2.5 Eigen­value Prob­lems

To an­a­lyze quan­tum me­chan­i­cal sys­tems, it is nor­mally nec­es­sary to find so-called eigen­val­ues and eigen­vec­tors or eigen­func­tions. This sec­tion de­fines what they are.

A nonzero vec­tor $\vec{v}$ is called an eigen­vec­tor of a ma­trix $A$ if $A\vec{v}$ is a mul­ti­ple of the same vec­tor:

$$A\vec v=a \vec v \mbox{ iff $\vec v$ is an eigenvector of $A$}$$ (2.13)

The mul­ti­ple $a$ is called the eigen­value. It is just a num­ber.

Fig­ure 2.8: Il­lus­tra­tion of the eigen­func­tion con­cept. Func­tion $\sin(2x)$ is shown in black. Its first de­riv­a­tive $2\cos(2x)$, shown in red, is not just a mul­ti­ple of $\sin(2x)$. There­fore $\sin(2x)$ is not an eigen­func­tion of the first de­riv­a­tive op­er­a­tor. How­ever, the sec­ond de­riv­a­tive of $\sin(2x)$ is $\vphantom{0}\raisebox{1.5pt}{$-$}$$4\sin(2x)$, which is shown in green, and that is in­deed a mul­ti­ple of $\sin(2x)$. So $\sin(2x)$ is an eigen­func­tion of the sec­ond de­riv­a­tive op­er­a­tor, and with eigen­value $\vphantom{0}\raisebox{1.5pt}{$-$}$4.

A nonzero func­tion $f$ is called an eigen­func­tion of an op­er­a­tor $A$ if $Af$ is a mul­ti­ple of the same func­tion:

$$Af=a f \mbox{ iff $f$ is an eigenfunction of $A$.}$$ (2.14)

For ex­am­ple, $e^x$ is an eigen­func­tion of the op­er­a­tor ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ with eigen­value 1, since ${\rm d}{e}^x$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1 $e^x$. An­other sim­ple ex­am­ple is il­lus­trated in fig­ure 2.8; the func­tion $\sin(2x)$ is not an eigen­func­tion of the first de­riv­a­tive op­er­a­tor ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$. How­ever it is an eigen­func­tion of the sec­ond de­riv­a­tive op­er­a­tor ${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$, and with eigen­value $\vphantom{0}\raisebox{1.5pt}{$-$}$4.

Eigen­func­tions like $e^x$ are not very com­mon in quan­tum me­chan­ics since they be­come very large at large $x$, and that typ­i­cally does not de­scribe phys­i­cal sit­u­a­tions. The eigen­func­tions of the first de­riv­a­tive op­er­a­tor ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$ that do ap­pear a lot are of the form $e^{{{\rm i}}kx}$, where ${\rm i}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{-1}$ and $k$ is an ar­bi­trary real num­ber. The eigen­value is ${{\rm i}}k$:

$$\frac{{\rm d}}{{\rm d}x} e^{{\rm i}kx} = {\rm i}k e^{{\rm i}kx}$$

Func­tion $e^{{{\rm i}}kx}$ does not blow up at large $x$; in par­tic­u­lar, the Euler for­mula (2.5) says:

$$e^{{\rm i}k x} = \cos(kx) + {\rm i}\sin(kx)$$

The con­stant $k$ is called the “wave num­ber.”

Key Points

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If a ma­trix turns a nonzero vec­tor into a mul­ti­ple of that vec­tor, then that vec­tor is an eigen­vec­tor of the ma­trix, and the mul­ti­ple is the eigen­value.

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If an op­er­a­tor turns a nonzero func­tion into a mul­ti­ple of that func­tion, then that func­tion is an eigen­func­tion of the op­er­a­tor, and the mul­ti­ple is the eigen­value.

2.5 Re­view Ques­tions

1.

Show that $e^{{{\rm i}}kx}$, above, is also an eigen­func­tion of ${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$, but with eigen­value $\vphantom{0}\raisebox{1.5pt}{$-$}$$k^2$. In fact, it is easy to see that the square of any op­er­a­tor has the same eigen­func­tions, but with the square eigen­val­ues.

So­lu­tion eigvals-a

2.

Show that any func­tion of the form $\sin(kx)$ and any func­tion of the form $\cos(kx)$, where $k$ is a con­stant called the wave num­ber, is an eigen­func­tion of the op­er­a­tor ${\rm d}^2$$\raisebox{.5pt}{$/$}$​${\rm d}{x}^2$, though they are not eigen­func­tions of ${\rm d}$$\raisebox{.5pt}{$/$}$​${\rm d}{x}$.

So­lu­tion eigvals-b

3.

Show that $\sin(kx)$ and $\cos(kx)$, with $k$ a con­stant, are eigen­func­tions of the in­ver­sion op­er­a­tor ${\mit\Pi}$, which turns any func­tion $f(x)$ into $f(-x)$, and find the eigen­val­ues.

So­lu­tion eigvals-c