to $\delta({\vec
p}_1-{\vec p}_2)$.

If the state $| \alpha \rangle $ is a wave packet described using the momentum state $|{\vec q} \rangle $ as $$\begin{aligned}
| \alpha \rangle=\int d{\vec q} F_{\alpha}({\vec q})|\vec q \rangle,\end{aligned}$$ then we have $$\begin{aligned}
C({\vec p}_1,{\vec p}_2)&=&\int d{\vec q}_1 F_{\alpha}^{*}({\vec q}_1) 
d{\vec q}_2 F_{\alpha}({\vec q}_2)\delta({\vec p}_1-{\vec q}_1)\delta({\vec p}_2-{\vec q}_2)\\
&=& F_{\alpha}^{*}({\vec p}_1) F_{\alpha}({\vec p}_2).
\nonumber\end{aligned}$$ Thus, the correlation function vanishes if $|{\vec p}_1-{\vec p}_2|$ is larger than the momentum width of the wave packet. When all the states of a complete set are added, $$\begin{aligned}
\sum_{\alpha} C({\vec p}_1,{\vec p}_2)=\sum_{\alpha } \langle {\vec p}_1|\alpha \rangle \langle \alpha|{\vec p}_2 \rangle
=\delta({\vec p}_1-{\vec p}_2),\end{aligned}$$ then the correlation function agrees with the delta function. For a state described using a wave packet, $C({\vec p}_1,{\vec p}_2)$ deviates from $\delta({\vec p}_1-{\vec p}_2)$. The deviation of the correlation function $C({\vec p}_1-{\vec p}_2)$ from the $\delta({\vec p}_1-{\vec p}_2)$ shows a feature of the state $|{\alpha }
\rangle$, and the width of ${\vec p}_1-{\vec p}_2$ is determined using the momentum width $\Delta p$. This correlation function will be used later.

We investigate the problems connected with the particle’s coherence in this series of papers. In a previous paper,[@Ishikawa-Shimomura] we showed the general features of 