In spontaneous emission, an electron in an energetically higher state jumps to an energetically lower state. This seems a simple phenomenon. Why is it called a quantum mechanical effect? What was so hard in this simple looking process that it needed someone like Dirac to explain?
There are two main types of emissions:
spontaneous, in which a QM system transitions to a lower energy state from an excited state (relaxation), and emits a quantized amount of energy, in a form of a photon
simulated, where an incoming photon that has a specific energy level can interact with a excited state QM system, and causing it to drop to a lower energy level
Spontaneous emission is ultimately responsible for most of the light we see all around us.
Spontaneous emission cannot be explained by classical EM (because it is a fundamentally QM process), because it is not explainable with the Scrodinger equations where the energy levels were quantized, but the EM field was not.
In this case, the overlap between the wavefunctions of the excited and ground state is 0, so without a quantized EM field, the excited state cannot decay into a ground state.
Now, if you want to explain spontaneous emission, you need to extend QM with QFT, where the EM field is quantized at every point in space (QED).
In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the QED vacuum, which can mix with the excited stationary states of the atom. As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon vacuum fluctuations to get started.
It is very important to understand that there are two ways of orbit:
In classical EM, the electron would orbit around the nucleus, like planets around a star, and this would mean constant acceleration, constant radiation, as the electron would spiral into the nucleus
in QM (QED), we use an orbital, which is as per the wavefunction, the probability distribution of the electron being at a certain energy level around the nucleus