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## Homework Statement

What is the matrix representation of ##\mathbb{\hat J}_z## using the states ##|+y\rangle## and ##|-y\rangle## as a basis?

## Homework Equations

##|\pm y\rangle =\frac{1}{\sqrt{2}}|+z\rangle \pm \frac{i}{\sqrt{2}}|-z\rangle##

## The Attempt at a Solution

A solution was given:

##\mathbb{\hat J}_z =\left[{\begin{array}{cc} \langle +y|+z\rangle & \langle +y|-z\rangle \\ \langle -y|+z\rangle & \langle -y|-z\rangle \\\end{array}}\right]\frac{\hbar}{2}\left[{\begin{array}{cc} 1 & 0 \\ 0 & -1 \\\end{array}}\right]

\left[{\begin{array}{cc} \langle +z|+y\rangle & \langle +z|-y\rangle \\ \langle -z|+y\rangle & \langle -z|-y\rangle \\\end{array}}\right] = \frac{\hbar}{2}\left[{\begin{array}{cc} 0 & 1 \\ 1 & 0 \\\end{array}}\right],##

but I am confused on what is going on? Can anyone explain exactly what is going on, please?

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