When an electron absorbs energy and jumps to the another excited state by absorbing the photon and why it is always said that the electron will come back to $\hbar\omega/2$?
Why doesn't the electron sit in the excited state ?
What is the actual reason for it to come back to ground state?
The reason is: Because it can. Also known as "Everything that can happen, will happen.". We already know from statistical mechanics that the lowest energy state is the most likely. In QM, the fact that all things drop into their lowest allowed energy state is even simpler: As long as there is a non-zero transition amplitude $e^- \to e^- + \gamma$ (where the $e^-$ has dropped into a lower energy level and radiated the energy difference as a photon), this transition will, sooner or later, happen. Since for an electron that has been raised into an excited state by hitting it with a photon the reverse process has certainly non-zero amplitude, the electron will, sooner or later, drop to the lower energy level. Once there, is has no chance to go anywhere without energy coming from the outside, so it stays there.
An excited atom in vacuum is being gently tugged by the electromagnetic field back into its ground state. This occurs even when the electromagnetic field is in its quantum mechanical ground state.
Should the electromagnetic field already contain photons with energies at the transition energy, this tugging will become greater, with the result that the atom will decay more quickly.
In simple terms, it is a phenomenon called 'spontaneous emission' as first (?) put by Einstein.
To me the key takeaway from https://en.wikipedia.org/wiki/Spontaneous_emission is that:
Spontaneous transitions were not explainable within the framework of the Schrödinger equation,Stimulated emission: how can giving energy to electrons make them decay to a lower state?
and:
The first person to derive the rate of spontaneous emission accurately from first principles was Dirac in his quantum theory of radiation
I haven't learnt Dirac or QED yet, but at least it clarifies that Schrodinger does not explain it, you have to go to more general equations.
It is also interesting to contrast spontaneous emission with stimulated emission, which Stimulated emission: how can giving energy to electrons make them decay to a lower state? suggests can be explained by the Schrodinger equation.
