Study the dynamics of the nonspreading Airy packets from the time evolution operator

Abstract

Berry and Balazs showed that an initial Airy packet Ai(b x) under time evolution is nonspreading in free space and also in a homogeneous time-varying linear potential V(x,t)=-F(t) x. We find both results can be derived from the time evolution operator U(t). We show that U(t) can be decomposed into ordered product of operators and is essentially a shift operator in x; hence, Airy packets evolve without distortion. By writing the Hamiltonian H as H=H_b+H_i, where H_b is the Hamiltonian such that Ai(b x) is its eigenfunction. Then, H_i is shown to be as an interacting Hamiltonian that causes the Airy packet into an accelerated motion of which the acceleration a=(-H_i/( x))/m. Nonspreading Airy packet then acts as a classical particle of mass m, and the motion of it can be described classically by H_i.