Give the physical interpretation of the expression

$$\langle A\rangle_{\psi}=\int \psi(x)^{*} \hat{A} \psi(x) d x$$

for an observable $A$, where $\hat{A}$ is a Hermitian operator and $\psi$ is normalised. By considering the norm of the state $(A+i \lambda B) \psi$ for two observables $A$ and $B$, and real values of $\lambda$, show that

$$\left\langle A^{2}\right\rangle_{\psi}\left\langle B^{2}\right\rangle_{\psi} \geqslant \frac{1}{4}\left|\langle[A, B]\rangle_{\psi}\right|^{2} .$$

Deduce the uncertainty relation

$$\Delta A \Delta B \geqslant \frac{1}{2}\left|\langle[A, B]\rangle_{\psi}\right|,$$

where $\Delta A$ is the uncertainty of $A$.

A particle of mass $m$ moves in one dimension under the influence of potential $\frac{1}{2} m \omega^{2} x^{2}$. By considering the commutator $[x, p]$, show that the expectation value of the Hamiltonian satisfies

$$\langle H\rangle_{\psi} \geqslant \frac{1}{2} \hbar \omega .$$