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

History

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.[8]

Equations

The Madelung equations are quantum Euler equations:[citation needed] where

  • is the flow velocity,
  • is the mass density,
  • is the Bohm quantum potential,
  • V is the potential from the Schrödinger equation.

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]

Derivation

The Madelung equations are derived by first writing the wavefunction in polar form[12][13] with and both real and the associated probability density. Substituting this form into the probability current gives: where the flow velocity is expressed as However, the interpretation of 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 first equation corresponds to the imaginary part of Schrödinger equation and can be interpreted as the continuity equation. The second equation corresponds to the real part and is also referred to as the quantum Hamilton-Jacobi equation.[15] Multiplying the first equation by and calculating the gradient of the second equation results in the Madelung equations: with quantum potential

Alternatively, the quantum Hamilton-Jacobi equation can be written in a form similar to the Cauchy momentum equation: with an external force defined as and a quantum pressure tensor[16]

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]

Quantum energies

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 and is related to the local Carl Friedrich von Weizsäcker correction.[19]

See also

Notes

  1. ^ Schönberg 1954, p. 103.
  2. ^ Madelung 1926.
  3. ^ Madelung 1927.
  4. ^ Kragh & Carazza 2000, p. 52.
  5. ^ Kragh & Carazza 2000, p. 53.
  6. ^ Wallstrom, Timothy C. (1994-03-01). "Inequivalence between the Schr\"odinger equation and the Madelung hydrodynamic equations". Physical Review A. 49 (3): 1613–1617. doi:10.1103/PhysRevA.49.1613.
  7. ^ Takabayasi, Takehiko (1952-08-01). "On the Formulation of Quantum Mechanics associated with Classical Pictures*". Progress of Theoretical Physics. 8 (2): 143–182. doi:10.1143/ptp/8.2.143. ISSN 0033-068X.
  8. ^ Reddiger, Maik; Poirier, Bill (2023-04-19). "Towards a mathematical theory of the Madelung equations: Takabayasi's quantization condition, quantum quasi-irrotationality, weak formulations, and the Wallstrom phenomenon". Journal of Physics A: Mathematical and Theoretical. 56 (19): 193001. doi:10.1088/1751-8121/acc7db. ISSN 1751-8113.
  9. ^ Białynicki-Birula, Cieplak & Kaminski 1992, p. 88.
  10. ^ Białynicki-Birula, Cieplak & Kaminski 1992, pp. 90–91.
  11. ^ Wyatt 2005, §13.5.
  12. ^ Białynicki-Birula, Cieplak & Kaminski 1992, pp. 88–89.
  13. ^ Wyatt 2005, §2.2.
  14. ^ Sakurai & Napolitano 2020, p. 95.
  15. ^ Wyatt 2005, p. 48.
  16. ^ Tsekov 2011.
  17. ^ Reginatto 1998.
  18. ^ Heifetz & Cohen 2015.
  19. ^ Tsekov 2009.

References

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