Hermiticity of Second Quantized Operators

As is well known, in the quantum mechanics physical observables correspond to Hermitian operators, that is  

It follows simply from Eq. (4.41) that the matrix elements of such an operator form a Hermitian matrix  

Accordingly, the list of one-electron integrals of the Hamiltonian h v of Eq. (4.40) form a symmetric matrix. 

The symmetry of the two-electron integral list is two-fold. On the one hand, it is based on the fact that renumbering electrons 1 and 2 in Eq. (4.31) does not change the value of the integral. That is  

On the other hand, it is based on the Hermiticity of the two-electron operator l/ri2, which, in the case of real orbitals, results in the Hermitian symmetry  

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