# Odd notation in a QM problem.

1. Jan 15, 2006

### inha

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
$$X=a_0+\sigma \cdot a$$,
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($$\sigma_k X$$ )
b. Obtain a_0 and a_k in terms of the matrix elements $$X_{ij}$$

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. Jan 15, 2006

### qbert

The main point here is that the Pauli matrices form a
complete set. The notation is shorthand for:
$$X = a_0 I + a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma 3$$
or
$$X = a_0 \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) + a_1 \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) + a_2 \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) + a_3 \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)$$

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. Jan 15, 2006

### inha

Thanks a lot! I didn't realize that the sigmas were supposed to be the Pauli matrices and that got me confused.