Cyclotron motion. Gauge invariance of U(1).

We represent two examples of different vector potentials. These examples provide same probability densities.
The Gauge transformation and initial states are
  1. \psi_{1} = \exp[-x^{2}-(y-\pi)^{2}+i\pi x]
    Ax=-y/2,Ay=x/2
  2. \psi_{2} = \exp[-ixy/2]\psi_{1}
    Ax=-y,Ay=0.

A corresponding classical problem is a cyclotron motion, where any particle shows a circle motion and this period $T_{\rm c}$ is $2\pi m/ eB $. In the quantum problem the circle motion and the periodicity are realized. Any wave packet shows a circle motion and comes back to the initial state after $2T_{\rm c}$. The absolute value $ |\psi |$ comes back after $T_{\rm c}$.

MPEG

  1. Wave function \psi_{1}(90,799 Byte)
  2. Wave function \psi_{2}(101,115 Byte)

GIF images of \psi_{2}