When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator:
Suppose that are two operators $L^{+}$ and $L^{-}$ and define $f_{n}$ such that
$L^{+}(f_{n})=\sqrt{n+1}f_{n+1}$
$L^{-}(f_{n})=\sqrt{n}f_{n-1}$
then $L^{+}L^{-}(f_{n})=L^{+}(\sqrt{n}f_{n-1})=nf_{n}$ $L^{-}L^{+}(f_{n})=L^{-}(\sqrt{n+1}f_{n+1})=(n+1)f_{n}$
Thus
$L^{+}L^{-}+L^{-}L^{+}=(2n+1)f_{n}$
Now since $H=-\frac{d^{2}}{dx^{2}}+x^{2}=\left(x+\frac{d}{dx}\right)\left(x-\frac{d}{dx}\right)$
$=\frac{1}{2}\left[\left(x+\frac{d}{dx}\right)\left(x-\frac{d}{dx}\right)+\left(x+\frac{d}{dx}\right)\left(x-\frac{d}{dx}\right)\right]$
$=\frac{1}{\sqrt{2}}\left(x+\frac{d}{dx}\right)\frac{1}{\sqrt{2}}\left(x-\frac{d}{dx}\right)+\frac{1}{\sqrt{2}}\left(x-\frac{d}{dx}\right)\frac{1}{\sqrt{2}}\left(x+\frac{d}{dx}\right)$
If we let $L^{+}=\frac{1}{\sqrt{2}}\left(x+\frac{d}{dx}\right)\quad\text{and}\quad L^{-}=\frac{1}{\sqrt{2}}\left(x-\frac{d}{dx}\right)$
then $H=L^{+}L^{-}+L^{-}L^{+}$
Since $L^{+}(f_{n})=\sqrt{n+1}f_{n+1}\implies$
$f_{n}=\sqrt{n!}L^{n}(f_{0})$
Thus if we can a $f_{0}$ such that
$f_{n}=\sqrt{n!}L^{n}(f_{0})\implies L^{-}(f_{n})=\sqrt{n}f_{n-1}\tag{1}$ would be satisfied, then we have found a set of eigenfunctions for H, which is $f_{n}$ and the corresponding eigenvalue is 2n+1.
Now my instructor says that if we let $f_{0}$ be in the kernel of $L^-$, meaning that:
$L^{-1}f_{0}=0\tag{2},$ then (1) would be true.
Specifically, this means that
$f_{0}(x)=e^{-\frac{1}{2}x^{2}}.$
As it is already verified (and commonly used in quantum mechanics) that this $f_{0}(x)$ does satisfy (1). But I can't really see how does (2) leads to (1). Is this choice $f_0$ really a lucky guess or actually a choice the generally works for all operators that can be expressed as $H=L^{+}L^{-}+L^{-}L^{+}$? A different way of asking this question is: are the ladder operator really just operators that tells us how to get from one state to another, or it is actually something that would be used as a general method of solving some differential equations?