I have read about an electron making a transition between two energy levels and electromagnatic radiation will be emitted. The problem is how and why e.m radiation is emitted.
When we are calculating the energy levels of an atom (or anything else) we are generally solving the Schrodinger equation to calculate the energy eigenstates. These eigenstates have the property that they are time independant i.e. they do not change with time. So if you consider the ground state and an excited state we end up with the surprising result that since they are both time independant there can be no transition between them.
This is perfectly true, but only as long as we are considering the atom in isolation. If we introduce a photon then the electric field of the photon changes the Schrodinger equation so that the ground state and the excited state that we calculated above are no longer solutions. In that case the excited state gets mixed up with the combined system of the ground state plus a photon and there is now a non-zero probability that the excited state will decay. We calculate this probability using Fermi's golden rule.
To understand exactly how the transition happens requires quantum field theory because it involves the creation of a new particle (the photon) and particle creation is not described by non-relativistic quantum mechanics. Basically the energy of the excited state is transferred to the photon quantum field where the energy appears as the creation of a photon.
I have read about an electron making a transition between two energy levels and electromagnetic radiation will be emitted. The problem is how and why e.m radiation is emitted.
An intuitive understanding can come from considering the Bohr atom. Bohr imposed the fixed energy levels for his hydrogen atom to explain why radiation from hydrogen gave distinct lines
An electron falling on a proton by classical electromagnetic theory would radiate continuously because of the acceleration as it was falling, there should not be the distinct lines that are observed. The distinct lines had to be imposed by hand in the model, and a ground level decided upon so that the electron would never fall on the proton and atoms could exist.
The Schrodinger equation and quantum mechanics gave a solid theoretical framework for the stability of the energy. Momentum, energy and angular momentum conservation in electromagnetic boundary conditions identified the energy leaving as a photon .