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## Homework Statement

I need to prove some relations but going round in circles.

## [\hat{J}_z, \hat{J}_+] = \hbar J_+ ##

I've got this:

##\left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)\left(a_+^{\dagger }a_-\right)-\left(a_+^{\dagger }a_-\right)\left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)##

## Homework Equations

##J_{\pm} = \hbar a_{\pm}^{\dagger} a_{\mp} ##

##J_z = \frac{\hbar}{2} \left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)##

## The Attempt at a Solution

I have these sort of obvious relations to work from

##[a_{\pm}^{\dagger}, a_{\pm}] = 1##

##[a_{\pm}^{\dagger}, a_{\mp}] = 0##

##[a_{\pm}^{\dagger}, a_{\mp}^{\dagger}] = 0##

##[a_{\pm}^{\dagger}, a_{\mp}] = 0##

##[a_{\pm}, a_{\mp}] = 0##

I start expanding and it gets too tough, what am I missing?