How are photons created?

Question

In boundary conditions electrons or other molecules are jumping to a lower energy level and the energy difference is radiated with a photon. But in electrodynamics when charges accelerate electromagnetic waves are created, which are also photons; so is there 2 ways of making photons/electromagnetic waves ?

Answer 1

The two ways of producing photons that you mention are essentially the same thing. Very naively, you can think of an electron “jumping” between energy levels as an accelerating electron.

To be slightly less naive, when an atom transitions from a higher-energy state to a lower-energy one, the charge distribution of its electron “cloud” changes. What typically happens is that the expectation value of the atom’s electric dipole moment oscillates sinusoidally. Classically, this oscillating dipole moment creates an electromagnetic wave. Quantum mechanically, it produces one or more photons.

Answer 2

electrons ... are jumping to a lower energy level and the energy difference is radiated with a photon.

In order to raise an electron to a higher energy level, energy must be applied. On the atomic level it is obvious, that photons are the carriers of this energy. So it’s natural that the energy transmitted to the electron will be released again in the form of photons during the electron relaxation.

But in electrodynamics when charges (get) accelerate(d) electromagnetic waves are created, which are also photons ...

On the atomic level we distinguish three different possibilities to accelerate free charges:

1. The charge is located inside a potential difference and the electric field accelerates the charge to the opposite pole.
2. The charge gets accelerated by a light beam. The photons in the beam have a momentum and this momentum is transferred to the electron.
3. A magnetic field is applied to a moving electron (non-parallel to the external field). The electron gets deflected in the plain perpendicular to both the direction of motion and the direction of the external magnetic field.

In the second case, when the electron is braked by photons, it loses kinetic energy in the form of photons. If the electron is accelerated by photons, some part of the absorbed energy will be re-emitted again.

In the latter case, the electron possesses kinetic energy and loses this energy in the form of photons when deflected in the magnetic field.

... so is there 2 ways of making photons ..?

Yes and no.
Yes, there are two ways, because the electron in an atom or molecule isn’t a free particle, the process of absorption and re-emission of photons depends from the overall energy content of the atom. The energy, which has to be applied and which gets re-emitted, depends from the temperature of the whole system. Not so for a free electron, where we have to do only with the kinetic energy of this electron.
And no, because in any case for an emission of photons always firstly an input of photons is needed. To say it briefly, an electron with a temperature above 0 Kelvin (and this is always the case) emits photons all the time (and since subatomic particles are doing so, the electron does not “cool” down).

It seems to be more important to distinguish between acceleration from photons and the immediate re-emission of photons and the deflection in magnetic fields with the transformation of the electrons kinetic energy into radiation (until the kinetic energy is exhausted and the electron comes to stand still in thee centre of its spiral path.