Question

Consider yet another entangled state of two spin-1/2 particles, the so-called singlet state\\ $|x\rangle = \frac{1}{\sqrt{2}}(|+\rangle|-\rangle - |-\rangle|+\rangle)$\\(note the minus sign).\\(a) What is the z-component of total spin in this entangled state, i.e. what is the eigenvalue of $S_z = \hat{S}_{1z} + \hat{S}_{2z}$ in this state?\\ Hint: Apply the technique used in problem 1.\\(b) Prove that the singlet state is rotationally invariant, i.e. that it has the same mathematical form regardless of the coordinate system used. In particular, show that $|x\rangle$ has the same form whether it is expressed in terms of eigenstates of $\hat{S}_z$ or $\hat{S}_x$. Proceed by expressing $|\pm\rangle$ in terms of $|\pm\rangle_x$, where $|\pm\rangle_x$ denote spin up and down states along the x-direction, then substituting for $|\pm\rangle$ in the expression for $|x\rangle$ and simplifying.\\ Hint: Does the overall sign of a state (more generally an overall phase factor) have a physical significance, e.g. does it affect calculations of probabilities?

          Consider yet another entangled state of two spin-1/2 particles, the so-called singlet state\\
$|x\rangle = \frac{1}{\sqrt{2}}(|+\rangle|-\rangle - |-\rangle|+\rangle)$\\(note the minus sign).\\(a) What is the z-component of total spin in this entangled state, i.e. what is the eigenvalue of $S_z = \hat{S}_{1z} + \hat{S}_{2z}$ in this state?\\
Hint: Apply the technique used in problem 1.\\(b) Prove that the singlet state is rotationally invariant, i.e. that it has the same mathematical form regardless of the coordinate system used. In particular, show that $|x\rangle$ has the same form whether it is expressed in terms of eigenstates of $\hat{S}_z$ or $\hat{S}_x$. Proceed by expressing $|\pm\rangle$ in terms of $|\pm\rangle_x$, where $|\pm\rangle_x$ denote spin up and down states along the x-direction, then substituting for $|\pm\rangle$ in the expression for $|x\rangle$ and simplifying.\\
Hint: Does the overall sign of a state (more generally an overall phase factor) have a physical significance, e.g. does it affect calculations of probabilities?

Consider yet another entangled state of two spin-1/2 particles, the so-called singlet state

|x⟩ = (1)/(√(2))(|+⟩|-⟩ - |-⟩|+⟩)
(note the minus sign).
(a) What is the z-component of total spin in this entangled state, i.e. what is the eigenvalue of Sz = Ŝ1z + Ŝ2z in this state?

Hint: Apply the technique used in problem 1.
(b) Prove that the singlet state is rotationally invariant, i.e. that it has the same mathematical form regardless of the coordinate system used. In particular, show that |x⟩ has the same form whether it is expressed in terms of eigenstates of Ŝz or Ŝx. Proceed by expressing |±⟩ in terms of |±, where |± denote spin up and down states along the x-direction, then substituting for |±⟩ in the expression for |x⟩ and simplifying.

Hint: Does the overall sign of a state (more generally an overall phase factor) have a physical significance, e.g. does it affect calculations of probabilities?

Added by Charles S.

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