I am aware this question has been asked before, but I am looking for a more technical answer than those given to the other questions.
I am aware the question in the title is a problem in classical mechanics that quantum mechanics solves. But it doesn't seem immediately obvious at first blush how QM accomplishes this. For the simple hydrogen atom case, you model the electron as experiencing a static potential and also coupled to the surrounding radiation field.
From this $H$ there will be certain stationary states (for the joint electron-field system), but it doesn't seem obvious that these stationary states will involve nice atomic orbitals. You could imagine that the vast majority of the stationary states involve almost all the energy being stored in the field, and so any initial state with the electron orbiting the atom will not be an eigenstate and so tend to dissipate its energy to the field over time.
Is there a "deep" reason why this isn't the case, or is it just something that comes out of the mathematics?-
