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 $-4\sin(2x)$, 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 $-4$.

A nonzero function $f$ is called an eigenfunction of an operator $A$ if $A f$ 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 ${\rm d}/{\rm d}x$ with eigenvalue 1, since ${\rm d}e^x/{\rm d}x = 1 e^x$. Another simple example is illustrated in figure 2.8; the function $\sin(2x)$ is not an eigenfunction of the first derivative operator ${\rm d}/{\rm d}x$. However it is an eigenfunction of the second derivative operator ${\rm d}^2/{\rm d}x^2$, and with eigenvalue $-4$.

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 ${\rm d}/{\rm d}x$ that do appear a lot are of the form $e^{{\rm i}k x}$, where ${\rm i}=\sqrt{-1}$ 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

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.

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 ${\rm d}^2/{\rm d}x^2$, but with eigenvalue $-k^2$. 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 ${\rm d}^2/{\rm d}x^2$, though they are not eigenfunctions of ${\rm d}/{\rm d}x$
Solution eigvals-b

3

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