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I Limit of large but finite

  1. Apr 8, 2016 #1
    I've calculated the eigenstates of the Hubbard Hamiltonian for two fermions.
    The ground state is (U2 - (U2 + 16t2)1/2)/2
    For U = infty, I get 0.
    For U >> t, I should get the exchange energy J = -4t2/U
    How do I get from the ground state equation to J?
  2. jcsd
  3. Apr 8, 2016 #2


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    Staff: Mentor

    Use the Taylor series expansion ##\sqrt{1 + x} \approx 1 + x/2## for small x.
  4. Apr 8, 2016 #3
    Too easy, thanks!
  5. Apr 8, 2016 #4


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    When I saw a thread called "limit of large but finite" under "calculus" I wasn't expecting something about the Hubbard Hamiltonian of two fermions. But, clearly, @DrClaude was prepared for anything!
  6. Apr 8, 2016 #5


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    Staff: Mentor

    I was actually more prepared for the Hubbard Hamiltonian than for regular calculus :wink:

    I hesitated to move the thread. @supernano, note that while your question was a mathematical one, you have a higher probability of getting an answer for such a question in the QM forum. It's more a question of knowing the tricks than knowing maths.
  7. Apr 8, 2016 #6


    Staff: Mentor

    For one thing, you can't just substitute ##\infty## into the expression. For another thing, the expression above has the form ##[\infty - \infty]##, which is one of several indeterminate forms, along with ##[\frac0 0]##, ##[\frac{\infty}{\infty}]## and a few others.
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