Decomposing the Schrödinger Equation I heard that the Schrödinger Equation is (naturally) to two coupled first-order real differential equations, where one is a continuity equation for the probability amplitude and the other is somehow the evolution of the phase. What is this decoupling and how does one do it?
 A: If I remember correctly this trick boils down to writing the wavefunction $\psi(x)$ as an amplitude times a phase factor $\psi(x) = \sqrt{\rho} e^{i \varphi(x)}$, and working on the new (real) variables $\rho(x)$ and $\overrightarrow{v} = \frac{\hbar}{m} \overrightarrow{\nabla} \varphi(x)$.
This is especially useful when working with an interacting Bose-Einstein condensate, which can be described by the Gross-Pitaevskii equation (GPE) at $T = 0$. This is the same as Schrödinger equation, but with an extra non-linear term $\propto |\psi|^2 \psi$. In both cases (regular Schrödinger equation or GPE), the continuity equation for the probability density can be written as:
$$\partial_t |\psi|^2 + \mathrm{div}(\overrightarrow{j}) = 0,$$
with $\overrightarrow{j} = \frac{\hbar}{2im} \left(\psi^* \overrightarrow{\nabla} \psi - \psi \overrightarrow{\nabla} \psi^*\right) = \rho \overrightarrow{v}$.
So the continuity equation writes:
$$\partial_t \rho + \mathrm{div}(\rho \overrightarrow{v}) = 0,$$
which is the same continuity equation as in fluid dynamics.
In the case of GPE, the analogy with fluid dynamics actually goes further than that, and you can find an equation on $\rho$ and $\overrightarrow{v}$ very similar to the equation of a perfectly inviscid fluid (also known as Euler equation). I won't do the calculation myself here but you can check for "Hydrodynamics of the Gross Pitaevskii equation" online.
From this tutorial:
$$m \partial_t \overrightarrow{v} + \overrightarrow{\nabla}\left( \frac{1}{2}m v^2 + U - \frac{\hbar^2}{2m} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} \right) = 0$$
The last term (which is not present in the usual Euler equation) is often reffered to as "quantum pressure term". It can be dropped if the condensate is strongly interacting (Thomas-Fermi approximation).
Edit : since you were talking about the Schrödinger equation rather than the GPE, it might be interesting to look at the notion of Quantum Potential, which seems to be the equivalent of the hydrodynamic treatment of GPE for the regular Schrödinger equation, as pointed out in a comment by Cosmas Zachos. I am not familiar with this version of the Schrödinger equation so I don't think I would be able to say a lot about it.
