Representation of spin matrices

[bznm]

I have just started to study quantum mechanics, so I have some doubts.

1) if I consider the base given by the eigenstates of s_z $$s_z | \pm >=\pm \frac{\hbar}{2} |\pm>$$ the spin operators are represented by the matrices

$$s_x= \frac {\hbar}{2} (|+><-|+|-><+|)$$
$$s_y= i \frac {\hbar}{2}(|-><+|-|+><-|)$$
$$s_z=\frac{\hbar}{2}(|+><+|-|-><-|)$$

but I don't have clear idea of what they correspond to concretely. Considering the Dirac formalism, can they be represented by Pauli's matrices?

$$s_x=\frac {\hbar}{2} \sigma_x; s_y= i \frac {\hbar}{2} \sigma_y; s_z=\frac {\hbar}{2} \sigma_z$$

2) I can write a vector using its components, e.g. v=(a,b)
but which are the components of the eigenstates of s_z?

3) If I have a state such as $$s_x |\phi>=\frac {\hbar}{2} |\phi>$$
and I want to write it using the base of the eigenstates of s_z,
can I write $$|\phi>=a|+>+b|->$$, with $$|a|^2+|b|^2=1?$$ (I need this condition to have a normalized vector)
Is it equal to $$\frac {e^{i\theta}}{\sqrt 2}(|+>+|->)?$$

Many thanks for your help!

 

[Orodruin]

Considering the Dirac formalism, can they be represented by Pauli's matrices?

Yes.

2) I can write a vector using its components, e.g. v=(a,b)
but which are the components of the eigenstates of s_z?

Yes, if you use the representation of the spin operators in terms of Pauli matrices.

can I write

Yes, what you quoted here was a general spin state. You need to fix a and b so that it is actually an eigenvector of s_x.

Is it equal to

... Which you did here. The exponential is an arbitrary phase factor and can be dropped.

 

[bznm]

Thanks a lot!