2.6 Her­mit­ian Op­er­a­tors

Most op­er­a­tors in quan­tum me­chan­ics are of a spe­cial kind called Her­mit­ian. This sec­tion lists their most im­por­tant prop­er­ties.

An op­er­a­tor is called Her­mit­ian when it can al­ways be flipped over to the other side if it ap­pears in a in­ner prod­uct:

$$\langle f \vert A g\rangle = \langle Af \vert g\rangle \mbox{ always iff $A$ is Hermitian.}$$ (2.15)

That is the de­f­i­n­i­tion, but Her­mit­ian op­er­a­tors have the fol­low­ing ad­di­tional spe­cial prop­er­ties:

• They al­ways have real eigen­val­ues, not in­volv­ing ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$. (But the eigen­func­tions, or eigen­vec­tors if the op­er­a­tor is a ma­trix, might be com­plex.) Phys­i­cal val­ues such as po­si­tion, mo­men­tum, and en­ergy are or­di­nary real num­bers since they are eigen­val­ues of Her­mit­ian op­er­a­tors {N.3}.
• Their eigen­func­tions can al­ways be cho­sen so that they are nor­mal­ized and mu­tu­ally or­thog­o­nal, in other words, an or­tho­nor­mal set. This tends to sim­plify the var­i­ous math­e­mat­ics a lot.
• Their eigen­func­tions form a com­plete set. This means that any func­tion can be writ­ten as some lin­ear com­bi­na­tion of the eigen­func­tions. (There is a proof in de­riva­tion {D.8} for an im­por­tant ex­am­ple. But see also {N.4}.) In prac­ti­cal terms, it means that you only need to look at the eigen­func­tions to com­pletely un­der­stand what the op­er­a­tor does.

In the lin­ear al­ge­bra of real ma­tri­ces, Her­mit­ian op­er­a­tors are sim­ply sym­met­ric ma­tri­ces. A ba­sic ex­am­ple is the in­er­tia ma­trix of a solid body in New­ton­ian dy­nam­ics. The or­tho­nor­mal eigen­vec­tors of the in­er­tia ma­trix give the di­rec­tions of the prin­ci­pal axes of in­er­tia of the body.

An or­tho­nor­mal com­plete set of eigen­vec­tors or eigen­func­tions is an ex­am­ple of a so-called “ba­sis.” In gen­eral, a ba­sis is a min­i­mal set of vec­tors or func­tions that you can write all other vec­tors or func­tions in terms of. For ex­am­ple, the unit vec­tors ${\hat\imath}$, ${\hat\jmath}$, and ${\hat k}$ are a ba­sis for nor­mal three-di­men­sion­al space. Every three-di­men­sion­al vec­tor can be writ­ten as a lin­ear com­bi­na­tion of the three.

The fol­low­ing prop­er­ties of in­ner prod­ucts in­volv­ing Her­mit­ian op­er­a­tors are of­ten needed, so they are listed here:

$$\mbox{If $A$ is Hermitian: }\quad \langle g \vert A f\ran... ...angle^*, \quad \langle f \vert A f\rangle \mbox{ is real.} %$$ (2.16)

The first says that you can swap $f$ and $g$ if you take the com­plex con­ju­gate. (It is sim­ply a re­flec­tion of the fact that if you change the sides in an in­ner prod­uct, you turn it into its com­plex con­ju­gate. Nor­mally, that puts the op­er­a­tor at the other side, but for a Her­mit­ian op­er­a­tor, it does not make a dif­fer­ence.) The sec­ond is im­por­tant be­cause or­di­nary real num­bers typ­i­cally oc­cupy a spe­cial place in the grand scheme of things. (The fact that the in­ner prod­uct is real merely re­flects the fact that if a num­ber is equal to its com­plex con­ju­gate, it must be real; if there was an ${\rm i}$ in it, the num­ber would change by a com­plex con­ju­gate.)

Key Points

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Her­mit­ian op­er­a­tors can be flipped over to the other side in in­ner prod­ucts.

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Her­mit­ian op­er­a­tors have only real eigen­val­ues.

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Her­mit­ian op­er­a­tors have a com­plete set of or­tho­nor­mal eigen­func­tions (or eigen­vec­tors).

2.6 Re­view Ques­tions

1.

A ma­trix $A$ is de­fined to con­vert any vec­tor ${\skew0\vec r}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $x{\hat\imath}+y{\hat\jmath}$ into ${\skew0\vec r}_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $2x{\hat\imath}+4y{\hat\jmath}$. Ver­ify that ${\hat\imath}$ and ${\hat\jmath}$ are or­tho­nor­mal eigen­vec­tors of this ma­trix, with eigen­val­ues 2, re­spec­tively 4.

