1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear algebra - Anti-Commutation Relations

  1. Jul 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Show that {x_i, x_j} = 2*y_ij* I for i = 1; 2; 3 and j = 1; 2; 3.
    where
    y_ij: N x N -> {0,1}, such that y_ij = {1, if i = j ; 0, if i not = j

    2. Relevant equations



    3. The attempt at a solution

    I'm confused about exactly what i'm supposed to do here. Do i do all the combinations between i and j or what? Can someone please show me an example or two so that I can do the rest. Thank you
     
  2. jcsd
  3. Jul 13, 2010 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Yes, you need to show it for all combinations of i and j.

    How are the xi's defined?
     
  4. Jul 13, 2010 #3
    x_1 =
    [0 1
    1 0]

    x_2 =
    [ 0 -i
    i 0]

    x_3 =
    [1 0
    0 -1]

    could you show me one of them just so that i know i'm doing them correctly please.
     
    Last edited: Jul 13, 2010
  5. Jul 13, 2010 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The easiest thing to do is just calculate the six possibilities. For example, for x1 and x2, you want to calculate

    [tex]\begin{align*}
    \{x_1,x_2\} & = x_1 x_2 + x_2 x_1 \\
    & = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\begin{bmatrix}0 & -i \\ i & 0\end{bmatrix} + \begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}
    \end{align*}
    [/tex]
     
  6. Jul 13, 2010 #5
    I calculate that then do the 2*y_ij* I and compare? how do i know {x1, x2} = x1x2 + x2x1?
     
    Last edited: Jul 13, 2010
  7. Jul 14, 2010 #6

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Yes.
    That's the definition of the anticommutator.
     
  8. Jul 14, 2010 #7
    oh, ok. I wish we had been told that. But it makes sense.

    I also have one similar to that

    Commutation Relations. Show that [x_i; x_j ] = 2[tex]\sqrt{-1}[/tex][tex]\sum[/tex]e_ijk * x_k
    from k= 1 to 3
    for i = 1,2,3 and j= 1,2,3

    where e_ijk = { 1 if(i,j,k) is (1,2,3),(2,3,1),(3,1,2)
    -1 if (i,j,k) is (3,2,1),(1,3,2) or (2,1,3)
    0 if i= j or j=k or k=1

    For this do i basically do the same thing? is the relationship the same for the x1x2 + x2x1? And also say i do x1 and x2, i'm not sure what to choose for k
     
  9. Jul 14, 2010 #8

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The commutator is defined as [A,B]=AB-BA.

    The righthand side of the equation is

    [tex]2\sqrt{-1}\sum_{k=1}^3 \epsilon_{ijk}x_k[/tex]

    You're summing over k. You don't get to choose it.

    Suppose i=1 and j=2, then [itex]\epsilon_{ijk}=0[/itex] if k=1 or k=2, so the only term that will survive is the k=3 term. So you want to show

    [tex][x_1,x_2] = 2\sqrt{-1}x_3[/tex]
     
    Last edited: Jul 14, 2010
  10. Jul 14, 2010 #9
    oh! ok that makes sense. Thank you so much
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook