The Riccati equation was used in [Bit00] as a numerically more suitable variant of the hydrodynamical formulation.

Set

(6.28) |

With this the stationary Schrödinger equation

becomes

By the substitution

(6.30) |

it reduces to a nonlinear first order equation. For constant mass and we get

This type of equation is known as Riccati equation [MR96]. The Riccati equation plays an important role in control theory and has a rich theory of its own. Separation of Equation 6.30 into real and imaginary part recovers the hydrodynamical equation for the phase as equation for the imaginary part of .

Another alternative formulation of the Schrödinger equation, which is closely related to the Riccati equation, is the Prüfer equation. The Prüfer transformation is a useful tool in the qualitative theory of second order Sturm-Liouville differential equations [BD01]. In the case of the Schrödinger equation with constant mass it is introduced in the following way: Define complex quantities

(6.32) | ||

(6.33) |

Introducing the transformation

(6.34) | ||

(6.35) |

with complex , and we get from the Schrödinger equation and from the defining equation for (6.34)

(6.36) | ||

(6.37) |

Respective multiplication of these equations with and and adding the resulting equations yields (after elimination of ):

So the equation for the angular variable separates. For we get:

(6.39) |

The link to the original Riccati idea is immediate. If is a solution to the Prüfer equation, then is a solution to the Riccati Equation 6.32. The advantage of the Prüfer equation (6.39) over the traditional Riccati equation is that it can be solved for arbitrary without leading to singularities, as is the case with nodes ( ) in the Riccati equation. The Schrödinger, Riccati and Prüfer equation will be investigated numerically in Section 7.1.3.

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