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Ground State Energy of Helium Atom - Integration problem

  1. Dec 15, 2011 #1
    I'm trying to follow a derivation of the ground state energy of Helium using perturbation theory. I've made it through most of the derivation but I'm stuck at the following integral

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

    Find the value of C where

    [itex]C=\frac{1}{(4\pi)^2}\int d\mathbf{z}_1d\mathbf{z}_2\frac{\exp(-(z_1+z_2))}{|\mathbf{z}_1-\mathbf{z}_2|}[/itex]

    To give context to this equation, the vectors z_1 and z_2 are dimensionless rescalings of the position vectors (r_1 r_2) of the two electrons.
    [itex]\mathbf{r}_1=\frac{a}{2}\mathbf{z}_1[/itex]
    [itex]\mathbf{r}_2=\frac{a}{2}\mathbf{z}_2[/itex]

    where a is the Bohr radius.

    The integral is to be taken all over space.

    2. Relevant equations

    Don't think there are any

    3. The attempt at a solution

    integrating wrt to z_1 first:

    [itex]\frac{1}{(4\pi)^2} \int z_1^2 \sin^2\theta_1 dz_1 d\theta_1 d\phi_1 \frac{\exp{z_1+z_2}}{\sqrt{z_{1}^2 + z_{2}^2 - 2z_1 z_2 \cos\theta_{12}}}d\mathbf{z}_2[/itex]

    I basically don't know where to go from here. My first idea to simplify was to make the integral relative and set the z_2 vector to always be the z-axis of the coordinate frame (thus eliminating an angle) but I still couldn't do the integral itself.

    Thanks,
     
  2. jcsd
  3. Dec 16, 2011 #2

    dextercioby

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    Science Advisor
    Homework Helper

    I think it is discussed in books. You should use certain coordinates (don't remember which), which would simplify the 6-tuple integration.

    Books? Cohen-Tannoudji 2nd volume. Bethe & Salpeter (which treat the Helium atom perturbatively), Bransden & Joachain, etc.
     
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