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For the quantum transmitting boundary method we implemented a finite difference Schrödinger solver and as equivalent alternatives a Riccati and a Prüfer solver. For the Riccati and the Prüfer solver we used a custom first order differential equation solver (the odesuite from Matlab, a sample of highly accurate ode solvers.)
The most important clue in the implementation of the
Prüfer equation was the insight that one has to solve
the angular equation and the equation for the amplitude
in parallel. Solving for the angle first and only then
for the amplitude results in jumps in the solution stemming
from the minima of 
which are not resolved, even if
the error tolerance is set to the machine epsilon.
However, when solving for both polar coordinates they
are correctly detected and the artificial jumps in the solution
disappear. In this respect the Riccati equation was better behaved,
using very stringent tolerances it was always solved
correctly. But it also improved when we solved for
and
in parallel.
All implemented methods are able
to solve the Schrödinger equation with high precision.
However, the Prüfer and especially the Riccati solver
require quite stringent error tolerances to achieve a correct
solution. This makes the solvers slow. They are at least
by a factor of 
slower than the Schrödinger solver
and not competitive in terms of efficiency. In this last
respect the Prüfer
equation was superior to the Riccati equation.
However, we made no attempts to implement either a special purpose
Riccati or Prüfer solver, so this study may not be the last word
with regard to speed comparison. But the discrete Schrödinger equation
is a tridiagonal system for which there are highly
optimized direct solvers. It is unlikely that one can
improve on it.
The main value in the Riccati and Prüfer equation is that they allow for a check of the implementation of open boundary conditions in the Schrödinger equation. It turns out, that accuracy is excellent. The simulation results are identical for all practical purposes.
With respect to the usage of hydrodynamical methods
the experiences with the Riccati equation raises some
doubt that these methods can be used successfully in
two dimensions. Already in one dimension it was very
difficult to resolve the spikes from the minima in 
of the wave function.
Previous: 7.1.2.2 Riccati and Prüfer:
Up: 7.1 Single Particle Transport
Next: 7.2 The Open Von