N.4 Are Her­mit­ian op­er­a­tors re­ally like that?

A math­e­mati­cian might choose to phrase the prob­lem of Her­mit­ian op­er­a­tors hav­ing or not hav­ing eigen­val­ues and eigen­func­tions in a suit­able space of per­mis­si­ble func­tions and then find, with some jus­ti­fi­ca­tion, that some op­er­a­tors in quan­tum me­chan­ics, like the po­si­tion or mo­men­tum op­er­a­tors do not have any per­mis­si­ble eigen­func­tions. Let alone a com­plete set. The ap­proach of this text is to sim­ply fol­low the for­mal­ism any­way, and then fix the prob­lems that arise as they arise.

More gen­er­ally, what this book tells you about op­er­a­tors is ab­solutely true for sys­tems with a fi­nite num­ber of vari­ables, but gets math­e­mat­i­cally sus­pect for in­fi­nite sys­tems. The func­tional analy­sis re­quired to do bet­ter is well be­yond the scope of this book and the ab­stract math­e­mat­ics a typ­i­cal en­gi­neer would ever want to have a look at.

In any case, when prob­lems are dis­cretized to a fi­nite one for nu­mer­i­cal so­lu­tion, the prob­lem no longer ex­ists. Or rather, it has been re­duced to fig­ur­ing out how the nu­mer­i­cal so­lu­tion ap­proaches the ex­act so­lu­tion in the limit that the prob­lem size be­comes in­fi­nite.