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 $\vphantom0\raisebox{1.5pt}{$-$}$$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 $\vphantom0\raisebox{1.5pt}{$-$}$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 $\vphantom0\raisebox{1.5pt}{$-$}$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 $\vphantom0\raisebox{1.5pt}{$-$}$$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