So­lu­tion herm-a

2.

A ma­trix $A$ is de­fined to con­vert any vec­tor ${\skew0\vec r}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(x,y)$ into the vec­tor ${\skew0\vec r}_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $(x+y,x+y)$. Ver­ify that $(\cos 45^\circ ,\sin 45^\circ)$ and $(\cos 45^\circ ,-\sin 45^\circ)$ are or­tho­nor­mal eigen­vec­tors of this ma­trix, with eigen­val­ues 2 re­spec­tively 0. Note: $\cos 45^\circ$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sin 45^\circ$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\frac 12\sqrt{2}$.

So­lu­tion herm-b

3.

Show that the op­er­a­tor $\widehat 2$ is a Her­mit­ian op­er­a­tor, but $\widehat{\rm i}$ is not.

So­lu­tion herm-c

4.

Gen­er­al­ize the pre­vi­ous ques­tion, by show­ing that any com­plex con­stant $c$ comes out of the right hand side of an in­ner prod­uct un­changed, but out of the left hand side as its com­plex con­ju­gate;

$$\langle f\vert cg\rangle = c \langle f\vert g\rangle\qquad\langle c f\vert g\rangle = c^* \langle f\vert g\rangle .$$

As a re­sult, a num­ber $c$ is only a Her­mit­ian op­er­a­tor if it is real: if $c$ is com­plex, the two ex­pres­sions above are not the same.

So­lu­tion herm-d

5.

Show that an op­er­a­tor such as ${\widehat{x}}^2$, cor­re­spond­ing to mul­ti­ply­ing by a real func­tion, is an Her­mit­ian op­er­a­tor.

So­lu­tion herm-e

6.

Show that the op­er­a­tor ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ is not a Her­mit­ian op­er­a­tor, but ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ is, as­sum­ing that the func­tions on which they act van­ish at the ends of the in­ter­val $a$ $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $b$ on which they are de­fined. (Less re­stric­tively, it is only re­quired that the func­tions are pe­ri­odic; they must re­turn to the same value at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $b$ that they had at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $a$.)

So­lu­tion herm-f

7.

Show that if $A$ is a Her­mit­ian op­er­a­tor, then so is $A^2$. As a re­sult, un­der the con­di­tions of the pre­vi­ous ques­tion, $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$ is a Her­mit­ian op­er­a­tor too. (And so is just ${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$, of course, but $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$ is the one with the pos­i­tive eigen­val­ues, the squares of the eigen­val­ues of ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$.)

So­lu­tion herm-g

8.

A com­plete set of or­tho­nor­mal eigen­func­tions of $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$ on the in­ter­val 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $\pi$ that are zero at the end points is the in­fi­nite set of func­tions

$$\frac{\sin(x)}{\sqrt{\pi /2}}, \frac{\sin(2x)}{\sqrt{\pi /2}... ...in(3x)}{\sqrt{\pi /2}}, \frac{\sin(4x)}{\sqrt{\pi /2}}, \ldots$$

Check that these func­tions are in­deed zero at $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and $x$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\pi$, that they are in­deed or­tho­nor­mal, and that they are eigen­func­tions of $\vphantom{0}\raisebox{1.5pt}{$-$}$${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$ with the pos­i­tive real eigen­val­ues

$$1, 4, 9, 16, \ldots$$

Com­plete­ness is a much more dif­fi­cult thing to prove, but they are. The com­plete­ness proof in the notes cov­ers this case.

So­lu­tion herm-h

9.

A com­plete set of or­tho­nor­mal eigen­func­tions of the op­er­a­tor ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ that are pe­ri­odic on the in­ter­val 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $2\pi$ are the in­fi­nite set of func­tions

$$\ldots , \frac{e^{-3{\rm i}x}}{\sqrt{2\pi}}, \frac{e^{-2{\rm... ... i}x}}{\sqrt{2\pi}}, \frac{e^{3{\rm i}x}}{\sqrt{2\pi}}, \ldots$$

Check that these func­tions are in­deed pe­ri­odic, or­tho­nor­mal, and that they are eigen­func­tions of ${\rm i}{\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ with the real eigen­val­ues

$$\ldots , 3, 2, 1, 0 , -1, -2, -3, \ldots$$

Com­plete­ness is a much more dif­fi­cult thing to prove, but they are. The com­plete­ness proof in the notes cov­ers this case.

So­lu­tion herm-i