Can Maxwell's Equations explain electromagnetic radiation emission in an atom? Can Maxwell's equations be used to explain the process of spontaneous emission when an electron drops from a higher energy level to a lower energy level? According the Maxwell equations, a changing electric field generates a changing magnetic field which generates a changing electric field and so on and so forth. This is propagation. As an electron drops from a higher energy level to a lower energy level, can it be modeled as a the continuous movement of a charged body, therefore causing a magnetic field to be generated around it? Is so, doesn't this imply that the magnetic field is constant in time (electric field is not changing) and therefore no magnetic field will be generated?
Thanks,
 A: The answer to 

As an electron drops from a higher energy level to a lower energy
  level, can it be modeled as a the continuous movement of a charged
  body, therefore causing a magnetic field to be generated around it?

is "Yes, but only trivially." That is, you could probably work backwards from the far-field radiation to some imagined moving source charge at the atom that would produce more or less the radiation you see, but it would have nothing to do with what actually happened in the atom. Certainly you can't do it by imagining a classical electron orbiting in an ellipse: it would radiate continuously as it spiraled into the nucleus.
However, you could treat the wave function of the electron orbit as a charge distribution, $\rho(\mathbf{r})=q|\Psi(\mathbf{r})|^2$. This distribution changes when changing states. But there is a major obstacle in trying to turn this into classical radiation: the QM transition is (as near as we can tell) instantaneous. 
As for this part:

Is so, doesn't this imply that the magnetic field is constant in time > (electric field is not changing) and therefore no magnetic field will be generated?

the magnetic field of an accelerating particle is not constant, because the electric field is not constant - it is moving in space.
