An infinitely long solenoid has a constant magnetic field in the interior. Outside, the magnetic field strength is zero, but the vector potential for this situation has to be nonzero (it is, however, a pure gauge potential). Classically, a pure gauge potential has no influence on the motion of a particle. But quantum mechanically, the vector potential influences the wave function.
For this movie, the solenoid is described in two dimensions by an inpenetrable small circle. The wave function is zero on the circle (Dirichlet boundary conditions). Although the wave packet has no contact with the magnetic field strength, there is a measurable interference effect (the black line behind the obstacle). Notice that the presence of the magnetic vector potential causes a spiral-like distortion of the lines with constant phase.
In "Visual Quantum Mechanics - Book One" this scattering process is investigated for different field strengths. The measurable influence on the wave function depends only on the flux of the magnetic field (i.e., on the circulation of the vector potential). For certain values of the magnetic field strength, the effect becomes unmeasurable (if the flux is an integer multiple of 2 pi).