Phase-Modulus Relations for a Reflected Particle

Abstract

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 {\it 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 ($\hbar K >> \frac{\hbar trajectory length} {(size of wave-packet)^2} >>> \frac{\hbar}{size of wave-packet}$) the rate of acquisition of the relative phase between different wp components is enormous (for a bullet it is typically $10^{14}$ GHertz) with also a very large value for the phase maximum.