Can spontaneous emission be explained using Schrödinger equation? Because Schrödinger equation dictates that an eigenstate of the Hamiltonian evolves only with a phase in time, is there a way to modify the Hamiltonian to incorporate spontaneous emission?
 A: Yes, there is.
In order to describe emission (or absorption) we need to include in our description the thing that is emitted or absorbed, namely the electromagnetic field. While a complete description of the coupling of the electron to the EM field is quite complicated, at the energies and setups involved in most experiments or to explain visible phenomena, it is sufficient to approximate it into a slightly simpler form.
One such way is to write the EM field as a classical force that acts on the electron. We just take a coupling of the electron to the EM field from classical mechanics, and then plug-in by hand the form of the field (as manifested in the vector potential).
If we take the EM field to be weak, this is a perturbation in the original Hamiltonian, which means that what we thought of as the stationary eigenstates (the original energy levels of the atom) are no longer stationary eigenstates! This will also solve the issue with the time-evolution. Now, if we prepare the system in a state where the electron has the wave-function described by the let's say $(n=4, l=2, m=1)$ quantum numbers, its time evolution is no longer just a phase, and it will get mixed with other states as time evolve. The transitions into these other states describe emission or absorption of energy from the EM field.
In this description, however, the EM field itself remains unchanged. It is a classical potential. We can go further, and quantize it (see for example my answer here). Then, we have a way to quantum-mechanically couple the electron to the field, and also describe the photons in the field. Done correctly, we can really "see" how spontaneous emission happens when the electron changes its state and "create" a photon in the process. Again, once we add the description of the EM field and the coupling to it, the "pure" states of the atomic orbitals are no longer eigenstates, and their time evolution according to the TDSE is no longer a pure phase.
Notice that this is still an approximation, as the full interaction of electrons and EM fields is described by the field theory of QED. However, this is enough in order to quite accurately describe and calculate emission and absorption of atoms and molecules.
