product of the variance of the momentum with that of the coordinate, $$\begin{aligned}
& &(\delta x)^2 \times (\delta p)^2= {(\hbar)^2 \over 4},  \end{aligned}$$ is independent from $\gamma $ and is the minimum allowed from the commutation relation.

The coherent state also satisfies the completeness condition,[@Ishikawa-Shimomura] $$\begin{aligned}
\int {d P_0 d X_0 \over 2 \pi \hbar}\langle x| P_0,X_0\rangle \langle  P_0,X_0|y\rangle 
&=& \int {d P_0 d X_0 \over 2 \pi \hbar} N_1^2 e^{ik_0(x-y)}e^{-{1 \over 2 \gamma }(x-X_0)^2-{1 \over 2 \gamma }(y-X_0)^2 } \nonumber\label{completeness}\\
&=&\delta (x-y).\end{aligned}$$ In higher dimensions, the products of the functions of each variable are used. They satisfy the minimum uncertainty relations and completeness conditions in higher dimensions.

For nonminimal wave packets, a function $ h_m(x-X_0) $ is multiplied to the right-hand side, $$\begin{aligned}
\langle x |P_0,X_0\rangle =N_1e^{i{ P_0 \over \hbar} (x-X_0)-{1 \over 2{\gamma}}(x-X_0)^2}h_m(x-X_0).
\label{non-minumum-wave packet}\end{aligned}$$ The completeness condition, Eq. $(\ref{completeness})$, is satisfied also in this case, but the product of $\delta
x$ and $\delta p$ is not minimum and is larger than the minimum. This packet should be used in realistic cases of having a larger value of the product of uncertainties. If the function $h_m(x-X_0)$ is a Hermite polynomial of order $m$, the product of uncertainties becomes $$\begin{aligned}
