The quantum mechanical harmonic oscillator has Hamiltonian

$$H=\frac{1}{2 m} p^{2}+\frac{1}{2} m \omega^{2} x^{2}$$

and is in a stationary state of energy $<H>=E$. Show that

$$E \geqslant \frac{1}{2 m}(\Delta p)^{2}+\frac{1}{2} m \omega^{2}(\Delta x)^{2},$$

where $(\Delta p)^{2}=\left\langle p^{2}\right\rangle-\langle p\rangle^{2}$ and $(\Delta x)^{2}=\left\langle x^{2}\right\rangle-\langle x\rangle^{2}$. Use the Heisenberg Uncertainty Principle to show that

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