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## Homework Statement

Given the expression

[tex] s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1>[/tex]

obtain the matrix representations of s

_{+/-}for spin 1/2 in the usual basis of eigenstates of s

_{z}

## Homework Equations

[tex] s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1>[/tex]

[tex] S_{+} = \hbar

\begin{bmatrix}

0 &1 \\

0 & 0

\end{bmatrix}

[/tex]

[tex] S_{-} = \hbar

\begin{bmatrix}

0 &0 \\

1 & 0

\end{bmatrix}

[/tex]

## The Attempt at a Solution

So I've gotten the first part. You just sub into s and m for spin up or spin down yielding

[tex] s_{+}|\downarrow> = \hbar |\uparrow>[/tex]

[tex] s_{-}|\uparrow> = \hbar |\downarrow>[/tex]

In most textbooks I've checked, they just skip from what I've gotten above straight to the matrix representations. But I'm totally confused as to how the matrix elements of the matrices are found as you go from one to the other.