In many textbooks, it is said that because operators in QM are hermitian, we can write:
$$\int \psi^*\hat{A}\phi\,\mathrm d\tau = \int\phi^*\hat{A}\psi\,\mathrm d\tau$$
An operator $\hat{A}$ is called hermitian iff $\hat{A}^\dagger = \hat{A}$. Using this definition, I tried to prove the above property but I can't. I started from:
$$\langle\psi|\hat{A}|\phi\rangle = {\langle\phi|\hat{A}|\psi\rangle}^\dagger$$
based on the fact that the operator is hermitian. But then I don't know how to continue. Any ideas?
I have found the first relation in the following textbook: Quantum Chemistry and Molecular Interactions, in page 81.
For any two wavefunctions $\psi_i$ and $\psi_j$ $$\int \psi_i^*\hat{H}\psi_j\,\mathrm d\tau = \int\psi_j^*\hat{H}\psi_i\,\mathrm d\tau$$