Gaussian in a magnetic field

A Gaussian wave function exp(-x^2/2) corresponds to a particle at rest if there is no external field. Did you know that in a constant magnetic field this wave packet describes a moving particle? The center of the wave packet moves on a circle which goes precisely through the origin (assuming that the vector potential is given in the Poincaré gauge).

The yellow arrow gives the average velocity of the wave packet at every instant of time. The average momentum is represented by the dark arrow. The white arrow (and the arrows in the background) symbolize the magnetic vector potential.The constant magnetic field is described by a vector potential with constant curl.

The actual motion of the Gaussian initial wave packet depends on the chosen gauge. "Visual Quantum Mechanics - Book One" explains why this is no contradiction to the requirement of gauge-invariance. Chapter 12 describes the quantum mechanics in a constant magnetic field in every detail.