Sakurai - Constructing 2x2 matrix from scalars?

[Adoniram]

Hello all, I am beginning a course in QM with Sakurai's 2nd Edition book on QM. In one of our problems, he defines a matrix as the sum of a scalar and a dot product... This seems like nonsense to me, but he uses the same notation in the next problem, so I am guessing this is some unorthodox notation he uses. Can someone please enlighten me how to construct a 2X2 matrix from this definition?

X = a_{o} + \overline{\sigma} \bullet \overline{a}

(btw, the problem is not to construct the matrix. The problem is 1.2 from the 2nd edition if you want to look it up, and involved relating the scalars to the trace of X. If I can define X, I can do the rest)

Anyway, to me it looks like X is a scalar itself, not a 2X2 matrix, but allegedly it's a matrix... My only guess is that he's using some bizarre rule where Det(X) = X, but again, that's nonsense... Help!

 

[DrClaude]

It's even worse than you think, since $\vec{\sigma}$ is actually a vector of matrices!

What it means, using correct mathematics, is
$$
X = a_0 \mathbf{1} + \sum_{i=1}^3 a_i \mathbf{\sigma}_i
$$
where $\mathbf{1}$ is the $(2 \times 2)$ identity matrix and $\mathbf{\sigma}_i$ the Pauli matrices
$$
\mathbf{\sigma}_1 = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)
$$
$$
\mathbf{\sigma}_2 = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)
$$
$$
\mathbf{\sigma}_3 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)
$$

 

[Adoniram]

Oh my gosh... that helps a LOT. Thank you!