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I am trying to show how singlet and triplet states mix via spin-orbit coupling, but I am having some trouble.

I've gotten most of the way through, but I am stuck on how an operator works. Sadly, I think my poor basis in quantum is going to show right here.

My starting point...

[itex]\left(s_{1z}-s_{2z}\right)\left|0,0\right\rangle =\hbar \left|1,0\right\rangle [/itex]

The book I am looking at sort of skims past this and assumes that I have a clue what is going on here.

So I know

[itex]\left|0,0\right\rangle =\frac{1}{\sqrt{2}}\left\{\left|-+\right\rangle -\left|+-\right\rangle \right\}[/itex]

[itex]s_z=\frac{\hbar }{2}\left\{\left|+\right\rangle \left\langle +\right|-\left|-\right\rangle \left\langle -\right|\right\}[/itex]

and I am thinking that

[itex]\left(s_{1z}-s_{2z}\right)\left|0,0\right\rangle = \frac{1}{\sqrt{2}}\left(s_{1z}\left\{\left|-+\right\rangle -\left|+-\right\rangle \right\}-s_{2z}\left\{\left|-+\right\rangle -\left|+-\right\rangle \right\}\right) [/itex]

After this it becomes a giant mess. Basically it hinges on my inability to evaluate things of the form

[itex]\left|+\right\rangle \left\langle +|-+\right\rangle[/itex]

I am thinking that this is heavily reliant on some other basics as well. In determining the s_z spinor I am not sure how to evaluate something like

[itex]\left\langle +|+\right\rangle \left\langle +|+\right\rangle - \left\langle +|-\right\rangle \left\langle -|+\right\rangle[/itex]

which is supposed to equal one and

[itex]\left\langle +|+\right\rangle \left\langle +|-\right\rangle - \left\langle +|-\right\rangle \left\langle -|-\right\rangle[/itex]

which is supposed to equal zero

I was hoping that someone here would be able to help me figure out how

[itex]\left(s_{1z}-s_{2z}\right)\left|0,0\right\rangle =\hbar \left|1,0\right\rangle [/itex]

and the above evaluations. Thanks a whole bunch!