Question

(a) [10 points] Check the Heisenberg-Robertson generalized uncertainty relation $\langle(\Delta A)^2\rangle \langle(\Delta B)^2\rangle \ge \frac{1}{4}|\langle[A, B]\rangle|^2$ (1) with $A \to S_x$, $B \to S_y$, and the expectation values taken for the $|S_z+\rangle$ state. (b) [10 points] Check (1) with $A \to S_z$, $B \to S_y$, and the expectation values taken for the $|S_z+\rangle$ state. (c) [10 points] Find the linear combination of $|+\rangle$ and $|-\rangle$ kets that maximizes the uncertainty product $\langle(\Delta S_x)^2\rangle \langle(\Delta S_y)^2\rangle$. Verify that the uncertainty relation for $S_x$ and $S_y$ is satisfied for this combination.

          (a) [10 points] Check the Heisenberg-Robertson generalized uncertainty relation
$\langle(\Delta A)^2\rangle \langle(\Delta B)^2\rangle \ge \frac{1}{4}|\langle[A, B]\rangle|^2$
(1)
with $A \to S_x$, $B \to S_y$, and the expectation values taken for the $|S_z+\rangle$ state.
(b) [10 points] Check (1) with $A \to S_z$, $B \to S_y$, and the expectation values taken
for the $|S_z+\rangle$ state.
(c) [10 points] Find the linear combination of $|+\rangle$ and $|-\rangle$ kets that maximizes the
uncertainty product $\langle(\Delta S_x)^2\rangle \langle(\Delta S_y)^2\rangle$. Verify that the uncertainty relation for
$S_x$ and $S_y$ is satisfied for this combination.

(a) [10 points] Check the Heisenberg-Robertson generalized uncertainty relation
⟨(Δ A)^2⟩⟨(Δ B)^2⟩≥(1)/(4)|⟨[A, B]⟩|^2
(1)
with A → Sx, B → Sy, and the expectation values taken for the |Sz+⟩ state.
(b) [10 points] Check (1) with A → Sz, B → Sy, and the expectation values taken
for the |Sz+⟩ state.
(c) [10 points] Find the linear combination of |+⟩ and |-⟩ kets that maximizes the
uncertainty product ⟨(Δ Sx)^2⟩⟨(Δ Sy)^2⟩. Verify that the uncertainty relation for
Sx and Sy is satisfied for this combination.

Added by Anthony H.

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