packet is a linear combination of plane waves of different energies, the wave packet is not a stationary state and varies with time. For a nonrelativistic particle of mass $m$, the energy, $E={{\vec p}^2 \over 2m}$, has the width $$\begin{aligned}
\Delta E={p \over m} \Delta p= v\Delta p,~v={p \over m},\end{aligned}$$ and the time 