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 $v$ into a dif­fer­ent vec­tor $A\vec{v}$:

$$\vec v \quad \begin{picture}(100,0) \put(50,15){\makebox... ...0,2){\vector(1,0){100}} \end{picture} \quad \vec w = A \vec v$$

Sim­i­larly, an op­er­a­tor trans­forms a func­tion into an­other func­tion:

$$f(x) \quad \begin{picture}(100,10) \put(50,15){\makebox(... ...put(0,2){\vector(1,0){100}} \end{picture} \quad g(x) = A f(x)$$

Some sim­ple ex­am­ples of op­er­a­tors:

$$f(x) \quad \begin{picture}(100,10) \put(50,13){\makebox(... ...put(0,2){\vector(1,0){100}} \end{picture} \quad g(x) = x f(x)$$

$$f(x) \quad \begin{picture}(100,23) \put(50,21){\makebox(... ...\put(0,2){\vector(1,0){100}} \end{picture} \quad g(x) = f'(x)$$

Note that a hat is of­ten used to in­di­cate op­er­a­tors; for ex­am­ple, ${\widehat x}$ is the sym­bol for the op­er­a­tor that cor­re­sponds to mul­ti­ply­ing by $x$. If it is clear that some­thing is an op­er­a­tor, such as ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$, 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 $f$ by a fac­tor, $Af$ in­creases by that same fac­tor. Also, for any two func­tions $f$ and $g$, $A(f+g)$ will be $(Af)$ + $(Ag)$. For ex­am­ple, dif­fer­en­ti­a­tion is a lin­ear op­er­a­tor:

$$\frac{{\rm d}\Big( c_1 f(x) + c_2 g(x)\Big)}{{\rm d}x} = c... ...frac{{\rm d}f(x)}{{\rm d}x} + c_2 \frac{{\rm d}g(x)}{{\rm d}x}$$

Squar­ing is not a lin­ear op­er­a­tor:

$$\Big(c_1 f(x) + c_2 g(x)\Big)^2 = c_1^2 f^2(x) + 2 c_1 c_2 f(x) g(x) + c_2^2 g^2(x) \ne c_1 f^2(x) + c_2 g^2(x)$$

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

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

Ma­tri­ces turn vec­tors into other vec­tors.

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

Op­er­a­tors turn func­tions into other func­tions.

2.4 Re­view Ques­tions

1.

So what is the re­sult if the op­er­a­tor ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ is ap­plied to the func­tion $\sin(x)$?

So­lu­tion math­ops-a

2.

If, say, $\widehat{x^2}\sin(x)$ is sim­ply the func­tion $x^2\sin(x)$, then what is the dif­fer­ence be­tween $\widehat{x^2}$ and $x^2$?

So­lu­tion math­ops-b

3.

A less self-ev­i­dent op­er­a­tor than the above ex­am­ples is a trans­la­tion op­er­a­tor like ${\cal T}_{\pi /2}$ that trans­lates the graph of a func­tion to­wards the left by an amount $\pi$$\raisebox{.5pt}{$/$}$​2: ${\cal T}_{\pi /2}f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $f\left(x+\frac 12\pi\right)$. (Cu­ri­ously enough, trans­la­tion op­er­a­tors turn out to be re­spon­si­ble for the law of con­ser­va­tion of mo­men­tum.) Show that ${\cal T}_{\pi /2}$ turns $\sin(x)$ into $\cos(x)$.

So­lu­tion math­ops-c

4.

The in­ver­sion, or par­ity, op­er­a­tor ${\mit\Pi}$ turns $f(x)$ into $f(-x)$. (It plays a part in the ques­tion to what ex­tent physics looks the same when seen in the mir­ror.) Show that ${\mit\Pi}$ leaves $\cos(x)$ un­changed, but turns $\sin(x)$ into $\vphantom{0}\raisebox{1.5pt}{$-$}$$\sin(x)$.

So­lu­tion math­ops-d