An electron in any orbital isn't going anywhere, because an orbital is a stationary state solution to the S.E.
However, the probability density does 'swirl' in practice because the wavefunction is a linear combination over all orbitals. That allows for interference between the phases of differing orbitals creating time dependent regions of low and high probability.
The 'swirls' are not much like planetary orbits, but there is movement.
This applet should convince you that particle wavefunctions do swirl in a potential well.
Edit: This applet is even better- it's for the H-atom
You can easily combine an S orbital together with a P orbital and watch oscillations occur.