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## Homework Statement

When calculating expectation values for spin states I encountered ##\langle \hat{\mathbb{S}}_+\rangle = \langle+z|\hat{\mathbb{S}}_+|+z\rangle = \frac12\langle+z|\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle.##

How do we compute ##\langle+z|\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle?##

Also, similary ##\langle+z|\left(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-\right)^2|+z\rangle?##

## Homework Equations

##\hat{\mathbb{S}}_x=\frac12(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-)##

## The Attempt at a Solution

I know that ##\langle+z|\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle = 0## but I am not sure how to compute it to get zero.

Do I compute ##\left(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle\right)## first, which gives ##\hbar|-z\rangle## and then use the bra ##\langle +z|## to get zero?