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Madelunggleichungen

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Die Madelunggleichungen sind eine von Erwin Madelung (1881–1972) formulierte Alternative der Schrödingergleichung.<ref name="Madelung1926"/>

Ersetzt man dort die komplexe Funktion <math>\psi</math> durch ihren Betrag <math>\rho</math> und ihre Phase <math>S</math> gemäß <math>\psi= \sqrt{\rho}\; e^{\frac{i}{\hbar} S} </math>, so erhält man die Madelunggleichungen:<ref name="Madelung1926"/>

  1. <math>\partial_t \rho +\frac{1}{m}\nabla(\rho\nabla S)= 0</math>
  2. <math>

\partial_t S +\frac{1}{2m}(\nabla S)^2 +V(x)- \frac{\hbar^2}{2m} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0,</math>

wobei <math>V</math> das Potential aus der Schrödingergleichung ist.

Die erste dieser beiden Gleichungen hat die Form einer Kontinuitätsgleichung,

die zweite ist eine Hamilton-Jacobi-Gleichung (siehe Kanonische Gleichungen).

Interpretation

<math>S</math> wird als Wirkung interpretiert, <math>\nabla S</math> als Impuls. Die Madelunggleichungen lassen sich als Quanten-Euler-Gleichungen (Strömungsmechanik) deuten wie folgt:<ref name="Madelung1927" /><ref name="Bialynicki1992"/>

  1. <math>\partial_t \rho_m + \nabla\cdot(\rho_m \vec v) = 0,</math>
  2. <math> \frac{d \vec v}{dt} = \partial_t\vec v + \vec v \cdot \nabla\vec v = -\frac{1}{m} \mathbf{\nabla}(Q + V),</math>

wobei

  • <math>\vec{v}(\vec{x}, t) = \mathbf{\nabla} S / m </math> (Strömungsgeschwindigkeit) bzw. <math>\nabla S = m \cdot \vec{v} </math> (Impuls)
    ​
  • <math>\rho_m = m \rho = m |\psi|^2</math> (Massedichte) mit Normierungsbedingung <math>\int \rho (\vec x,t) d^3 x = 1</math> bzw. <math>\int \rho_m (\vec x,t) d^3 x = m</math> zu jeder Zeit <math>t</math>
  • <math>Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}} = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho_m}}{\sqrt{\rho_m}}</math> (Bohmsches Quantenpotential).

Bedeutung

Aufgrund ihrer Nichtlinearität sind die Madelunggleichungen schwierig zu handhaben, zeigen aber, dass es nichtlineare Gleichungen gibt, die sich auf lineare Gleichungen zurückführen lassen.

Siehe auch

Literatur

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Einzelnachweise

<references>

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</ref>

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</references>

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