Calculate the eigenfunctions for a spin half particle

[borndisaster]

Homework Statement

Spin can be represented by matrices. For example, a spin half particle can be described by the following Pauli spin matrices
$$s_x = \frac{\hbar} {2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , s_y = \frac{\hbar} {2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} , s_z = \frac{\hbar} {2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Calculate the corresponding eigenfunctions which we will denote as ɑ- and β-eigenfunctions corresponding to spin 1/2 particles. Further show that s_j can be determined by the commutation of the other two matrices s_n and s_m , n,m≠j.

Homework Equations

The Attempt at a Solution

I have calculated the eigenvalues (first section of the question) of all three matrices to be ±ħ/2

I think the diagonal matrix s_z has eigenfunctions |α> = (1; 0) and |β> = (0; 1)
From that I found the eigenfunctions of s_x and s_x to be
|x₊> = |α> + |β> & |x_-> = -|α> + |β> and
|y₊> = -i|α> + |β> & |y_-> = i|α> + |β> respectively

But I'm not entirely sure I'm correct...

 

[DrClaude]

From that I found the eigenfunctions of s_x and s_x to be
|x₊> = |α> + |β> & |x_-> = -|α> + |β> and
|y₊> = -i|α> + |β> & |y_-> = i|α> + |β> respectively

Apart from normalization, these are correct.

 

[borndisaster]

Thanks @DrClaude

Any ideas as to how I show the commutation part of the question?

 

[DrClaude]

Any ideas as to how I show the commutation part of the question?

Calculate the commutator for each pair of operators and see what you get.