1.5 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:

$$A\vec v=a \vec v \mbox{ iff $\vec v$ is an eigenvector of $A$}$$ (1.13)

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

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

$$Af=a f \mbox{ iff $f$ is an eigenfunction of $A$.}$$ (1.14)

For example, $e^x$ is an eigenfunction of the operator ${\rm d}/{\rm d}x$ with eigenvalue 1, since ${\rm d}e^x/{\rm d}x = 1 e^x$.

However, 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 ${\rm d}/{\rm d}x$ 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 (1.5) says:

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

The constant $k$ is called the “indexwave number!simplewave number.”

Key Points

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

1.5 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. Answer

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 ${\rm d}/{\rm d}x$ Answer

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. Answer