Quantum theory of particle motion in a rapidly oscillating field

Abstract

We show that the motion of a particle of mass m in a high-frequency time-dependent potential V(x→)=v(x→)cos(Ωt) is governed by a Schrödinger equation with time-independent effective potential $V_{\mathrm{eff}}$(x→)=∇→v(x→)⋅∇→v(x→) /4m${\Omega }^2$. The validity of this approximation and an exact formal solution based on the Wigner function are discussed for the case in which v(x→) is quadratic.