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\ensuremath{\rightarrow})=v(x\ensuremath{\rightarrow})cos(\ensuremath{\Omega}t) is governed by a Schr\"odinger equation with time-independent effective potential ${V}_{\mathrm{eff}}$(x\ensuremath{\rightarrow})=\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow})\ensuremath{\cdot}\ensuremath{\nabla}\ensuremath{\rightarrow}v(x\ensuremath{\rightarrow}) /4m${\ensuremath{\Omega}}^{2}$. The validity of this approximation and an exact formal solution based on the Wigner function are discussed for the case in which v(x\ensuremath{\rightarrow}) is quadratic.