We must, of course, define the physical interpretation of f. The 'probabilistic' approach dictated by the uncertainty principle is therefore built into quantum mechanics. The next step is to formulate an equation that allows us to calculate f. Here the pioneers of quantum mechanics were most ingenious. They were able to take one of the fundamental equations of classical physics, that representing conservation of total energy, and recast it in a form whose solutions are consistent with the requirement of the new mechanics and, of course, with observation.
Atoms are positive nuclei which have trapped electrons.
The planetary model of orbiting electrons has its conceptual base in classical physics and is inappropriate. The quantum mechanical model is less easy to envisage but it is more elegant and equally intuitive. This profound and beautiful model of atomic structure is, moreover, in agreement with experiment.