1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Position wave function of two electrons

  1. Oct 23, 2014 #1
    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. jcsd
  3. Oct 23, 2014 #2

    DrClaude

    User Avatar

    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.
     
  4. Oct 24, 2014 #3
    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?
     
  5. Oct 24, 2014 #4

    DrClaude

    User Avatar

    Staff: Mentor

    Yes to both.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted