In QM a wave $\psi$ strangely gives the full information of particle and $|\psi|^2$ represents probability for finding that particle (just like $|I|^2$ is probability in EM-theory). When total probability is unity, $||\psi||_2=1$, we have the first connection to $L^2$ functions. Since QM is fundamentally probabilistic, we can only deal with expectations values: $$<f(x)>=\int f(x)|\psi|^2 dx$$ Again, since physical expectations value must be finite, we demand $$\int f(x)|\psi|^2 dx \leqq ||f||_2||\psi||_2=||f||_2 <\infty$$ So again $f\in L^2$ is utmost important for theory.
E.g. for Coulomb potential: $$<V(x)>=<\frac{-ke^2}{|x|}>=-ke^2 \int_{R^3} \frac{1}{|x|}| \psi|^2dx$$
Fourier transform in $L^2$ is important for QM, since that gives us operators. If one considers particle as a wave packet then we know that transform $\hat \psi$ is the probability amplitude for momentum and $|\hat \psi|^2$ is the corresponding probability (since isomorphy $\psi \rightarrow \hat \psi$ ensures that $\hat \psi$ contains the same informations as $\psi$ and due to Parseval's relation $||\hat \psi||_2=||\psi||_2=1$). Now we can calculate those tricky expectations values for momentum $p$: $$<p>=\int_{R^3} p|\hat \psi|^2dp=\int_{R^3} -i\hbar\frac{d}{dx}\psi \centerdot \overline{\psi} dx$$ $$<T>=<\frac{p^2}{2m} >=\int_{R^3} \frac{p^2}{2m}|\hat \psi|^2dp=\int_{R^3} -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi \centerdot \overline{\psi} dx=\frac{{\hbar}^2}{2m}\int_{R^3} |\nabla \psi|^2dx$$
To get some physical results we again need $L^2$ theory:
The lowest expectation value for hydrogen atom, that is the sum of expectation value for kinetic energy and Coulomb potential: $$\text {inf}<T+V(x)>=\text {inf}(\frac{{\hbar}^2}{2m}\int_{R^3} |\nabla \psi|^2dx-ke^2 \int_{R^3} \frac{1}{|x|}| \psi|^2dx)$$
Theorem . If both $f$ and $\nabla f$ belong to space $L^2 (R^3)$ then following inequality holds:$$\int_{R^3} \frac{1}{|x|}| f|^2dx \leqq ||\nabla f||_2 || f||_2$$
[You will have much fun in proving that!]
When you insert that theorem to $\text {inf}<T+V(x)>$ everything boils down to minimizing expression: $$\frac{{\hbar}^2}{2m}||\nabla \psi||^2_2 -ke^2||\nabla \psi||_2$$
Keeping $||\nabla \psi||_2$ as a variable you can use elementary calculus to find the fundamental physical result, the ground state energy for hydrogen: $$\text {inf}<T+V(x)>=-k^2me^4/2\hbar^2.$$
Now you have $||\nabla \psi||_2=kme^2/\hbar^2$ and $\text {inf}<T+V(x)>=-k^2me^4/2\hbar^2$, so you can put them back into expectation value equality: $$-k^2me^4/2\hbar^2=\frac{{\hbar}^2}{2m}\centerdot (kme^2/\hbar^2)^2 - ke^2 \int_{R^3} \frac{1}{|x|}| \psi|^2dx$$
You have $$<\frac{1}{ |x|}>=\int_{R^3} \frac{1}{|x|}| \psi|^2dx=kme^2/\hbar^2=1/a_0$$
That is, the Bohr's radius! $$a_0=\frac{\hbar^2}{kme^2}.$$