Question

Operator on two spin-\frac{1}{2} particles Consider the following operator acting on a system of two spin-\frac{1}{2} particles: $\hat{A} = 3(\hat{\vec{s}}_1 \cdot \vec{n})(\hat{\vec{s}}_2 \cdot \vec{n}) - \hat{\vec{s}}_1 \cdot \hat{\vec{s}}_2$ where $\vec{n}$ is the unit vector connecting the two particles (pointing, say, from particle 1 to particle 2). Suppose that the particles are in the state $|\psi\rangle = \frac{1}{\sqrt{2}}(|+z, -z\rangle - |-z, +z\rangle)$, Known as the singlet state, what is the expectation value $\langle \hat{A} \rangle$? Hint: You may want to consider the fact that the singlet state has total spin $S=0$ and total spin projection $M_z=0$

          Operator on two spin-\frac{1}{2} particles
Consider the following operator acting on a system of two spin-\frac{1}{2} particles:
$\hat{A} = 3(\hat{\vec{s}}_1 \cdot \vec{n})(\hat{\vec{s}}_2 \cdot \vec{n}) - \hat{\vec{s}}_1 \cdot \hat{\vec{s}}_2$
where $\vec{n}$ is the unit vector connecting the two particles (pointing, say, from particle 1 to particle 2).
Suppose that the particles are in the state
$|\psi\rangle = \frac{1}{\sqrt{2}}(|+z, -z\rangle - |-z, +z\rangle)$,
Known as the singlet state, what is the expectation value $\langle \hat{A} \rangle$?
Hint: You may want to consider the fact that the singlet state has total spin $S=0$ and total spin projection $M_z=0$

Operator on two spin-(1)/(2) particles
Consider the following operator acting on a system of two spin-(1)/(2) particles:
Â = 3(ŝ⃗̂1 ·n⃗)(ŝ⃗̂2 ·n⃗) - ŝ⃗̂1 ·ŝ⃗̂2
where n⃗ is the unit vector connecting the two particles (pointing, say, from particle 1 to particle 2).
Suppose that the particles are in the state
|ψ⟩ = (1)/(√(2))(|+z, -z⟩ - |-z, +z⟩),
Known as the singlet state, what is the expectation value ⟨Â⟩?
Hint: You may want to consider the fact that the singlet state has total spin S=0 and total spin projection Mz=0

Added by Anthony R.

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