On the quantum mechanics of particles with classically assigned trajectories

We consider the state vector of a system containing a particle, in the approximation treating the particle (mass = m) as if it followed a classically prescribed trajectory R(t). Since a real particle has position and momentum operators r and p of its own, one must find a way effectively to demote these from dynamical variables to mere time-dependent parameters. The traditional method is to subject the Hamiltonian system to a formal constraint r — R(t) = 0. Here we consider instead the particle trapped in a sufficiently tight-binding potential V(r — R(t)); the conclusions remain the same but the reasoning becomes much more accessible. The state vector acquires an overall phase factor exp{—i ft dt' (-1/2mR2(t'))} surprising in that the integrand features not the enforced kinetic energy, but its negative. We elucidate the physics of this phase factor, sketch a scenario for verifying it experimentally and show that it persists essentially unchanged when the trapped particle is coupled to the Maxwell field. The standard adiabatic approximation routinely applied misses this factor altogether, but a more careful modification recovers the right result.