Concerning azimuthal component of the probability current in Hydrogen atom Jϕ, the azimuthal component of the probability flow is nonzero in Hydrogen atom for the stationary states with m=1.
This should corresponds to orbital motion of electrons. But here is my doubt. How is it possible to have a stationary wave with an orbital motion, shouldn’t this orbit emit radiation?
 A: If we look at the expectation value of the dipole moment of the atom, $\langle \vec{p} \rangle = - q \langle \vec{r} \rangle$, it is not too hard to show that this quantity is zero for any energy eigenstate.  And if we believe that radiation is created by a change in the dipole moment of a system, then we must conclude that an energy eigenstate cannot emit radiation.
We can think of a uniformly charged ring, rotating about its axis, as an analogy to this.  Each individual piece of the ring is accelerating towards the rotation axis, and the whole system has a non-zero angular momentum.  But the net dipole moment of the ring is constant (it vanishes), because contribution to the dipole moment $\vec{p}$ from a particular charge on one side of the ring exactly cancels the contribution to the dipole moment from the charge exactly opposite on the ring.
With the caveat that thinking of an electron orbital as a "charge distribution" is not completely accurate, the hydrogen atom can be thought of in the same way.  The electron's charge and mass distributions are rotating about the proton in such a way that it does have a net angular momentum, but its net dipole moment is constant, and so no radiation is emitted.
