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Perturbation Theory: Calculating for the correction on the ground state energy

  1. Dec 1, 2012 #1
    1. The problem statement, all variables and given/known data
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    2. Relevant equations

    [itex]E_{1}=<ψ_{1}|V(r)|ψ_{1}>[/itex]

    3. The attempt at a solution

    That is equal to the integral ∫ψVψd^3r

    So I'll just perform the integral, correct ? But r is not constant here right? So, I' ll keep it inside the integral? How should I continue? Please help. Thanks. :))
     
  2. jcsd
  3. Dec 2, 2012 #2

    TSny

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    Right, ##r## is a variable of integration.

    Something to think about: In the integral should you use the potential energy ##V(r)## as stated in the problem or the perturbation ##\delta V(r)## of the potential energy (due to switching from the potential energy of a point nucleus to the potential energy of a finite-sized nucleus)?
     
  4. Dec 2, 2012 #3
    I really don't get your point sorry. I'm guessing I should use the perturbation of the potential. But how can I get that?
     
  5. Dec 2, 2012 #4

    TSny

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    The hydrogen atom is usually solved treating the nucleus as concentrated in a point. The ground state wavefunction that you specified was derived under this assumption.

    Now you want to treat the nucleus more realistically as having a finite size and calculate a correction to the ground state energy in going from a point nucleus to the finite nucleus. The Hamiltonian for a finite nucleus can be thought of as the Hamiltonian for the point nucleus plus a "perturbation". So, the perturbation is just the difference between the Hamiltonian for a finite nucleus and the Hamiltonian for a point nucleus. You should convince yourself that the perturbation is just the change ##\delta V(r)## in the potential energy function when going from the point nucleus to the finite nucleus.

    Can you find a mathematical expression for ##\delta V(r)##? The potential energy for a finite nucleus is given in the problem. So, you need to remember what the potential energy function is for a point nucleus.
     
    Last edited: Dec 2, 2012
  6. Dec 2, 2012 #5
    you should just break up the integral at r=R.see if it works.
     
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