Question

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.

          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.

1. (5 points) Spin-0 entangled state
In class, we often use the entangled two-particle state
| ψ⟩ = (1)/(√(2))( | + ⟩1 | - ⟩2 - | - ⟩1 | + ⟩2 )
where we leave out the ⊗ between the particle-1 and particle-2 parts of the state.
(a) Show that | ψ⟩ is properly normalized.
(b) Is | ψ⟩ an eigenstate of Sz,1 (the operator associated with measuring the z-component
of the spin of particle-1 only)? If so, what is the eigenvalue?
(c) Is | ψ⟩ an eigenstate of the total spin operator Sz,1 + Sz,2? If so, what is the
eigenvalue?
(d) Show that | ψ⟩ is mathematically identical to the more general state
(1)/(√(2))( | + ⟩n̂,1| - ⟩n̂,2 - | - ⟩n̂,1| + ⟩n̂,2)
where n̂ is an arbitrary-direction unit vector first defined in Ch. 2. (but note that
it's the same n̂ for both particles). Hint: try writing the n̂-basis states back in
terms of the z-basis.

Added by Miguel -Ngel J.

Question

Please give Ace some feedback