We formulate analytically the reflection of a one-dimensional, expanding free wave packet (wp) from an infinite barrier. Three types of wp's are considered, representing an electron, a molecule, and a classical object. We derive a threshold criterion for the values of the dynamic parameters so that reciprocal (Kramers-Kronig) relations hold in the time domain between the log-modulus of the wp and the (analytic part of its) phase acquired during the reflection. For an electron, in a typical case, the relations are shown to be satisfied. For a molecule the modulus-phase relations take a more complicated form, including the so-called Blaschke term. For a classical particle characterized by a large mean momentum {K h[trajectory length/(size of wave packet)~2] > h/size of wave packet}, the rate of acquisition of the relative phase between different wp components is enormous (for a bullet it s typically 10~(14) GHz) with also a very large value for the phase maximum.
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机译：我们分析性地制定了来自无限屏障的一维扩展自由波包（wp）的反射。考虑三种类型的wp，分别代表电子，分子和经典物体。我们为动态参数的值导出一个阈值标准，以便在wp的对数模量与反射过程中获取的（其解析部分）相位之间的时域中保持相互的（Kramers-Kronig）关系。对于电子，在典型情况下，显示满足关系。对于分子，模数-相位关系采用更复杂的形式，包括所谓的Blaschke项。对于具有大的平均动量{K h [轨迹长度/（波包的大小）〜2] > h /波包的大小}的经典粒子，不同wp之间相对相位的获取速率分量非常大（对于子弹而言，通常为10〜（14）GHz），并且最大相位值也非常大。
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