Position wave function of two electrons

  • Thread starter Faust90
  • Start date
  • #1
20
0
Hi,

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

1. Homework Statement


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')

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
 

Answers and Replies

  • #2
DrClaude
Mentor
7,601
3,997
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)
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
20
0
Hi,

thanks for your answer :-)

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
DrClaude
Mentor
7,601
3,997
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?
Yes to both.
 

Related Threads on Position wave function of two electrons

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
0
Views
951
Replies
26
Views
667
Replies
1
Views
1K
  • Last Post
Replies
11
Views
3K
Replies
9
Views
5K
  • Last Post
Replies
1
Views
1K
Top