# Sakurai Quantum Mechanics problems

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 $$$$X = a_0 + \mathbf{\sigma} \cdot \mathbf{a},$$$$ where $$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

• s are the pauli matrices. Each s_i is a 2 by 2 Matrix. The identity 2x2 Matrix is implicit with the a_0. May 31 '19 at 9:26
• 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? May 31 '19 at 9:32
• Please use Mathjax for mathematics expressions and variables. It is the site standard. May 31 '19 at 9:34
• 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. May 31 '19 at 9:35
• Thanks all, sorry for unexperience about expression coding, I'll use next time. May 31 '19 at 9:41

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: $$$$X = a_0 I_{2\times 2} + a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3$$$$ where $$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.