The Madelung equations are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation.

## Equations

The Madelung equations are quantum Euler equations:

$\partial_t \rho+\nabla\cdot(\rho\bold u)=0,$
$\partial_t\bold u+\bold u\cdot\nabla\bold u=-\frac{1}{m}\nabla\left(\frac{1}{\sqrt{\rho}}\hat H\sqrt{\rho}\right)=-\frac{1}{\rho}\nabla\cdot\bold p_Q-\frac{1}{m}\nabla U,$

where $\bold u$ is the hydrodynamic-like velocity in the quantum probability space with mass density $\bold \rho=m|\psi|^2$. The circulation of the velocity field along any closed path obeys the auxiliary condition[1][2] $\begin{matrix} \Gamma \doteq \oint{d\mathbf{l}\cdot \left( m\mathbf{u} \right)}=2\pi n\hbar , & n\in \mathbb{Z} \\ \end{matrix}$. The term in the brackets represents a quantum chemical potential. The kinetic energy operator from the Hamiltonian $\hat H$ results in a non-local quantum pressure tensor

$\bold p_Q=-(\hbar/2m)^2 \rho \nabla\otimes\nabla \ln \rho$

or alternatively to the Bohm quantum potential. While the latter is the icon of the de Broglie–Bohm theory, the quantum symbol of the Madelung hydrodynamics is $\bold p_Q.$ The integral energy stored in the quantum pressure tensor is proportional to the Fisher information, which accounts for the quality of measurements.