Matrix representation in QM Assignment -- Need some help please

[Ashish Somwanshi]

Homework Statement

In the lecture, we used the eigenstates |z+⟩ and |z−⟩ of Sz^, we obtained the matrices for spin operators

Sx^=ℏ/2(0,1,1,0) Sy^=ℏ/2(0,i,−i,0) Sz^=ℏ/2(1,0,0,−1)
note: the numbers in brackets are 2×2 matrices!!!


Now use the eigenstates of |x+⟩ and |x−⟩ of Sx^,as a new basis, construct matrices for the spin operators Sx^, Sy^ and Sz^.

Relevant Equations

Both Question and Relevant equations are posted below in attempt.

This screenshot contains the original assignment statement and I need help to solve it. I have also attached my attempt below. I need to know if my matrices were correct and my method and algebra to solve the problem was correct...

 

[berkeman]

Homework Statement:: In the lecture, we used the eigenstates |z+⟩ and |z−⟩ of Sz^, we obtained the matrices for spin operators

Sx^=ℏ/2(0,1,1,0) Sy^=ℏ/2(0,i,−i,0) Sz^=ℏ/2(1,0,0,−1)
note: the numbers in brackets are 2×2 matrices!Now use the eigenstates of |x+⟩ and |x−⟩ of Sx^,as a new basis, construct matrices for the spin operators Sx^, Sy^ and Sz^.
Relevant Equations:: Both Question and Relevant equations are posted below in attempt.

I have also attached my attempt below.

Please make it a habit to post your work using LaTeX, and not in blurry attached pictures. There is a "LaTeX Guide" link below the Edit window to help you learn LaTeX. Thank you.

 

[Ashish Somwanshi]

My solution to above assignment goes like this:

Since
| ×±> = 1/ sqrt(2) |z+> ± 1/sqrt(2) |z->

|x+> = 1/sqrt (2) |z+> + 1/sqrt(2) |z->
|x-> = 1/sqrt(2) |z+> - 1/sqrt(2) |z->

So eigenvalue equations are:

Sx |x+> = 1/sqrt(2) { |z+> + |z->}
Sx |x-> = 1/sqrt(2) { |z+> - |z->}

So we can represent Sx operator in matrix form as:

Sx = 1/sqrt(2)*

|z+> + |z->	0
0	|z+> - |z->

Is my method to evaluate Sx operator from the relationship between x and z correct? Also how can I evaluate Sy and Sz operators for spin.?