What is the Madelung transformation and how is it used?

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I am interested in an answer but perhaps it would be nice to give a little motivation why you are interested in this? – Marek Nov 26 '10 at 11:00
Indeed this is intended to be a Google-magnet; I know something about it because of my superfluid helium work. – mbq Nov 26 '10 at 11:26
What do you mean by "Google-magnet" here? Are you looking for an answer or not? If so, you should address the concerns of Marek. – j.c. Nov 27 '10 at 2:33
Well, this question is hit #8 on Google for "Madelung transformation," I guess that counts for something ;-) But I agree with Marek that expanding on your motivation would make this a better question. – David Z Nov 27 '10 at 4:19
Your comments looks like I'm asking about building a nuclear bomb ;-) . I think this question is quite straightforward. I can and probably I'll answer it myself if no one answers it within a week, yet I hope this won't be the case and I'll learn something interesting from those answer(s). The fact it may attract few more people here is just an additional gain. – mbq Nov 27 '10 at 18:03

The Schrödinger equation is an axiom of quantum physics that is difficult to interpret i.a. because of its usage of complex numbers: $i\hbar \frac{\partial}{\partial t}\Psi = \mathcal{H} \Psi$.
In the Kopenhagen interpretation $\Psi\Psi^*$ is known to describe the probability distribution of a particle. Erwin Madelung tried (already in 1926) to understand the nature of the Schrödinger equation from a different point of view. He substituted $\Psi = ae^{i\phi}$ and split the Schrödinger equation in its real part and its imaginary part (first and second Madelung equation).
Because $\Psi\Psi^* = a^2$, the second Madelung equation (see above reference) becomes a continuity equation of the probability density $a^2$. This equation can thus be interpreted in a more physical way than the Schrödinger equation and that's the point. Similar arguments hold for the first equation but these arguments are more involved (see above reference).