Question about Eq. 10.9 in Ashcorft and Mermin

Question

I have a doubt concerning the assumptions made in deriving Eq. 10.9 in Ashcroft and Mermin's Solid State Physics text. We have two entities in the equation: $\psi_m (\textbf{r})$ and $\psi(\textbf{r})$ referring to a localized and bloch wavefunction, respectively. The first equality amounts to interchanging the atomic Hamiltonian operator on the bloch and atomic wavefuctions. The relevant equality is reproduced below: $$ \int \psi_m^* (\textbf{r}) H_{at} \psi (\textbf{r})d\textbf{r} = \int(H_{at}\psi_m(\textbf{r}))^*\psi(\textbf{r})d\textbf{r}. \tag{10.9} $$ Presumably the matrix elements of the Hamiltonian operator is equal when $\psi_m (\textbf{r})$ and $\psi(\textbf{r})$ are interchanged. But how can it be justified. I am new to this forum so I apologize if this question is too naive.

Answer 1

The Hamiltonian operator is Hermitian, so $H_{at} = H_{at}^{\dagger}$. So looking at the relevant part inside the integral, we have:

$$(RHS) \quad (H_{at}\psi_m(\vec{r}))^{*} = \psi_m^*(\vec{r})H_{at}^{\dagger} = \psi_m^*(\vec{r})H_{at} \quad (LHS)$$

The first equal sign is because if you think in terms of matrices, $(AB)^\dagger=B^\dagger A^\dagger$.