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I just began studying QM on Sakurai's "Modern Quantum Mechanics" and just finished chapter 1. I am now approaching the exercises. On exercise 2 there is a notation I can't understand:

A 2x2 square matrix X is written as \begin{equation} X = a_0 + \mathbf{\sigma} \cdot \mathbf{a}, \end{equation} where $a_0$ and $a_{1,2,3}$ numbers.

I can't understand this notation: it is clear that $\mathbf{\sigma}$ and $\mathbf{a}$ are vectors with the same dimensionality, and $\mathbf{a} = (a_1, a_2, a_3)$ so I guess $\mathbf{\sigma} = (\sigma_1, \sigma_2, \sigma_3)$. How can their product produce a 2x2 matrix? And what kind of product does Sakurai intend with $\mathbf{\sigma} \cdot \mathbf{a}$?

Thanks

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    $\begingroup$ s are the pauli matrices. Each s_i is a 2 by 2 Matrix. The identity 2x2 Matrix is implicit with the a_0. $\endgroup$
    – lalala
    Commented May 31, 2019 at 9:26
  • $\begingroup$ Oh, this is an epiphany! (Even though, in the chapter text it is generally explicited when eye is implicit). But given what you said, there is som sum implicit in s · a or the X operator is a set of 3 operators? And if so, why isn't it fomatted as bold? $\endgroup$
    – Dario Ligori
    Commented May 31, 2019 at 9:32
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    $\begingroup$ Please use Mathjax for mathematics expressions and variables. It is the site standard. $\endgroup$
    – StephenG - Help Ukraine
    Commented May 31, 2019 at 9:34
  • $\begingroup$ There is no X in your Question, so not sure. sigma consists of three Matrices. And then a dot s is meant as a_1 s_1+a_2s_2+ a_3 s_3. Why the formatting is different then expected, we have to ask the Editor of the book. $\endgroup$
    – lalala
    Commented May 31, 2019 at 9:35
  • $\begingroup$ Thanks all, sorry for unexperience about expression coding, I'll use next time. $\endgroup$
    – Dario Ligori
    Commented May 31, 2019 at 9:41

1 Answer 1

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As mentioned by others in the comment, $\sigma$ refers to a set of three $2\times 2$ matrices $\sigma_x,\sigma_y,\sigma_z$ (or $\sigma_1,\sigma_2,\sigma_3$) known as Pauli matrices. So, what "$X = a_0 + \sigma\cdot\mathbf{a}$" means is this: \begin{equation} X = a_0 I_{2\times 2} + a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3 \end{equation} where $ I_{2\times 2} $ is the $2\times 2$ identity matrix (which was implicitly assumed to be understood and hence, left out in the original expression).

Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ are introduced only in chapter 3 of the book (even though spin-$\frac{1}{2}$ matrices are introduced in chapter 1 itself!). So, problems 2 and 3 of chapter 1 actually belong to chapter 3. This is the reason for your confusion.

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