Optimally focusing wave packets

Abstract

An appropriately prepared real-valued wave packet moving in one space dimension will focus during a brief period of time even in the absence of any force. We illustrate this phenomenon by considering the time evolution of the elementary superposition of the ground state and the second excited state of a harmonic oscillator. Moreover, we show that a variation of the superposition parameter leads us from a domain of enhanced spreading via a point of suppressed spreading to a region where the wave packets focuses before it spreads again. We determine the points of maximal spreading and optimal focusing. Our analysis of this unusual behavior of a free quantum particle rests on the time dependence of (i) the average separation of the wave packet from the origin, (ii) the probability density in position space, and (iii) the Wigner phase space distribution. We conclude our search for optimally focusing wave packets by solving the corresponding variational problem with respect to a family of measures expressing the width of the wave packet.