### D.24 Hy­dro­gen mol­e­cule ground state and spin

The pur­pose of this note is to ver­ify that the in­clu­sion of spin does not change the spa­tial form of the ground state of the hy­dro­gen mol­e­cule. The low­est ex­pec­ta­tion en­ergy , char­ac­ter­iz­ing the cor­rect ground state, only oc­curs if all spa­tial com­po­nents of the ground state with spin,

are pro­por­tional to the no-spin spa­tial ground state .

The rea­son is that the as­sumed Hamil­ton­ian (5.3) does not in­volve spin at all, only spa­tial co­or­di­nates, so, for ex­am­ple,

and the same for the other three terms in . So the ex­pec­ta­tion value of en­ergy be­comes

Be­cause of the or­tho­nor­mal­ity of the spin states, this mul­ti­plies out into in­ner prod­ucts of match­ing spin states as

In ad­di­tion, the wave func­tion must be nor­mal­ized, 1, or

Now when , , , and are each pro­por­tional to the no-spin spa­tial ground state with the low­est en­ergy , their in­di­vid­ual con­tri­bu­tions to the en­ergy will be given by , the low­est pos­si­ble. Then the to­tal en­ergy will be . Any­thing else will have more en­ergy and can there­fore not be the ground state.

It should be pointed out that to a more ac­cu­rate ap­prox­i­ma­tion, spin causes the elec­trons to be some­what mag­netic, and that pro­duces a slight de­pen­dence of the en­ergy on spin; com­pare ad­den­dum {A.39}. This note ig­nored that, as do most other de­riva­tions in this book.