Position wave function of two electrons

1. Oct 23, 2014

Faust90

Hi,

I want to calculate the position-wave-function of a system of two free electrons with momenta k1 and k2 (vectors).

1. The problem statement, all variables and given/known data

So, I want to have Psi_(k1,k2)(x1,x2) for a state |k1,k2>

I also know that <k'|k> = (2Pi)^3 Delta(k-k')
3. The attempt at a solution

I tried the following:

Psi(x1,x2)=<x1,x2|k1,k2>=<x1,x2|1|k1,k2>=Integral<x1,x2|k1',k2'><k1',k2'|k1,k2>dk1dk2

The first term <x1,x2|k1',k2'> are the momenta eigenfunction in Space presentation, the second term is the given delta-function:

= Integral Exp(ik1'x1)Exp(ik2'x2) Delta(k1-k1')Delta(k2-k2')
=Exp(ik1x1)Exp(ik2x2)

Is that right?
I'm a bit confused about k1,k2 and k1',k2'

Best regards :-)
Faust

2. Oct 23, 2014

Staff: Mentor

Aren't you running in circles here? If you can replace <x1,x2|k1',k2'> by Exp(ik1'x1)Exp(ik2'x2), why can't you do that with <x1,x2|k1,k2>?

If you take definite values of $k$, then the wave function is not localized: the uncertainty on position is infinite. You need to take a more physically acceptable starting point, such as a wave packet centered around $k_1$ and $k_2$. You can then Fourier transform it to get the wave function in position space.

Remember also that the total wave function has to obey the Pauli principle.

3. Oct 24, 2014

Faust90

Hi,

I'm still a bit confused. Let me summary again what I have:

Two electrons characterized by a wave vector k, where I know that the normalization is :<k|k'>=(2Pi)^3 Delta(k-k').
Now I have the state |k1,k2> and I shall construct the space wave function.

My first questions:

1. The |k>,|k'> are the eigenvectors of the momentum operator, or?
2. Does the state |k1,k2> mean that I have definite values of k?

4. Oct 24, 2014

Yes to both.