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## Homework Statement

Spin can be represented by matrices. For example, a spin half particle can be described by the following Pauli spin matrices

[tex] 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} [/tex]

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...