Here are the list of quantum-mechanical potentials. My question is how did scientists physically model these potentials, what were the parameters they consider before mathematically constructing a potential? Few of these potentials were devised when technology was not available to observe the behaviour and properties of quantum systems, so were these potentials mere ansatzes?
For example, while solving the Kronig-Penny model, we use a periodic square potential and the Bloch function (to account for the periodicity) to obtain solutions. For solving the Gamov's alpha particle tunnelling model, we use the barrier potential. These approximations are easy to deal with mathematically. If, for instance, I want to model my own potential without observation of experimental evidence, then how do I make an ansatz about the potential I want to construct?