Quantum mechanics says that if a system is in an eigenstate of the Hamiltonian, then the state ket representing the system will not evolve with time. So if the electron is in, say, the first excited state then why does it change its state and relax to the ground state (since it was in a Hamiltonian eigenstate it should not change with time)?
Answer
The atomic orbitals are eigenstates of the Hamiltonian $$ H_0(\boldsymbol P,\boldsymbol R)=\frac{\boldsymbol P^2}{2m}+\frac{e}{R} $$
On the other hand, the Hamiltonian of Nature is not $H_0$: there is a contribution from the electromagnetic field as well $$ H(\boldsymbol P,\boldsymbol R,\boldsymbol A)=H_0(\boldsymbol P+e\boldsymbol A,\boldsymbol R)+\frac12\int_\mathbb{R^3}\left(\boldsymbol E^2+\boldsymbol B^2\right)\,\mathrm d\boldsymbol x $$ (in Gaussian units, and where $\boldsymbol B\equiv\nabla \times\boldsymbol A$ and $\boldsymbol E\equiv \dot{\boldsymbol A}-\nabla\phi$)
Therefore, atomic orbitals are not stationary: they depend on time and you get transitions from different states.
The problem is that what determines time evolution is the total Hamiltonian of the system, and in Nature, the total Hamiltonian includes all forms of interactions. We usually neglect most interactions to get the overall description of the system, and then add secondary effects using perturbation theory. In this sense, the atom is very accurately described by $H_0$, but it is not the end of the story: there are many more terms that contribute to the real dynamics.
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