##### 5.5.4.2 So­lu­tion com­plexsb-b

Ques­tion:

A more con­cise way of un­der­stand­ing the or­tho­nor­mal­ity of the two-par­ti­cle spin states is to note that an in­ner prod­uct like equals , where the first in­ner prod­uct refers to the spin states of par­ti­cle 1 and the sec­ond to those of par­ti­cle 2. The first in­ner prod­uct is zero be­cause of the or­thog­o­nal­ity of and , mak­ing zero too.

To check this ar­gu­ment, write out the sums over and for and ver­ify that it is in­deed the same as the writ­ten out sum for given in the an­swer for the pre­vi­ous ques­tion.

The un­der­ly­ing math­e­mat­i­cal prin­ci­ple is that sums of prod­ucts can be fac­tored into sep­a­rate sums as in:

This is sim­i­lar to the ob­ser­va­tion in cal­cu­lus that in­te­grals of prod­ucts can be fac­tored into sep­a­rate in­te­grals:

An­swer:

and writ­ten out

and mul­ti­ply­ing out, and re­order­ing the sec­ond and third fac­tor in each term, you see it is the same as the ex­pres­sion ob­tained in the an­swer to the pre­vi­ous ques­tion,