In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation.
The Madelung equations[1] are quantum Euler equations:[2]
The circulation of the flow velocity field along any closed path obeys the auxiliary condition for all integers n.[3]
The Madelung equations are derived by writing the wavefunction in polar form:
The flow velocity is defined by
The quantum force, which is the negative of the gradient of the quantum potential, can also be written in terms of the quantum pressure tensor:
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 of measurements. The thermodynamic definition of the quantum chemical potential