Question

Another hermitian operators are defined as $\delta_A \equiv A - \bar{A}$ and $\delta_B \equiv B - \bar{B}$, where $\bar{A} = \langle A \rangle$. (a) Show that $\langle \delta_A^2 \rangle_\phi \langle \delta_B^2 \rangle_\phi \ge |\langle \delta_A \phi | \delta_B \phi \rangle|^2$ (b) Show that $|\langle \delta_A \phi | \delta_B \phi \rangle|^2 = \frac{1}{2}(\langle \delta_A \delta_B \rangle_\phi + \langle \delta_B \delta_A \rangle_\phi) + \frac{1}{2}|[\delta_A, \delta_B]|^2 \ge \frac{1}{2}|[\delta_A, \delta_B]|^2$ (c) Show that $[\delta_A, \delta_B] = [A, B] = iC$ (d) So far you have shown that $(\Delta A)^2 (\Delta B)^2 \ge \frac{1}{2}|\langle C \rangle|^2$. Now insert $x \to A$ and $p \to B$, then confirm yourself $\Delta x \Delta p \ge \frac{h}{2}$

          Another hermitian operators are defined as $\delta_A \equiv A - \bar{A}$ and $\delta_B \equiv B - \bar{B}$, where $\bar{A} = \langle A \rangle$.
(a) Show that $\langle \delta_A^2 \rangle_\phi \langle \delta_B^2 \rangle_\phi \ge |\langle \delta_A \phi | \delta_B \phi \rangle|^2$
(b) Show that $|\langle \delta_A \phi | \delta_B \phi \rangle|^2 = \frac{1}{2}(\langle \delta_A \delta_B \rangle_\phi + \langle \delta_B \delta_A \rangle_\phi) + \frac{1}{2}|[\delta_A, \delta_B]|^2 \ge \frac{1}{2}|[\delta_A, \delta_B]|^2$
(c) Show that $[\delta_A, \delta_B] = [A, B] = iC$
(d) So far you have shown that $(\Delta A)^2 (\Delta B)^2 \ge \frac{1}{2}|\langle C \rangle|^2$. Now insert $x \to A$ and $p \to B$,
then confirm yourself $\Delta x \Delta p \ge \frac{h}{2}$

Another hermitian operators are defined as ≡ A - A̅ and ≡ B - B̅, where A̅ = ⟨ A ⟩.
(a) Show that ⟨^2 ⟩ϕ⟨^2 ⟩ϕ≥ |⟨ϕ | ϕ⟩|^2
(b) Show that |⟨ϕ | ϕ⟩|^2 = (1)/(2)(⟨⟩ϕ + ⟨⟩ϕ) + (1)/(2)|[, ]|^2 ≥(1)/(2)|[, ]|^2
(c) Show that [, ] = [A, B] = iC
(d) So far you have shown that (Δ A)^2 (Δ B)^2 ≥(1)/(2)|⟨ C ⟩|^2. Now insert x → A and p → B,
then confirm yourself Δ x Δ p ≥(h)/(2)

Added by Laura T.

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