1. (5 points) Spin-0 entangled state
In class, we often use the entangled two-particle state
$\left| \psi \right\rangle = \frac{1}{\sqrt{2}} \left( \left| + \right\rangle_1 \left| - \right\rangle_2 - \left| - \right\rangle_1 \left| + \right\rangle_2 \right)$
where we leave out the $\otimes$ between the particle-1 and particle-2 parts of the state.
(a) Show that $\left| \psi \right\rangle$ is properly normalized.
(b) Is $\left| \psi \right\rangle$ an eigenstate of $S_{z,1}$ (the operator associated with measuring the $z$-component
of the spin of particle-1 only)? If so, what is the eigenvalue?
(c) Is $\left| \psi \right\rangle$ an eigenstate of the total spin operator $S_{z,1} + S_{z,2}$? If so, what is the
eigenvalue?
(d) Show that $\left| \psi \right\rangle$ is mathematically identical to the more general state
$\frac{1}{\sqrt{2}} \left( \left| + \right\rangle_{\hat{n},1} \left| - \right\rangle_{\hat{n},2} - \left| - \right\rangle_{\hat{n},1} \left| + \right\rangle_{\hat{n},2} \right)$
where $\hat{n}$ is an arbitrary-direction unit vector first defined in Ch. 2. (but note that
it's the same $\hat{n}$ for both particles). Hint: try writing the $\hat{n}$-basis states back in
terms of the $z$-basis.
