### D.55 Why the Fock op­er­a­tor is Her­mit­ian

To ver­ify that the Fock op­er­a­tor is Her­mit­ian, first note that is Her­mit­ian since it is an Hamil­ton­ian. Next if you form the in­ner prod­uct , the first term in , the Coulomb term, can be taken to the other side since it is just a real func­tion. The sec­ond term, the ex­change one, pro­duces the in­ner prod­uct,

and if you take the op­er­a­tor to the other side, you get

and writ­ing out these in­ner prod­ucts as six-di­men­sion­al spa­tial in­te­grals and sums over spin, you see that they are the same.