### 2.2 Func­tions as Vec­tors

The sec­ond math­e­mat­i­cal idea that is cru­cial for quan­tum me­chan­ics is that func­tions can be treated in a way that is fun­da­men­tally not that much dif­fer­ent from vec­tors.

A vec­tor (which might be ve­loc­ity , lin­ear mo­men­tum , force , or what­ever) is usu­ally shown in physics in the form of an ar­row:

How­ever, the same vec­tor may in­stead be rep­re­sented as a spike di­a­gram, by plot­ting the value of the com­po­nents ver­sus the com­po­nent in­dex:

(The sym­bol for the com­po­nent in­dex is not to be con­fused with .)

In the same way as in two di­men­sions, a vec­tor in three di­men­sions, or, for that mat­ter, in thirty di­men­sions, can be rep­re­sented by a spike di­a­gram:

And just like vec­tors can be in­ter­preted as spike di­a­grams, spike di­a­grams can be in­ter­preted as vec­tors. So a spike di­a­gram with very many spikes can be con­sid­ered to be a sin­gle vec­tor in a space with a very high num­ber of di­men­sions.

In the limit of in­fi­nitely many spikes, the large val­ues of can be rescaled into a con­tin­u­ous co­or­di­nate, call it . For ex­am­ple, might be de­fined as di­vided by the num­ber of di­men­sions. In any case, the spike di­a­gram be­comes a func­tion of a con­tin­u­ous co­or­di­nate :

For func­tions, the spikes are usu­ally not shown:

In this way, a func­tion is just a sin­gle vec­tor in an in­fi­nite di­men­sional space.

Note that the in does not mean “mul­ti­ply by .” Here the is only a way of re­mind­ing you that is not a sim­ple num­ber but a func­tion. Just like the ar­row in is only a way of re­mind­ing you that that is not a sim­ple num­ber but a vec­tor.

(It should be noted that to make the tran­si­tion to in­fi­nite di­men­sions math­e­mat­i­cally mean­ing­ful, you need to im­pose some smooth­ness con­straints on the func­tion. Typ­i­cally, it is re­quired that the func­tion is con­tin­u­ous, or at least in­te­grable in some sense. These de­tails are not im­por­tant for this book.)

Key Points
A func­tion can be thought of as a vec­tor with in­fi­nitely many com­po­nents.

This al­lows quan­tum me­chan­ics do the same things with func­tions as you can do with vec­tors.

2.2 Re­view Ques­tions
1.

Graph­i­cally com­pare the spike di­a­gram of the 10-di­men­sion­al vec­tor with com­po­nents (0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) with the plot of the func­tion 0.5 .

2.

Graph­i­cally com­pare the spike di­a­gram of the 10-di­men­sion­al unit vec­tor , with com­po­nents (0,0,1,0,0,0,0,0,0,0), with the plot of the func­tion 1. (No, they do not look alike.)