Solution
= Solution
For a free particle, the <Hamiltonian operator> is
$$
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}.
$$
Since $\chi''=-k^2\chi$,
$$
H\chi=\frac{\hbar^2k^2}{2m}\chi.
$$
Thus
$$
\boxed{E=\frac{\hbar^2k^2}{2m}\geq0,
\qquad p=\hbar k},
$$
where the momentum follows because $-i\hbar\,d\chi/dx=\hbar k\chi$.