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What is the Madelung transformation and how is it used?

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I googled "Madelung Transformation" in German and found the following answer in an article about the theory of super conductivity.

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).

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  • $\begingroup$ This is extremely interesting - does the paper contain the expressions for the Madelung equations? $\endgroup$ – Sklivvz Nov 27 '10 at 21:07
  • $\begingroup$ Isn't this essentially the path followed in the Bohmian approach towards quantum mechanics? ... To answer my own question, the wikipedia page on Bohm's theory says: "Around this time Erwin Madelung[29] also developed a hydrodynamic version of Schrödinger's equation which is incorrectly considered as a basis for the density current derivation of the de Broglie–Bohm theory. The Madelung equations, being quantum Euler equations (fluid dynamics), differ philosophically from the de Broglie–Bohm mechanics[30] and are the basis of the hydrodynamic interpretation of quantum mechanics." $\endgroup$ – user346 Nov 27 '10 at 23:45
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    $\begingroup$ For a recent paper which compares the Bohmian and Madelung approaches see this. Also contains all the math ! Cheers. $\endgroup$ – user346 Nov 27 '10 at 23:58
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    $\begingroup$ First, Madelung equations. Second, this is not the transformation that is relevant to quantum fluid dynamics (at which mbq was hinting in the first comment under the question). The interesting one comes from nonlinear Schroedinger equation and is apparently somehow also related to KdV equation. Just google for papers. Still, I'll won't add this as an answer because I don't quite understand the stuff (I just looked at papers for a little while). And for the same reason I think Gerard's answer is pretty unsatisfactory. $\endgroup$ – Marek Nov 28 '10 at 15:01
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    $\begingroup$ @Marek It is related, you just do the same trick on Gross-Pitaevskii equation. Related paper: springerlink.com/content/0v2l261ru255n721 $\endgroup$ – user68 Nov 29 '10 at 12:30

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