What is the physical significance of the expectation value

$$\langle Q\rangle=\int \psi^{*}(x) Q \psi(x) d x$$

of an observable $Q$ in the normalised state $\psi(x)$ ? Let $P$ and $Q$ be two observables. By considering the norm of $(Q+i \lambda P) \psi$ for real values of $\lambda$, show that

$$\left\langle Q^{2}\right\rangle\left\langle P^{2}\right\rangle \geqslant \frac{1}{4}|\langle[Q, P]\rangle|^{2} .$$

Deduce the generalised uncertainty relation

$$\Delta Q \Delta P \geqslant \frac{1}{2}|\langle[Q, P]\rangle|,$$

where the uncertainty $\Delta Q$ in the state $\psi(x)$ is defined by

$$(\Delta Q)^{2}=\left\langle(Q-\langle Q\rangle)^{2}\right\rangle$$

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

$$E \geqslant \frac{1}{2} \hbar \omega$$