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:
- Write down the given information and translate this information from words into equations.
- Write down what is to be proved in the form of an equation or equations.
- (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 .