How to Derive Matrix Representations for Spin Operators?

[gabriellelee]

Homework Statement

Find the matrix representation of \hat{S_x}, \hat{S_y}, \hat{S_z} for s = 1 spin one (electroweak Z-boson) in the basis of |sm> eigenstates.

Relevant Equations

Hint: do it for \hat{S_\pm} first.

$$\hat{S_+} = \hbar \begin{bmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{bmatrix}$$
$$\hat{S_-} = \hbar \begin{bmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{bmatrix}$$
$$\hat{S_x} = \hbar/\sqrt{2} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$
$$\hat{S_y} = \hbar/\sqrt{2} \begin{bmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{bmatrix}$$
$$\hat{S_z} = \hbar \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix}$$

 

[PeroK]

Correct!