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I'm stuck on a problem, banging of head etc. Basically, I've got a time-evolution problem restricted to the positive quadrant of complex space, where the state of the system [tex]\psi \in \mathbb{C}[/tex] is described by the following type of ODE:

[tex]

\frac{{d\psi \left( t \right)}}{{dt}} = i\overline {\psi \left( t \right)}

[/tex]

i.e, the time evolution of the state depends on the conjugate, not the state itself (this is determined by the first principles of my system).

Numerically, I can easily generate the quiver diagrams of this first difference, and then use it to simulate the time evolution of the system. The norm seems to grow roughly exponentially, while the argument exponentially approaches [tex]\pi /4 [/tex] , which is what I expect.

My problem is that I am at a loss of ideas for how to actually integrate the expression, in order to obtain the expression for the time evolution of [tex]\psi \left( 0 \right)[/tex] for any given starting point in the space.

Any assistance on this would be very much appreciated.

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# Stuck on a complex ODE (conjugate issues )

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