---
author:
- 'Kenzo <span style="font-variant:small-caps;">ISHIKAWA</span> and Yutaka <span style="font-variant:small-caps;">TOBITA</span>'
title:  On Coherence Lengths of Wave Packets 
---

Introduction
============

When particles are identified using detectors, they show classical trajectories. These classical trajectories are observed because the particle’s wave functions have finite sizes and the probability for the particle to be observed within this width becomes unity.

A position $x$ and a momentum $p$ satisfy the canonical commutation relation, $$\begin{aligned}
[x,p]=i \hbar,
\label{commutation-relation}\end{aligned}$$ and a momentum eigenstate is extended in space. Thus, the momentum eigenstate does not show a classical trajectory, and a linear combination of momentum eigenstates, a wave packet, which has a finite spatial size, shows a classical trajectory. Wave packets are necessary for describing states with finite spatial extensions.

Wave packet behaves like a particle and is convenient for studying the connection between quantum mechanics and classical mechanics. Uncertainty relations and other properties of quantum mechanics are described well using wave packets and are explained in many textbooks of quantum mechanics [@yeazell].

In scattering experiments, an overlap between the initial and final states is studied. Since the final states are determined using detectors of finite sizes, they are described using wave packets. [@Goldberger] Their sizes are normally 