Quantising a Hamiltonian curl force

Abstract

Classical curl forces are position-dependent Newtonian forces (accelerations) that are not the gradient of a scalar potential, and in general cannot be described by Hamiltonians. However, a special class of curl forces can be described by Hamiltonians, with the unusual feature that the kinetic energy is anisotropic in the momentum components. Therefore they can be quantised conventionally. We quantise the simplest such case: motion in the plane, with a curl force azimuthally directed and linear. As expected, the quantum propagator, and the way this drives Gaussian wavepackets, directly reflects the spiralling classical curl force dynamics. Two classes of stationary states—eigenfunctions of a continuous spectrum for the unbounded Hamiltonian—are described. They possess unusual singularities and an unfamiliar quantisation condition; their explanation requires asymptotics and unfamiliar singularities in the underlying families of classical trajectories. The analysis is supported and illustrated numerically.