Example 6 Given the function $f(x)=x^{2}+x-1, \alpha 、 \beta$ $(\alpha>\beta)$ are the two roots of the equation $f(x)=0$, and $f^{\prime}(x)$ is the derivative function.
Let $a_{1}=1, a_{n+1}=a_{n}-\frac{f\left(a_{n}\right)}{f^{\prime}\left(a_{n}\right)}(n=1,2, \cdots)$.
(1) Find the values of $\alpha 、 \beta$;
(2) It is known that for any positive integer $n$, $a_{n}>\alpha$, and let $b_{n}=\ln \frac{a_{n}-\beta}{a_{n}-\alpha}(n=1,2, \cdots)$. Find the sum of the first $n$ terms $S_{n}$ of the sequence $\left\{b_{n}\right\}$.
(2007, National College Entrance Examination (Guangdong Paper))