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 $\vec{f}$ (which might be ve­loc­ity $\vec{v}$, lin­ear mo­men­tum ${\skew0\vec p}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $m\vec{v}$, force $\vec{F}$, or what­ever) is usu­ally shown in physics in the form of an ar­row:

Fig­ure 2.1: The clas­si­cal pic­ture of a vec­tor.

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

Fig­ure 2.2: Spike di­a­gram of a vec­tor.

(The sym­bol $i$ for the com­po­nent in­dex is not to be con­fused with ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$.)

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:

Fig­ure 2.3: More di­men­sions.

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 $i$ can be rescaled into a con­tin­u­ous co­or­di­nate, call it $x$. For ex­am­ple, $x$ might be de­fined as $i$ di­vided by the num­ber of di­men­sions. In any case, the spike di­a­gram be­comes a func­tion $f$ of a con­tin­u­ous co­or­di­nate $x$:

Fig­ure 2.4: In­fi­nite di­men­sions.

For func­tions, the spikes are usu­ally not shown:

Fig­ure 2.5: The clas­si­cal pic­ture of a func­tion.

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

Note that the $(x)$ in $f(x)$ does not mean “mul­ti­ply by $x$.” Here the $(x)$ is only a way of re­mind­ing you that $f$ is not a sim­ple num­ber but a func­tion. Just like the ar­row in $\vec{f}$ is only a way of re­mind­ing you that that $f$ 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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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

$\begin{picture}(15,5.5)(0,-3) \put(2,0){\makebox(0,0){\scriptsize\bf0}} \put(12... ...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}} \end{picture}$

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 $\vec{v}$ with com­po­nents (0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) with the plot of the func­tion $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.5 $x$.

So­lu­tion funcvec-a

2.

Graph­i­cally com­pare the spike di­a­gram of the 10-di­men­sion­al unit vec­tor ${\hat\imath}_3$, with com­po­nents (0,0,1,0,0,0,0,0,0,0), with the plot of the func­tion $f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1. (No, they do not look alike.)

So­lu­tion funcvec-b