A Positive Energy Wave Packet Solution of the Dirac Equation

Abstract

A wave packet solution of the Dirac equation is obtained by superposition of positive energy solutions only. The solution involves an arbitrary parameter, and it is shown that this can be chosen so that all three dimensions of the packet are less than $\frac{\hslash }{\mathrm{mc}}$. The possibility of constructing such a packet had been doubted. It is also shown that in the limit $\Delta x=0\mathrm{}$, $\Delta x\Delta p_x=1.49\mathrm{}\hslash$. It is not known whether this is the smallest value for the uncertainty product, subject to the condition that only positive energy solutions are used in the construction of the packet, or whether some other solution might yield a still smaller value.