extremely heavy particles, the spreading velocities are negligible and they are localized at certain positions of the initial states. Translational invariance is violated for these objects.

Potential scatterings 
======================

In this section, we study the scatterings of the wave packets using simple rectangular potentials. We see that the sizes of extensions in the positions and momenta have strong correlations with the velocities of the wave packets and the products of both sizes are approximately adiabatic invariants. Minimum wave packets are changed to nonminimum wave packets in certain reactions.

Potential wall in one dimension
-------------------------------

We study the wave packets first in a simple potential, i.e., in a constant potential wall of a height or depth $V_0$ described by $$\begin{aligned}
U(x) = \begin{cases}
    V_0, \ x \leq 0, \\
    0, \ \ 0 < x.
       \end{cases}\end{aligned}$$ We obtain a solution of Schrödinger equation, $$\begin{aligned}
& & i\hbar {\partial \over \partial t} \psi(x,t)=H\psi(x,t),\\
& &H={p^2 \over 2m}+U(x),\end{aligned}$$ of the following form, $$\begin{aligned}
\psi(x,t)&=&e^{Et \over i\hbar} \psi(x), \\
\psi(x) &=& \begin{cases}
       e^{i k x}+ B_{-}e^{-i k x},~x < 0, \\     
       C_{+}e^{i k'x},~0 < x,
      \end{cases}\end{aligned}$$ where at $x<0$, a right-moving plane wave comes in toward the wall and a left-moving wave of magnitude 