Comprehensive Study Notes

Suppose that the effect of operating on some function $f(x)$ with the linear operator $\hat{A}$ is simply to multiply $f(x)$ by a certain constant $k$. We then say that $f(x)$ is an eigenfunction of $\hat{A}$ with eigenvalue $k$. (Eigen is a German word meaning characteristic.) As part of the definition, we shall require that the eigenfunction $f(x)$ is not identically zero. By this we mean that, although $f(x)$ may vanish at various points, it is not everywhere zero. We have

\(
\begin{equation}
\hat{A} f(x)=k f(x) \tag{3.14}
\end{equation}
\)

As an example of (3.14), $e^{2 x}$ is an eigenfunction of the operator $d / d x$ with eigenvalue 2:

\(
(d / d x) e^{2 x}=2 e^{2 x}
\)

However, $\sin 2 x$ is not an eigenfunction of $d / d x$, since $(d / d x)(\sin 2 x)=2 \cos 2 x$, which is not a constant times $\sin 2 x$.

EXAMPLE

If $f(x)$ is an eigenfunction of the linear operator $\hat{A}$ and $c$ is any constant, prove that $c f(x)$ is an eigenfunction of $\hat{A}$ with the same eigenvalue as $f(x)$.

A good way to see how to do a proof is to carry out the following steps:

1. Write down the given information and translate this information from words into equations.
2. Write down what is to be proved in the form of an equation or equations.
3. (a) Manipulate the given equations of step 1 so as to transform them to the desired equations of step 2. (b) Alternatively, start with one side of the equation that we want to prove and use the given equations of step 1 to manipulate this side until it is transformed into the other side of the equation to be proved.

We are given three pieces of information: $f$ is an eigenfunction of $\hat{A} ; \hat{A}$ is a linear operator; $c$ is a constant. Translating these statements into equations, we have [see Eqs. (3.14), (3.9), and (3.10)]

\(
\begin{gather}
\hat{A} f=k f \tag{3.15}\
\hat{A}(f+g)=\hat{A} f+\hat{A} g \quad \text { and } \quad \hat{A}(b f)=b \hat{A} f \tag{3.16}\
c=\text { a constant }
\end{gather}
\)

where $k$ and $b$ are constants and $f$ and $g$ are functions.
We want to prove that $c f$ is an eigenfunction of $\hat{A}$ with the same eigenvalue as $f$, which, written as an equation, is

\(
\hat{A}(c f)=k(c f)
\)

Using the strategy of step $3(\mathrm{~b})$, we start with the left side $\hat{A}(c f)$ of this last equation and try to show that it equals $k(c f)$. Using the second equation in the linearity definition (3.16), we have $\hat{A}(c f)=c \hat{A} f$. Using the eigenvalue equation (3.15), we have $c \hat{A} f=c k f$. Hence

\(
\hat{A}(c f)=c \hat{A} f=c k f=k(c f)
\)

which completes the proof.

EXAMPLE

(a) Find the eigenfunctions and eigenvalues of the operator $d / d x$. (b) If we impose the boundary condition that the eigenfunctions remain finite as $x \rightarrow \pm \infty$, find the eigenvalues.
(a) Equation (3.14) with $\hat{A}=d / d x$ becomes

\(
\begin{align}
\frac{d f(x)}{d x} & =k f(x) \tag{3.17}\
\frac{1}{f} d f & =k d x
\end{align}
\)

Integration gives

\(
\begin{align}
\ln f & =k x+\text { constant } \
f & =e^{\text {constant }} e^{k x} \
f & =c e^{k x} \tag{3.18}
\end{align}
\)

The eigenfunctions of $d / d x$ are given by (3.18). The eigenvalues are $k$, which can be any number whatever and (3.17) will still be satisfied. The eigenfunctions contain an arbitrary multiplicative constant $c$. This is true for the eigenfunctions of every linear operator, as was proved in the previous example. Each different value of $k$ in (3.18) gives a different eigenfunction. However, eigenfunctions with the same value of $k$ but different values of $c$ are not independent of each other.
(b) Since $k$ can be complex, we write it as $k=a+i b$, where $a$ and $b$ are real numbers. We then have $f(x)=c e^{a x} e^{i b x}$. If $a>0$, the factor $e^{a x}$ goes to infinity as $x$ goes to infinity. If $a<0$, then $e^{a x} \rightarrow \infty$ in the limit $x \rightarrow-\infty$. Thus the boundary conditions require that $a=0$, and the eigenvalues are $k=i b$, where $b$ is real.

In the first example in Section 3.1, we found that $\left[z^{3}, d / d z\right] g(z)=-3 z^{2} g(z)$ for every function $g$, and we concluded that $\left[z^{3}, d / d z\right]=-3 z^{2}$. In contrast, the eigenvalue equation $\hat{A} f(x)=k f(x)$ [Eq. (3.14)] does not hold for every function $f(x)$, and we cannot conclude from this equation that $\hat{A}=k$. Thus the fact that $(d / d x) e^{2 x}=2 e^{2 x}$ does not mean that the operator $d / d x$ equals multiplication by 2 .