Density changes and flux

An often misunderstood concept (at least by me) is the relation between density (ρ) and flux (j), which is essential in many kinds of dynamical processes. In the absence of sources and sinks these are related according to the continuity equation

From this equation it obviously follows that

i.e. if the flux vanishes, the density has to be constant. The opposite is not true

Only the "divergence" of the flux has to vanish but there may still be a "rotation". Take the example of stirring a pot: there is motion inside the pot even though none of the water actually moves. The same idea holds for ring currents in a molecule like benzene.

The first time I came across this problem was in a somewhat different context, related to my diploma thesis. The question was whether the excited state intramolecular double proton transfer in bipyridyldiol occured in a concerted, sequential or mixed fashion.

We were claiming case (b) was correct, while (a) was the previously accepted view. I do not want to discuss the methodological details now, but there was one thing that bothered me: In our trajectory simulations, you could clearly see whether a single or double proton transfer occured. On the other hand there was a quantum dynamics paper, which showed the evolution of the probability density. And it was not clear to me at all, how one could interpret these density fluctuations in terms of single or double proton transfers. Did the classical trajectories show something that did not exist in the quantum world?

I would now claim the opposite is true: When you analyse quantum dynamics only in terms of probabilty densities, you miss important information (just as you would not know I was stirring the pot of water only from the density). The above continuity equation for quantum dynamics becomes

This means that the flux is coded in the phase of the wavefunction. For a full analysis of the dynamical process it is necessary to consider the phase as well as the amplitude. Another obvious consequence of this equation is that there is no density change in purely real wavefunctions (there may however be a phase change followed by a density change).

A similar question also arises in surface hopping dynamics: Is it enough to plot the state amplitudes against time or does it give additional information to consider also the state-to-state transition probabilities? You again need the transition probabilites for the whole picture, e.g. if you want to identify individual relaxation pathways as is done in this paper.