The Madelung equations are quantum Euler equations:
where is the hydrodynamic-like velocity in the quantum probability space with mass density . The circulation of the velocity field along any closed path obeys the auxiliary condition . The term in the brackets represents a quantum chemical potential. The kinetic energy operator from the Hamiltonian 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, which accounts for the quality of measurements.
- Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Z. Phys. 40 (3–4): 322–326. Bibcode:1927ZPhy...40..322M. doi:10.1007/BF01400372.
- Schönberg, M. (1954). "On the hydrodynamical model of the quantum mechanics". Il Nuovo Cimento 12 (1): 103–133. doi:10.1007/BF02820368.
- Tsekov, R. (2009). "Bohmian Mechanics versus Madelung Quantum Hydrodynamics". arXiv:0904.0723.