particle moves with a velocity $v$, the time spent by this particle crossing one position is given using the mean free path over the particle’s velocity, $$\begin{aligned}
\tau ={l \over v  }.\end{aligned}$$ This state has an uncertainty of time $\tau$ and an uncertainty of energy $\delta E$. $\delta E$ is given by $$\begin{aligned}
\delta E={\hbar \over \tau}={\hbar \over l}v.
\label{delta-energy-3}\end{aligned}$$ The energy width of Eq. $(\ref{delta-energy-3})$ agrees with those of Eqs. $(\ref{delta-energy-1})$ and $(\ref{delta-energy-2})$.

Thus, in matter, a particle wave has a finite spatial size. Consequently, a wave behaves as a wave packet of this spatial size. The momentum width is determined from the inverse of the spatial width, and the wave is approximately given using the minimum wave packet.

Long mean free path: finite momentum spreads
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When there is a degeneracy in one particle energy, a superposition of the states with the same energy is also an eigenstate. Which eigenstate is realized depends on the production process. In this section, we study wave packets that have origins in momentum spreading of the produced particles and are connected with the energy degeneracy.

### Many-particle states: transmutation of momentum spreads

In a system of many particles with long mean free 