Phase-modulus relations for a reflected particle

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 ≫ ℏ[trajectory length/(size of wave packet)²] > > > ℏ/size of wave packet}, the rate of acquisition of the relative phase between different wp components is enormous (for a bullet it is typically 10¹⁴ GHz) with also a very large value for the phase maximum.