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Action of spin operators on entangled spin states

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Hello.
I need some help on this question.
The basic assumption is that the background material is that for a spin 1/2 particle.

I have attached the spin matrices.

Thanks.
Fred.

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Solution Summary

We explain how to compute the action of spin operators on states involving two (or more) spins. We then compute the desired expressions for the actions of the spin operators on the state and the expectation values in detail.

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Let's denote the spin up state as |+> and spin down as |->. We can then write:

|chi> = 1/sqrt(2) [|+ -> - |- +>]

The spin operator with subscript 1 operates only on the first component of a product state, it acts as the identity on the second component. So, e.g because for one spin we have:

S^{z} |+> = hbar/2 |+>

S^{z} |->= - hbar/2 |->

we have:

S^{z}_1 |+ - >= hbar/2 |+ ->

and:

S^{z}_2 |+ - > = - hbar/2 |+ ->

We can simplify things a bit by ...

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