In basic atomic physics of light-matter interaction, very often we assume the atom is quantum mechanical with quantized atomic levels but assume the electromagnetic field to be a classical wave. It is true that this cannot explain the phenomenon of spontaneous emission and many other effects but is this framework mathematically consistent?
The reason I am asking this is as follows. Someone told me that one cannot live with a quantum theory of electrons and a classical theory of gravity. His reasoning was this. On one hand, an electron is a quantum mechanical object but since it has mass, in principle, it also gravitates (no matter how minuscule its gravitational influence be). Therefore, it is not internally consistent to treat the electron quantum mechanically and its gravitational interaction classically. Does this argument has any merit? If it has any, then treating the atom quantum mechanically and electromagnetic field classically must also be internally inconsistent. Am I wrong?
