Solution
= Solution
Apply the <adjoint operator> identity of part (c) with $\chi=\hat A\phi$ and use part (a):
$$
\|\hat A\phi\|^2=\langle\phi,\hat B\hat A\phi\rangle
=2m(E-\tfrac12\hbar\omega)\|\phi\|^2.
$$
For a nonzero <eigenfunction>, positivity of the <norm> in particular forces $E\ge\hbar\omega/2$. Hence \b[the requested norm is]
$$
\boxed{\|\hat A\phi\|=\sqrt{2m(E-\tfrac12\hbar\omega)}\,\|\phi\|.}
$$
It vanishes exactly at the ground-state energy.