= Solution
With the <inner product> $\langle\chi,\psi\rangle=\int\chi^*\psi\,dx$, the assumed Hermiticity of $\hat x$ and $\hat p$ gives
$$
\hat A^\dagger=\hat p+im\omega\hat x=\hat B.
$$
Consequently \b[the adjoint relation is]
$$
\boxed{\int_{-\infty}^{\infty}\chi^*(\hat A\psi)\,dx
=\langle\chi,\hat A\psi\rangle
=\langle\hat B\chi,\psi\rangle
=\int_{-\infty}^{\infty}(\hat B\chi)^*\psi\,dx.}
$$
As with the stated Hermiticity assumption, these identities concern wavefunctions in the appropriate operator domains. Square integrability alone does not guarantee that derivatives or multiplication by $x$ are square integrable; one may first work with smooth rapidly decaying wavefunctions, where integration by parts has no boundary term, and then use the operator domains.
Back to article page
