Solution
= Solution
The <Hamiltonian> of the <quantum harmonic oscillator> is the sum of two positive quadratic forms. For a nonzero <eigenfunction> in its domain,
$$
E\|\phi\|^2=\langle\phi,\hat H\phi\rangle
=\frac{\|\hat p\phi\|^2}{2m}+\frac{m\omega^2}{2}\|\hat x\phi\|^2\ge0.
$$
Thus $\boxed{E\ge0}$. Part (d) supplies the stronger bound $E\ge\hbar\omega/2$.