On the quantum mechanics of particles with classically assigned trajectories

G. Barton, A. Calogeracos
1996 Proceedings of the Royal Society A  
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 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 Hamil tonian system to a formal constraint r -) = 0. Here we consider instead the particle
more » ... 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 f 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. In trod uction and preview When considering the interaction between a quantized field and a non-relativistically moving object, it can prove convenient to postpone the decision whether to treat the object either (a) quantum mechanically, or (b) as a source whose position ) is a classically prescribed function of the time, i.e. a c-number parameter rather than a dynamical variable. Hence one would like to interrelate the Hamiltonians that govern the time evolution in these two cases. In a recent study of quantum radiation from a dispersive mirror we have found this relation to be both surprising and unexpectedly subtle: the obvious guess about (b) based on (a) turns out to be wrong, and to discover the demonstrably correct result (Calogeracos & Barton 1995) we had to start with the quanta Posl tion variable, and then to tackle it in a manner developed from Dirac's ideas about constraints (Dirac 1962 (Dirac , 1964 . Essentially the same problem arises already for the familiar case of a charged particle coupled to the Maxwell field (where, likewise, one is interested in the radiation it emits and in the radiative reaction force that it feels), and even for a constrained particle coupled to no field at all. The surprise one meets is, roughly, the following. From the Hamiltonian p2/2m of a free particle, an o**1 p = mr, it might seem plausible to infer that, on demoting r from quantum varia to a prescribed parameter, the Hamiltonian should reduce simply to Heft
doi:10.1098/rspa.1996.0059 fatcat:hzwx2qqoozc2bng77pbe6hzqby