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I was wondering why the wave-function in quantum mechanics is complex. There are a lot of threads in the physics section and I've downloaded a lot of papers, but they seem quite technical. So I'd like to examine the following idea (sorry if I use sloppy terms ;) ):

I have an orthonormal basis of vectors/functions which can be labeled with two indices [itex]f_{E,k}[/itex] and which are "two-dimensional". It's not just a column vector, but rather [itex]f_{E,k}(x,t)[/itex]. The vector algebra is undetermined (could be any linear algebra).

Now I have two conditions

[tex]k^2f_{E,k}+V(x,t)f_{E,k}=Ef_{E,k}[/tex]

[tex]\forall a: f_{E,k}(x+Ea,t+ka)=f_{E,k}(x,t)[/tex]

(btw, the second is in a way equivalent to [itex]E=mc^2[/itex])

Is it now possible to proof that the only orthonormal solution for this is a complex algebra with

[tex]f_{E,k}(x,t)=\exp(i(kx-Et))[/tex]?

Please add definitions (for scalar product and so on) as appropriate!

Or which other definitions/conditions do I need to get that complex solution uniquely?