Which algebra vectors satisfy this (Trying to derive Schrödinger)

[Gerenuk]

Hi,

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 $f_{E,k}$ and which are "two-dimensional". It's not just a column vector, but rather $f_{E,k}(x,t)$. The vector algebra is undetermined (could be any linear algebra).

Now I have two conditions
$$k^2f_{E,k}+V(x,t)f_{E,k}=Ef_{E,k}$$
$$\forall a: f_{E,k}(x+Ea,t+ka)=f_{E,k}(x,t)$$
(btw, the second is in a way equivalent to $E=mc^2$)

Is it now possible to proof that the only orthonormal solution for this is a complex algebra with
$$f_{E,k}(x,t)=\exp(i(kx-Et))$$?

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?

 

[Gerenuk]

I notice he above factors k^2 and E have to be replaced by operators.

Anyway, the task is to find some axioms similar to the above, which yield the complex algebra as the only solution.

 

[IttyBittyBit]

What is V(x,t)? Because I have a feeling you mean something specific, not just a placeholder for an arbitrary function.

 

[Gerenuk]

Basically the above problem comes from the Schrödinger equation. Sometimes I might be missing concepts, but maybe you can add them.

The function V(x,t) is supposed to be an arbitrary real-valued function. k^2 and E are a real linear operators. So now I'm wondering which other conditions I need to add to make the vector basis of the algebraic solution isomorphic to the above complex solution.

Please formulate this more mathematically correctly whoever can. The aim is to show the vector basis must be complex numbers.

 

[ice109]

what are you talking about? this barely makes sense. is english your first language?

 

[Gerenuk]

Ice, If you have trouble with both language and maths, please devote your time to complaining in other forums.

 

[IanC89]

It might just be me, but you have specified anything about your "K²" being a second derivative with respect to position. Also, what you are sort of writing is the time independent Schrodinger equation so having a time dependent potential function V is not correct.