Transition dipole moment integral relation to probabilities derivation

I want to learn about selection rules and so have to understand transition dipole moment.
Here is a nice text on it:

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Quantum_States_of_Atoms_and_Molecules_(Zielinksi_et_al)/04%3A_Electronic_Spectroscopy_of_Cyanine_Dyes/4.05%3A_The_Transition_Dipole_Moment_and_Spectroscopic_Selection_Rules

However it cops out on the derivation: "The transition dipole moment integral and its relationship to the absorption coefficient and transition probability can be derived from the time-dependent Schrödinger equation. Here we only want to introduce the concept of the transition dipole moment and use it to obtain selection rules and relative transition probabilities for the particle-in-a-box. Later it will be applied to other systems that we will be considering."

Can somebody help me find this derivation or show it to me?