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The radiation of an atom was interpreted by time-independent schrodinger equation:electron was transformed from high energy level state to lower and emit a photon.

Could we treat this process through a wavefunction ${\psi}(t)$? Before emiting,the system's wavefunction is ${\psi}(0)$ and after emiting photon,it is ${\psi}(t_0)$. ${\psi}(t)$ is constrained by time-dependent schrodinger equation and contain all information of the system.

Is there any papers incorporate photon emiting in wavefunction as well as the method to define ${\psi}(t=0)$?

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We don't describe the radiation with the TISE. We describe the state of the system between radiation events that way, and simply note energy must be conserved to understand the radiation. It's an approximation, of course, but because the time to decay or excitation is generally much longer than period of the radiation involved (in both the final and initial states) this is a pretty good approximation.1

To describe the detailed evolution of the system as it radiates requires a time dependent formalism, but it has to be a formalism that includes the electromagnetic field: you end up with quantum electrodynamics.


1 A situation in which the approximation is not reasonable is in multi-photon excitation processes such as those I discuss in an earlier answer.

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