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

## Equations

The Madelung equations[1] are quantum Euler equations:[2]

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

where $\bold u$ is the flow velocity in the quantum probability space with mass density $\bold \rho=m|\psi|^2$. The circulation of the flow velocity field along any closed path obeys the auxiliary condition $\begin{matrix}\Gamma \doteq \oint{d\mathbf{l}\cdot \left( m\mathbf{u} \right)}=2\pi n\hbar , & n\in \mathbb{Z} \\\end{matrix}$.[3] The term in the brackets represents a quantum chemical potential in vacuum. 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, $\bold p_Q$ is the quantum symbol of the Madelung hydrodynamics.[4] The integral energy stored in the quantum pressure tensor is proportional to the Fisher information, which accounts for the quality of measurements. Thus, according to the Cramér–Rao bound, the Heisenberg Uncertainty principle is equivalent to a standard inequality for the efficiency (statistics) of measurements.