What's the deep reason for the velocity field of a wave function to be irrotational? In quantum mechanics, the current of probability is
$$\vec{v}=\frac{\hbar}{m}\mathrm{Im}\left(\frac{\nabla \psi}{\psi}\right)=\frac{\hbar}{m}\nabla S$$
where $\psi=R e^{i S}$ and $R$, $S$ are real functions. So the velocity field is irrotational.
My questions are:


*

*What's the deep reason behind this result? 

*Since now that the probabilistic current is irrotational, why are there still vortex or eddy in hydromechanics and many kinds of rotational velocity field? 
 A: One should be careful in comparing a probability current with an ordinary fluid. The two are not the same. A fluid can flow between two regions of equal density, because we can say that the fluid in one region is "distinct from" the fluid in the other region. Because of our modern understanding, we reason this as due to the fluid being made of atoms. On the other hand, a probability field can't flow between regions of equal probability, because the substance, if you will, in one region is not distinct from the substance in the other region. In both cases, the substance is just probability. You can't talk about the probability over here and the probability over there.
Another way of saying this is that the velocity that you wrote down is the group velocity for a wave-packet, but not the phase velocity.
On the other hand, it may happen that at some points $\Psi(\vec{x}) = 0$. Here the phase is not defined, and so while locally $\nabla S$ must be irrotational, it's line integral around some loop that encloses a singularity might be non-zero.
