 Uniform magnetic field with axial harmonic confinement

The cylindrical vector potential $A_{\theta }=\mu r$ gives a uniform axial magnetic field $2\mu$, while scalar potential $\lambda z^2$ supplies harmonic axial confinement for positive charge and positive $\lambda$. Transverse motion is cyclotron motion about a fixed guiding centre, and axial motion is an independent harmonic oscillation. A circle centred on the cylindrical origin has $\dot{\theta }=-{\Omega }_c$ for positive charge.