Madelung equations
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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.
Equations[edit]
The Madelung equations[1] are quantum Euler equations:[2]
- u is the flow velocity,
- is the mass density,
- is the Bohm quantum potential,
- V is the potential from the Schrödinger equation.
The circulation of the flow velocity field along any closed path obeys the auxiliary condition for all integers n.[3]
Derivation[edit]
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
See also[edit]
References[edit]
- ^ Madelung, E. (1926). "Eine anschauliche Deutung der Gleichung von Schrödinger". Naturwissenschaften (in German). 14 (45): 1004. Bibcode:1926NW.....14.1004M. doi:10.1007/BF01504657. S2CID 39430240.
- ^ Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form". Z. Phys. (in German). 40 (3–4): 322–326. Bibcode:1927ZPhy...40..322M. doi:10.1007/BF01400372. S2CID 121537534.
- ^ I. Bialynicki-Birula; M. Cieplak; J. Kaminski (1992), Theory of Quanta, Oxford University Press, ISBN 0195071573.
- ^ Tsekov, R. (2009). "Dissipative Time Dependent Density Functional Theory". International Journal of Theoretical Physics. 48 (9): 2660–2664. arXiv:0903.3644. Bibcode:2009IJTP...48.2660T. doi:10.1007/s10773-009-0054-6. S2CID 119252668.
Further reading[edit]
- Schönberg, M. (1954). "On the hydrodynamical model of the quantum mechanics". Il Nuovo Cimento. 12 (1): 103–133. Bibcode:1954NCim...12..103S. doi:10.1007/BF02820368. S2CID 123655548.
- Wyatt, Robert E.; Trahan, Corey J. (2005). "The Bohmian Route to the Hydrodynamic Equations". Quantum Dynamics with Trajectories : Introduction to Quantum Hydrodynamics. New York: Springer. pp. 40–61. ISBN 0-387-22964-7.