We study the motion of a particle in a particular magnetic field configuration both classically and quantum mechanically. For flux-free radially symmetric magnetic fields defined on circular regions, we establish that particle escape speeds depend, classically, on a gauge-fixed magnetic vector potential, and demonstrate some trajectories associated with this special type of magnetic field. Then we show that some of the geometric features of the classical trajectory (perpendicular exit from the field region, trapped and escape behavior) are reproduced quantum mechanically using a numerical method that extends the norm-preserving Crank-Nicolson method to problems involving magnetic fields. While there are similarities between the classical trajectory and the position expectation value of the quantum mechanical solution, there are also differences, and we demonstrate some of these.
We probe the dynamics of a modified form of the Schrödinger-Newton system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of "collapse" (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrödinger ground state energies match the Dirac ones with an error of ∼ 10%. At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, m ≈ 3.3 Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the Schrödinger-Newton system. Unlike the Schrödinger-Newton dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case. * jfrankli@reed.edu 1 arXiv:1603.03380v1 [gr-qc]
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