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## Homework Statement

Find the matrix representation of [itex]S_z[/itex] in the [itex]S_x[/itex] basis for spin [itex]1/2[/itex].

## Homework Equations

I have the Pauli matrices, and I also have the respective kets derived in each basis. There aren't really any relevant equations, other than the eigenvalue equations for the operators.

## The Attempt at a Solution

In the [itex]S_x[/itex] basis, the [itex]S_x[/itex] operator is just

[tex]

\frac{\hbar}{2}

\begin{pmatrix}

1 & 0\\

0 & -1

\end{pmatrix}

[/tex]

So isn't the [itex]S_z[/itex] operator in the [itex]S_x[/itex] basis just equal to the [itex]S_x[/itex] operator in the [itex]S_z[/itex] basis? That is,

[tex]

S_z=

\frac{\hbar}{2}

\begin{pmatrix}

0 & 1\\

1 & 0

\end{pmatrix}

[/tex]

in the [itex]S_x[/itex] basis? If so, I don't really understand how to show this in any meaningful way... I've also tried solving the eigenvalue equations for [itex]S_z[/itex] and converting the kets to the [itex]S_x[/itex] basis but that just lead me in circles. I can see that the question is a very basic one but something is just not letting me get a satisfactory answer. Any help would be much appreciated, thanks!