2.4 Op­er­a­tors

This sec­tion de­fines op­er­a­tors, which are a gen­er­al­iza­tion of ma­tri­ces. Op­er­a­tors are the prin­ci­pal com­po­nents of quan­tum me­chan­ics.

In a fi­nite num­ber of di­men­sions, a ma­trix A can trans­form any ar­bi­trary vec­tor into a dif­fer­ent vec­tor :

Sim­i­larly, an op­er­a­tor trans­forms a func­tion into an­other func­tion:

Some sim­ple ex­am­ples of op­er­a­tors:

Note that a hat is of­ten used to in­di­cate op­er­a­tors; for ex­am­ple, is the sym­bol for the op­er­a­tor that cor­re­sponds to mul­ti­ply­ing by . If it is clear that some­thing is an op­er­a­tor, such as , no hat will be used.

It should re­ally be noted that the op­er­a­tors that you are in­ter­ested in in quan­tum me­chan­ics are lin­ear op­er­a­tors. If you in­crease a func­tion by a fac­tor, in­creases by that same fac­tor. Also, for any two func­tions and , will be + . For ex­am­ple, dif­fer­en­ti­a­tion is a lin­ear op­er­a­tor:

Squar­ing is not a lin­ear op­er­a­tor:

How­ever, it is not some­thing to re­ally worry about. You will not find a sin­gle non­lin­ear op­er­a­tor in the rest of this en­tire book.

Key Points
Ma­tri­ces turn vec­tors into other vec­tors.

Op­er­a­tors turn func­tions into other func­tions.

2.4 Re­view Ques­tions
1.

So what is the re­sult if the op­er­a­tor is ap­plied to the func­tion ?

2.

If, say, is sim­ply the func­tion , then what is the dif­fer­ence be­tween and ?

3.

A less self-ev­i­dent op­er­a­tor than the above ex­am­ples is a trans­la­tion op­er­a­tor like that trans­lates the graph of a func­tion to­wards the left by an amount ​2: . (Cu­ri­ously enough, trans­la­tion op­er­a­tors turn out to be re­spon­si­ble for the law of con­ser­va­tion of mo­men­tum.) Show that turns into .

4.

The in­ver­sion, or par­ity, op­er­a­tor turns into . (It plays a part in the ques­tion to what ex­tent physics looks the same when seen in the mir­ror.) Show that leaves un­changed, but turns into .