Sub­sec­tions

This sub­sec­tion de­scribes a few fur­ther is­sues of im­por­tance for this book.

#### 2.7.1 Dirac no­ta­tion

Physi­cists like to write in­ner prod­ucts such as in Dirac no­ta­tion:

since this con­forms more closely to how you would think of it in lin­ear al­ge­bra:

The var­i­ous ad­vanced ideas of lin­ear al­ge­bra can be ex­tended to op­er­a­tors in this way, but they will not be needed in this book.

One thing will be needed in some more ad­vanced ad­denda, how­ever. That is the case that op­er­a­tor is not Her­mit­ian. In that case, if you want to take to the other side of the in­ner prod­uct, you need to change it into a dif­fer­ent op­er­a­tor. That op­er­a­tor is called the “Her­mit­ian con­ju­gate” of . In physics, it is al­most al­ways in­di­cated as . So, sim­ply by de­f­i­n­i­tion,

Then there are some more things that this book will not use. How­ever, you will al­most surely en­counter these when you read other books on quan­tum me­chan­ics.

First, the dag­ger is used much like a gen­er­al­iza­tion of com­plex con­ju­gate,

etcetera. Ap­ply­ing a dag­ger a sec­ond time gives the orig­i­nal back. Also, if you work out the dag­ger on a prod­uct, you need to re­verse the or­der of the fac­tors. For ex­am­ple

In words, putting into the left side of an in­ner prod­uct gives .

The sec­ond point will be il­lus­trated for the case of vec­tors in three di­men­sions. Such a vec­tor can be writ­ten as

Here , , and are the three unit vec­tors in the ax­ial di­rec­tions. The com­po­nents , and can be found us­ing dot prod­ucts:

Sym­bol­i­cally, you can write this as

In fact, the op­er­a­tor in paren­the­ses can be de­fined by say­ing that for any vec­tor , it gives the ex­act same vec­tor back. Such an op­er­a­tor is called an “iden­tity op­er­a­tor.”

The re­la­tion

is called the “com­plete­ness re­la­tion.” To see why, sup­pose you leave off the third part of the op­er­a­tor. Then

The -​com­po­nent is gone! Now the vec­tor gets pro­jected onto the -plane. The op­er­a­tor has be­come a “pro­jec­tion op­er­a­tor” in­stead of an iden­tity op­er­a­tor by not sum­ing over the com­plete set of unit vec­tors.

You will al­most al­ways find these things in terms of bras and kets. To see how that looks, de­fine

Then

so the com­plete­ness re­la­tion looks like

If you do not sum over the com­plete set of kets, you get a pro­jec­tion op­er­a­tor in­stead of an iden­tity one.

In many cases, the func­tions in­volved in an in­ner prod­uct may de­pend on more than a sin­gle vari­able . For ex­am­ple, they might de­pend on the po­si­tion in three-di­men­sion­al space.

The rule to deal with that is to en­sure that the in­ner prod­uct in­te­gra­tions are over all in­de­pen­dent vari­ables. For ex­am­ple, in three spa­tial di­men­sions:

Note that the time is a some­what dif­fer­ent vari­able from the rest, and time is not in­cluded in the in­ner prod­uct in­te­gra­tions.