Odd notation in a QM problem.

  • Thread starter inha
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  • #1
575
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?
 

Answers and Replies

  • #2
185
4
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).
 
  • #3
575
1
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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