A Hamiltonian yielding damped motion in an homogeneous magnetic field: quantum treatment

Abstract

In earlier work, a Hamiltonian describing the classical motion of a particle moving in two dimensions under the combined influence of a perpendicular magnetic field and of a damping force proportional to the particle velocity, was indicated. Here we derive the quantum propagator for the Hamiltonian in different representations, one corresponding to momentum space, the other to position, and the third to a natural choice of "velocity" variables. We call attention to the following noteworthy fact: the Hamiltonian contains three parameters which do not in any way influence the motion of the position of the particle. However, at the quantum level, the propagator, even in the position representation, depends in an intricate way on these classically irrelevant parameters. This creates considerable doubt as to the validity of such a quantization procedure, as the physical results predicted differ for various Hamiltonians, all of which describe the dissipative dynamics equally well.

The κ-Deformed Calogero–Leyvraz Lagrangians and Applications to Integrable Dynamical Systems

“…Recently, Calogero and Leyvraz [ 1 , 2 ] demonstrated a time-independent Hamiltonian description of the motion of a charged particle moving in a two dimensional space under the influence of a magnetic field perpendicular to the plane of motion and a frictional force proportional to the velocity. This motion may be viewed as a dynamics of cyclotron motion with friction; this model arises through the coupling of a particle to a large number of external degrees of freedom.…”