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Odd notation in a QM problem.

  1. Jan 15, 2006 #1
    I need help with deciphering notation from the second excercise of Sakurai's Modern QM's first chapter. Here's how it's presented in the book:

    Suppose a 2x2 matrix X (not neccessarily Hermitian, nor unitary) is written as
    [tex]X=a_0+\sigma \cdot a [/tex],
    where a_0 and a_k (k=1,2,3) are numbers.

    a. How are a_0 and a_k related to tr(X) and tr([tex]\sigma_k X[/tex] )
    b. Obtain a_0 and a_k in terms of the matrix elements [tex]X_{ij}[/tex]

    Now I have no idea what the matrix X is supposed to look like. Nor can I even figure out how a 2x2 matrix could be written like that. I remember seeing someone ask something about the same excercise here but I couldn't find that thread via search. I can't really present any work here since I don't know what the matrix is supposed to look like but could someone help me get started with this anyway?
     
  2. jcsd
  3. Jan 15, 2006 #2
    The main point here is that the Pauli matrices form a
    complete set. The notation is shorthand for:
    [tex] X = a_0 I + a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma 3 [/tex]
    or
    [tex] X = a_0 \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)
    + a_1 \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)
    + a_2 \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)
    + a_3 \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)
    [/tex]

    Now to solve the problem you'll want to use the following facts
    Tr(A+B) = Tr(A) + Tr(B)
    Tr(sigma_i) = 0
    sigma_i . sigma_i = I
    sigma_1. sigma_2 = i sigma_3 (and even permutations).
     
  4. Jan 15, 2006 #3
    Thanks a lot! I didn't realize that the sigmas were supposed to be the Pauli matrices and that got me confused.
     
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