The Madelung equations are Erwin Madelung

Erwin Madelung

Erwin Madelung was a German physicist.He was born in 1881 in Bonn. His father was the surgeon Otto Wilhelm Madelung. He earned a doctorate in 1905 from the University of Göttingen, specializing in crystal structure, and eventually became a professor...

's alternative formulation of the Schrödinger equation

Schrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

.

Equations

The Madelung equations are quantum Euler equations:



where is the hydrodynamic

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

-like velocity in the quantum probability space with mass density The term in the brackets represents a quantum chemical potential. The kinetic energy operator from the Hamiltonian

Hamiltonian system

In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....

  results in a non-local quantum pressure tensor


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 The integral energy stored in the quantum pressure tensor is proportional to the Fisher information

Fisher information

In mathematical statistics and information theory, the Fisher information is the variance of the score. In Bayesian statistics, the asymptotic distribution of the posterior mode depends on the Fisher information and not on the prior...

, which accounts for the quality of measurements.