to be minimum and satisfies the minimum uncertainty relation, and the reflected wave and accelerated wave have uncertainty relations of about twice the minimum. In particular, the value of $\delta x \delta p$ for the accelerated wave depends on the potential depth, as given in Fig. [\[cliff\]]{}. The product of uncertainties becomes $\frac{1}{2}\hbar$ at $V_0 \rightarrow \infty$ and $\frac{3}{2}\hbar$ at $V_0 \rightarrow 0$. This is because the wave function behaves in these regions as $$\begin{aligned}
 \psi_{wp}^3 \propto
  \begin{cases}
   k\times e^{-\frac{k^2}{2\sigma}} ,\ \ V_0 \ll E_0,\\
   \text{const}\times e^{-\frac{k^2}{2\sigma}}, \ \   V_0 \gg E_0,
  \end{cases}\end{aligned}$$ just like a wave function for a ground state and a first excited state of a harmonic oscillator in Eq. (\[Harmonic\]). The product of uncertainties of the accelerated wave is changed smoothly as seen in Fig. \[cliff\].

\[cliff\]

Potential barrier 
------------------

We study wave packets next in a simple potential, i.e., in a constant potential barrier of finite width $a$ and height or depth $V_0$ described as $$\begin{aligned}
U(x)=\begin{cases}
      0, ~x \leq 0, \\
      -V_0 ,~     0 \leq x \leq a,\\
      0, ~a\leq x.
     \end{cases}\end{aligned}$$ We obtain a wave function of the following form: $$\begin{aligned}
\psi(x)=
\begin{cases}
 e^{i{1 \over \hbar}\sqrt{2mE}x}+ B_{-}e^{-i{1 \over \hbar}\sqrt{2mE}x},~x<0,\\
=A_{+}e^{{i \over \hbar}\sqrt{2m(E+V_0)}x}+A_{-}e^{-{i \over 