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Interacting Fermion System Commutation

  1. Apr 25, 2015 #1
    Problem Question
    My question isn't an entire homework problem, but rather for a certain mathematical step in the problem which I assume to be very simple.

    The problem is dealing with interacting fermion systems using second quantization formulas. I am essentially following my notes from class for this problem and the part I am stuck on is this.

    [tex]<0,0|a_{2 \uparrow} a_{1 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}a_{1 \uparrow}a_{2 \uparrow}a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=-1[/tex]

    My question is, why is this equal to -1?

    Attempt at Solution
    Here is my logic on how to evaluate this.

    [tex]a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>[/tex]
    [tex]a_{1 \uparrow}a_{2 \uparrow}|\uparrow \uparrow>=|0,0>[/tex]
    [tex]a_{1 \uparrow}^{\dagger}a_{2 \uparrow}^{\dagger}|0,0>=|\uparrow \uparrow>[/tex]
    [tex]a_{2 \uparrow} a_{1 \uparrow}|\uparrow \uparrow>=|0,0>[/tex]
    Thus all together I get 1, not -1. Why is this equal to -1???
  2. jcsd
  3. Apr 25, 2015 #2


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    Homework Helper
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    2017 Award

    I believe the mistake is in the second equation above. There are sign conventions for Fermion operators operating on Fock states.

    See equations 4, 5, and 6 here: http://www.phys.ufl.edu/~kevin/teaching/6646/03spring/2nd-quant.pdf
  4. Apr 25, 2015 #3
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