How Does an Electron Move in Real Space on an Open Orbit in a Magnetic Field?

[ehrenfest]

Homework Statement

An open orbit in a monovalent tetragonal metal connects opposite faces of a Brillioun zone. A magnetic field is normal to the plane of the open orbit. Describe in real space the motion of the electron on this orbit in the presence of the magnetic field.

Homework Equations

The Attempt at a Solution

I am having loads of difficulty translating what goes on in k-space to what goes on in real space. The picture in k-space is very clear, but I am just really confused about what it means. It is not even clear to me what how the velocity vector is changing!

 

[genneth]

I don't know if the text says so, but all this is predicated on working in a semi-classical regime. So the particles can be localised with small uncertainties in space and momentum. In that case, remember that the group velocity is the gradient of the dispersion relation. So for an open orbit, you would get oscillations about zero in the group velocity in some direction, but in the other direction it would always be +ve or -ve. So you would get a macroscopic current that flows across the entire sample (or you would, if it wasn't for the fact there's an identical current flowing the other way...)