2.3 The Dot, oops, IN­NER Prod­uct

The dot prod­uct of vec­tors is an im­por­tant tool. It makes it pos­si­ble to find the length of a vec­tor, by mul­ti­ply­ing the vec­tor by it­self and tak­ing the square root. It is also used to check if two vec­tors are or­thog­o­nal: if their dot prod­uct is zero, they are. In this sub­sec­tion, the dot prod­uct is de­fined for com­plex vec­tors and func­tions.

The usual dot prod­uct of two vec­tors and can be found by mul­ti­ply­ing com­po­nents with the same in­dex to­gether and sum­ming that:

(The em­phatic equal, , is com­monly used to in­di­cate “is by de­f­i­n­i­tion equal” or is al­ways equal.) Fig­ure 2.6 shows mul­ti­plied com­po­nents us­ing equal col­ors.

Note the use of nu­meric sub­scripts, , , and rather than , , and ; it means the same thing. Nu­meric sub­scripts al­low the three term sum above to be writ­ten more com­pactly as:

The is called the sum­ma­tion sym­bol.

The length of a vec­tor , in­di­cated by or sim­ply by , is nor­mally com­puted as

How­ever, this does not work cor­rectly for com­plex vec­tors. The dif­fi­culty is that terms of the form are no longer nec­es­sar­ily pos­i­tive num­bers. For ex­am­ple, 1.

There­fore, it is nec­es­sary to use a gen­er­al­ized “in­ner prod­uct” for com­plex vec­tors, which puts a com­plex con­ju­gate on the first vec­tor:

 (2.7)

If the vec­tor is real, the com­plex con­ju­gate does noth­ing, and the in­ner prod­uct is the same as the dot prod­uct . Oth­er­wise, in the in­ner prod­uct and are no longer in­ter­change­able; the con­ju­gates are only on the first fac­tor, . In­ter­chang­ing and changes the in­ner prod­uct’s value into its com­plex con­ju­gate.

The length of a nonzero vec­tor is now al­ways a pos­i­tive num­ber:

 (2.8)

Physi­cists take the in­ner prod­uct bracket ver­bally apart as

and re­fer to vec­tors as bras and kets.

The in­ner prod­uct of func­tions is de­fined in ex­actly the same way as for vec­tors, by mul­ti­ply­ing val­ues at the same -​po­si­tion to­gether and sum­ming. But since there are in­fi­nitely many val­ues, the sum be­comes an in­te­gral:

 (2.9)

Fig­ure 2.7 shows mul­ti­plied func­tion val­ues us­ing equal col­ors:

The equiv­a­lent of the length of a vec­tor is in the case of a func­tion called its “norm:”

 (2.10)

The dou­ble bars are used to avoid con­fu­sion with the ab­solute value of the func­tion.

A vec­tor or func­tion is called “nor­mal­ized” if its length or norm is one:

 (2.11)

(“iff” should re­ally be read as if and only if.)

Two vec­tors, or two func­tions, and , are by de­f­i­n­i­tion or­thog­o­nal if their in­ner prod­uct is zero:

 (2.12)

Sets of vec­tors or func­tions that are all

• mu­tu­ally or­thog­o­nal, and
• nor­mal­ized

oc­cur a lot in quan­tum me­chan­ics. Such sets should be called “or­tho­nor­mal”, though the less pre­cise term or­thog­o­nal is of­ten used in­stead. This doc­u­ment will re­fer to them cor­rectly as be­ing or­tho­nor­mal.

So, a set of func­tions or vec­tors is or­tho­nor­mal if

and

Key Points
For com­plex vec­tors and func­tions, the nor­mal dot prod­uct be­comes the in­ner prod­uct.

To take an in­ner prod­uct of vec­tors,
• take com­plex con­ju­gates of the com­po­nents of the first vec­tor;
• mul­ti­ply cor­re­spond­ing com­po­nents of the two vec­tors to­gether;
• sum these prod­ucts.

To take an in­ner prod­uct of func­tions,
• take the com­plex con­ju­gate of the first func­tion;
• mul­ti­ply the two func­tions;
• in­te­grate the prod­uct func­tion.

To find the length of a vec­tor, take the in­ner prod­uct of the vec­tor with it­self, and then a square root.

To find the norm of a func­tion, take the in­ner prod­uct of the func­tion with it­self, and then a square root.

A pair of vec­tors, or a pair of func­tions, is or­thog­o­nal if their in­ner prod­uct is zero.

A set of vec­tors forms an or­tho­nor­mal set if every one is or­thog­o­nal to all the rest, and every one is of unit length.

A set of func­tions forms an or­tho­nor­mal set if every one is or­thog­o­nal to all the rest, and every one is of unit norm.

2.3 Re­view Ques­tions
1. Find the fol­low­ing in­ner prod­uct of the two vec­tors:

2. Find the length of the vec­tor

3. Find the in­ner prod­uct of the func­tions and on the in­ter­val 0 1.
4. Show that the func­tions and are or­thog­o­nal on the in­ter­val 0 .
5. Ver­ify that is not a nor­mal­ized func­tion on the in­ter­val 0 , and nor­mal­ize it by di­vid­ing by its norm.
6. Ver­ify that the most gen­eral mul­ti­ple of that is nor­mal­ized on the in­ter­val 0 is where is any ar­bi­trary real num­ber. So, us­ing the Euler for­mula, the fol­low­ing mul­ti­ples of are all nor­mal­ized: , (for 0), , (for ), and , (for ​2).
7. Show that the func­tions and are an or­tho­nor­mal set on the in­ter­val 0 1.