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## Homework Statement

Write down the 3×3 matrices that represent the operators [itex]\hat{L}_x[/itex], [itex]\hat{L}_y[/itex], and [itex]\hat{L}_z[/itex] of angular momentum for a value of [itex]\ell=1[/itex] in a basis which has [itex]\hat{L}_z[/itex] diagonal.

## The Attempt at a Solution

Okay, so my basis states [itex]\left\{\left|\ell,m\right\rangle\right\}[/itex] are [itex]\left|1,-1\right\rangle[/itex], [itex]\left|1,0\right\rangle[/itex], and [itex]\left|1,1\right\rangle[/itex]. [itex]\hat{L}_z\left|\ell, m\right\rangle=\hbar m\left|\ell,m\right\rangle[/itex], so the matrix representation of [itex]\hat{L}_z[/itex] is easy: [tex]\hat{L}_z \doteq \left( \begin{array}{ccc} -\hbar & & \\ & 0 & \\ & & \hbar \end{array} \right).[/tex] But I don't know what to do in order to determine [itex]\hat{L}_x[/itex] and [itex]\hat{L}_y[/itex].

## Homework Equations

The commutation relations [itex]\left[ \hat{L}_x, \hat{L}_y \right] = i\hbar \hat{L}_z[/itex], etc., could maybe be useful but I'm not sure how.