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A fermionic 2-state system

  1. Jul 19, 2007 #1
    1. The problem statement, all variables and given/known data

    A fermionic 2-state system is [tex]\left\{\left|00\right\rangle, \left|10\right\rangle, \left|01\right\rangle, \left|11\right\rangle\right\}[/tex], where
    [tex]\left|ab\right\rangle = \left|a\right\rangle_{1}\left|b\right\rangle_{2}[/tex] and [tex]\left\langle ab\left|cd\right\rangle = \delta_{ac}\delta_{bd}[/tex].

    2. Relevant equations

    What are the creation and annihilation operators [tex]f_{1}, f_{1}^{\dagger}, f_{2}, f_{2}^{\dagger}[/tex] in this base?

    3. The attempt at a solution

    I just do not know, how to get started. I just cannot find the theory and I do not know how to use it in this problem anyway. Can you give me some hints? Please, do not give right away the correct answers/results, just the hints to get started.
     
    Last edited: Jul 19, 2007
  2. jcsd
  3. Jul 19, 2007 #2

    Gokul43201

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    How would you find the matrix elements <ab|f|cd> of an operator f?
     
  4. Jul 19, 2007 #3
    So, I tried to calculate the matrix elements of an annihilation operator [tex]f_{1}[/tex]. There are 16 of them, I think. Is this correct?
    [tex]
    \left\langle 00\left|f_{1}\left|10\right\rangle = \left\langle 00\left|00\right\rangle = \delta_{00}\delta_{00} = 1\cdot 1 = 1.
    [/tex]
    [tex]
    \left\langle 00\left|f_{1}\left|01\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 00\left|f_{1}\left|11\right\rangle = \left\langle 00\left|01\right\rangle = \delta_{00}\delta_{01} = 1\cdot 0 = 0.
    [/tex]
    [tex]
    \left\langle 00\left|f_{1}\left|00\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 10\left|f_{1}\left|10\right\rangle = \left\langle 10\left|00\right\rangle = \delta_{10}\delta_{00} = 0\cdot 1 = 0.
    [/tex]
    [tex]
    \left\langle 10\left|f_{1}\left|01\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 10\left|f_{1}\left|11\right\rangle = \left\langle 10\left|01\right\rangle = \delta_{10}\delta_{01} = 0\cdot 0 = 0.
    [/tex]
    [tex]
    \left\langle 10\left|f_{1}\left|00\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 01\left|f_{1}\left|10\right\rangle = \left\langle 01\left|00\right\rangle = \delta_{00}\delta_{10} = 1\cdot 0 = 0.
    [/tex]
    [tex]
    \left\langle 01\left|f_{1}\left|01\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 01\left|f_{1}\left|11\right\rangle = \left\langle 01\left|01\right\rangle = \delta_{00}\delta_{11} = 1\cdot 1 = 1.
    [/tex]
    [tex]
    \left\langle 01\left|f_{1}\left|00\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 11\left|f_{1}\left|10\right\rangle = \left\langle 11\left|00\right\rangle = \delta_{10}\delta_{10} = 0\cdot 0 = 0.
    [/tex]
    [tex]
    \left\langle 11\left|f_{1}\left|01\right\rangle = 0.
    [/tex]
    [tex]
    \left\langle 11\left|f_{1}\left|11\right\rangle = \left\langle 11\left|01\right\rangle = \delta_{10}\delta_{11} = 0\cdot 1 = 0.
    [/tex]
    [tex]
    \left\langle 11\left|f_{1}\left|00\right\rangle = 0.
    [/tex]

    So, [tex]
    f_{1} =
    \begin{pmatrix}
    1 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 0 \\
    \end{pmatrix}?
    [/tex]
     
  5. Jul 21, 2007 #4

    Gokul43201

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    You've determined the non-zero matrix elements correctly, but put them in the wrong positions. The row and column indices come from the left side of those equations. Also, you haven't determined possible normalization constants.

    [tex]f_1|11 \rangle = c |01 \rangle [/tex]

    What is c?
     
    Last edited: Jul 21, 2007
  6. Jul 21, 2007 #5
    So, something like this
    [tex]f_{1} = \begin{pmatrix}0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\\end{pmatrix}?[/tex]

    Hmm. I have no idea. Can you give a hint?
     
  7. Jul 21, 2007 #6

    Gokul43201

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    Yes, that looks better.

    Have you come across the number operator (e.g., [itex]n_1=f_1^{\dagger}f_1[/itex])?
     
  8. Jul 21, 2007 #7
    Hi and thanks!

    Yes, I think so. It gives the number of particles in the ground state. Right?
     
  9. Jul 21, 2007 #8

    olgranpappy

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    Also, n.b., this is not a two-state system.
     
  10. Jul 22, 2007 #9
    Hmm. It is not? Oh boy. This was a exam problem. I failed badly in that exam. :confused:
     
  11. Jul 22, 2007 #10

    olgranpappy

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    well, there are four states, |00>, |01>, |10>, |11>... So it's a four-state system.
     
  12. Jul 22, 2007 #11

    olgranpappy

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    ...two times two-states.
     
  13. Jul 22, 2007 #12

    Gokul43201

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    I think the original question may have called it a "2-site" problem.
     
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