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Suppose we have a system that has discrete energy levels (e.g. hydrogen atom, potential well) and the stationary solutions for the wave function are $\psi_n$. I would assume that there should be a way in which one can model the transition $2\rightarrow 1$ by using as initial condition the stationary state $\psi_2$ for the Schrodinger equation (temporal version). However I did not manage to find materials that cover this approach.

I assume that the Schrodinger eq. should not be changed (for a spontaneous transition there is no need of a photon to trigger it). From this, it should follow that a stationary wave function should evolve in another one of lower energy, or, if the transition is possible on multiple lower energy levels, I assume that the solution of the Schrodinger eq. is a superposition of the possible lower energy states.

So, in order to conclude the question, are there any materials on this approach that I did not manage to find? If so, I would appreciate suggestions on the topic.

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    $\begingroup$ If the system is by itself, it seems to me that it will never transition from $\psi_2$. That is because energy eigenstates do not evolve in time (except by a global phase which has no impact on measurement outcomes). Some interaction is needed with another system, or the initial state must not be a perfect eigenstate, for the transition to occur. $\endgroup$ – doublefelix Oct 12 '20 at 11:20
  • $\begingroup$ Ok, but this rises 2 questions: First, does this mean that the spontaneous transition is not random but a result of the interaction between the system and its environment? Second, if the state is not perfect, the transition to the $n=1$ state should give a wave function that is not stable? $\endgroup$ – Victor Palea Oct 12 '20 at 11:36
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On place to look for information is by reading about Rabi oscillations, which arise when a two-level system is driven by a near-resonant electromagnetic field. For some situations one could find solutions in terms of wave functions that are exact or nearly exact.

More generally, the way to approach this problem is by considering a joint wave function of the two-level system (or atom) and the electromagnetic field. The electromagnetic field is the missing component in the reasoning, and it is also the presence of an infinite number of modes of this field with the energies close to the level splitting that is responsible for the spontaneous emission $$ |1\rangle_{atom}\prod_k|0\rangle_{k} \rightarrow |0\rangle_{atom}\sum_k\left[|1\rangle_k\prod_{q, q\neq k}|0\rangle_q\right] $$ The infinite number of the field modes here means that one needs to take thermodynamic limit, since otherwise one would expect the wave function revival, when the atom returns to its excited state.

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  • $\begingroup$ Ok so if I understand this correctly, the established way of treating this is by assuming the existence of an electromagnetic field, and the case of an atom in vacuum is not enough to get a transition (what doublefelix said). $\endgroup$ – Victor Palea Oct 12 '20 at 11:45
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    $\begingroup$ If your atom is isolated from the world, there will be no transition - for the reasons that you outlined in your question. Transition means emitting a photon, so the field should be there. Einstein coefficients for spontaneous and induces processes are a phenomenological way of approaching the problem - without adding the field in the description, and without treating the mechanism of the emission. $\endgroup$ – Roger Vadim Oct 12 '20 at 11:50

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