Hello
Assume we have a Stern Gerlach setup which its field is directed toward z direction. A ray of Ag atoms(N atoms) comes out of oven and pass through of our setup. when atoms feel magnetic field, the half of atoms collapse to high level of energy and the rest to low level. in this moment we remove high level atoms and immediately a EM wave with frequency of f whose photon energy is equal to energy levels difference meet atoms and some of them go to the upper level. after this setup, we have a SG-x setup and N/4 will be in Sx+ and N/4 in Sx-.
Now, we repeat mentioned process without EM wave as a result, Ag atoms will be same but we have not energy loss because there is no wave.
what happened to photons?
Abbas, you lost a step in your experiment. You say
==> in this moment we remove high level atoms and immediately a EM wave . . .
Take into account that not all the atoms with spin down absorb a photon. The absorption probability is not equal to 1. So, you need to pass your spin-down bean through an additional SG apparatus with z orientation of the magnetic field, and thereby select those atoms which have absorbed a photon.
Only after that it's worthy to pass them through the SG with magnetic field along x.
Now, in a magnetic field an atom acquires a potential energy. The potential energy level is according to the orientation of the spin of the atom versus the field direction.
As to the absorbed photon, it is inside the atom. If, in the Hz field, you would have selected the spin-up atoms, the behavior in continuation would have been the same as that of the spin-down atoms that absorbed a photon.
The energy of 2 magnetic sublevels of 5s electron of Ag in field free space are the same, the energy difference appears due to Zeeman splitting in the magnetic field. So, there is no need in the second SG, the atom can absorb a photon, and after leaving the field region it has the same energy of 5s electron. Proper estimation of the mechanical energy of Ag atom , interaction of the induced magnetic moment and field creating magnet should give a precise energy balance, energy conservation can not be violated.
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