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Second Quantization for Fermions: Creation Operator

  1. Dec 8, 2012 #1
    2crrgiq.jpg

    So, I'm studying Second Quantization for fermions and came across this equation. I was just wondering why there is a summation needed? And why do we do it with (i≠p).???? Please can someone explain this to me?

    Reply and help is much appreciated.
     
  2. jcsd
  3. Dec 8, 2012 #2
    This is basically a mathematical expression for the Pauli Exclusion Principle.
    Two ways to state the Pauli exclusion principle are:
    1) Two or more fermion particles can not share the same quantum state.
    2) A specific quantum state can have either 0 or 1 fermion particles.
     
  4. Dec 8, 2012 #3
    Ow! Yes, I've read that! But why the need for Np? And why is -1 raised to Np if we just want to keep its sign?
     
  5. Dec 8, 2012 #4
    i'll post the whole page so you would have an idea about my question

    30joh9w.jpg
    11tmnue.jpg

    So here, for those I put check on. Why is there a (-1)^Np? I know that's part of the creation and annihilation operator, but why is it there?

    Second, the equation I encircled has Nq+1. How did that happen? Thanks.
     
  6. Dec 9, 2012 #5
    It just arises because of the antisymmetry required,which introduces a factor of -1.
     
  7. Dec 9, 2012 #6
    Hmmm.. Because wavefunction for fermions are antisymmetric? Ok. Got that! But how was the encircled eqn derived? And the last equation is also giving me a hard time. Im trying to derive everything here.
     
  8. Dec 9, 2012 #7
    Hy, you seem to be lost.
    Perhaps, this reference will be helpful:
    www.scholarpedia.org/article/Second_quantization
    (please respect the recommandations at the bottom of the page)
    These operators seem to be spinors and have a typical anticommutative behavior.
    I cannot help you directly, sorry.
    Good luck
     
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