The Riccati equation was used in [Bit00] as a numerically more suitable variant of the hydrodynamical formulation.
With this the stationary Schrödinger equation
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
Introducing the transformation
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:
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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