Madelung equations

In theoretical physics, the Madelung equations, or the equations of quantum hydrodynamics, are Erwin Madelung's alternative formulation of the Schrödinger equation for a spinless non relativistic particle, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics.

In both cases the hydrodynamic interpretations are not equivalent to Schrodinger's equation without the addition of a quantization condition.

In the fall of 1926, Erwin Madelung reformulated[2][3] Schrödinger's quantum equation in a more classical and visualizable form resembling hydrodynamics.

His paper was one of numerous early attempts at different approaches to quantum mechanics, including those of Louis de Broglie and Earle Hesse Kennard.

[4] The most influential of these theories was ultimately de Broglie's through the 1952 work of David Bohm[5] now called Bohmian mechanics.

In 1994 Timothy C. Wallstrom showed[6] that an additional ad hoc quantization condition must be added to the Madelung equations to reproduce Schrodinger's work.

His analysis paralleled earlier work[7] by Takehiko Takabayashi on the hydrodynamic interpretation of Bohmian mechanics.

The mathematical foundations of the Madelung equations continue to be a topic of research.

where The Madelung equations answer the question whether

obeys the continuity equations of hydrodynamics and, subsequently, what plays the role of the stress tensor.

[9] The circulation of the flow velocity field along any closed path obeys the auxiliary quantization condition

for all integers n.[10][11] The Madelung equations are derived by first writing the wavefunction in polar form[12][13]

Substituting this form into the probability current gives:

where the flow velocity is expressed as

as a "velocity" should not be taken too literal, because a simultaneous exact measurement of position and velocity would necessarily violate the uncertainty principle.

[14] Next, substituting the polar form into the Schrödinger equation

and performing the appropriate differentiations, dividing the equation by

and separating the real and imaginary parts, one obtains a system of two coupled partial differential equations:

The second equation corresponds to the real part and is also referred to as the quantum Hamilton-Jacobi equation.

Alternatively, the quantum Hamilton-Jacobi equation can be written in a form similar to the Cauchy momentum equation:

with an external force defined as

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.

[17][18] The thermodynamic definition of the quantum chemical potential

follows from the hydrostatic force balance above:

According to thermodynamics, at equilibrium the chemical potential is constant everywhere, which corresponds straightforwardly to the stationary Schrödinger equation.

Therefore, the eigenvalues of the Schrödinger equation are free energies, which differ from the internal energies of the system.

The particle internal energy is calculated as

ε = μ − tr ⁡ (

and is related to the local Carl Friedrich von Weizsäcker correction.