2. Math­e­mat­i­cal Pre­req­ui­sites


Quan­tum me­chan­ics is based on a num­ber of ad­vanced math­e­mat­i­cal ideas that are de­scribed in this chap­ter.

First the nor­mal real num­bers will be gen­er­al­ized to com­plex num­bers. A num­ber such as ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$ is an in­valid real num­ber, but it is con­sid­ered to be a valid com­plex one. The math­e­mat­ics of quan­tum me­chan­ics is most eas­ily de­scribed in terms of com­plex num­bers.

Clas­si­cal physics tends to deal with num­bers such as the po­si­tion, ve­loc­ity, and ac­cel­er­a­tion of par­ti­cles. How­ever, quan­tum me­chan­ics deals pri­mar­ily with func­tions rather than with num­bers. To fa­cil­i­tate ma­nip­u­lat­ing func­tions, they will be mod­eled as vec­tors in in­fi­nitely many di­men­sions. Dot prod­ucts, lengths, and or­thog­o­nal­ity can then all be used to ma­nip­u­late func­tions. Dot prod­ucts will how­ever be re­named to be “in­ner prod­ucts” and lengths to be norms.

Op­er­a­tors will be de­fined that turn func­tions into other func­tions. Par­tic­u­larly im­por­tant for quan­tum me­chan­ics are eigen­value cases, in which an op­er­a­tor turns a func­tion into a sim­ple mul­ti­ple of it­self.

A spe­cial class of op­er­a­tors, Her­mit­ian op­er­a­tors will be de­fined. These will even­tu­ally turn out to be very im­por­tant, be­cause quan­tum me­chan­ics as­so­ciates phys­i­cal quan­ti­ties like po­si­tion, mo­men­tum, and en­ergy with cor­re­spond­ing Her­mit­ian op­er­a­tors and their eigen­val­ues.