Sakurai Problems - strange notation

[Hargoth]

Hello!

I'm just doing the Problems of Chapter 1 of Sakurai: Modern Quantum Mechanics. On page 60, problem 2 he writes:

"Suppose a 2x2 matrix X, (not necessary Hermitian, nor unitary) is written as

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

where $a_0$ and $a_{1,2,3}$ are numbers."

which confuses me a bit, because you can't add a number and a matrix, and what is $\sigma$ anyway? Would be nice if someone knows what is meant by this (and tells me ).

 

[dextercioby]

I'm sure he means that $\vec{\sigma}$ is a short name for the Pauli matrices and a_{0} is multiplied by the 2*2 unit matrix.

Daniel.

 

[inha]

dexter is correct. I was confused with this problem too and actually asked the same question about it here some time ago.

 

[Hargoth]

Thanks.

That $\mathbf{\sigma}$ is one of the Pauli-Matrices seems strange to me from the context of the book, because he never defined this during the first chapter. Nevertheless, I'll try to figure it out this way.

 

[Daverz]

I think you're just supposed to take the sigmas abstractly as some 2x2 (complex-valued) matrices such that any 2x2 (complex-valued) matrix can be written as

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

with no assumptions about the form of the $\mathbf{\sigma}$s. In math speak, the $\mathbf{\sigma}$s together with $I$ are some basis for the vector space of 2x2 complex-valued matrices.

 

[Daverz]

Eh, now that I try to actually do the problem, I think he (or the editor) is just assuming you know the Pauli matrices from undergrad QM.