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The scalar product of two quantum states gives the probability of transition between those two states. In particular, for two stationary (eigen) states, the orthogonality implies that the probabillity of transition is zero:
$$(\Psi,\Phi)=\int \Psi^* \Phi dv=0 \tag 1\\$$
That being said, I have a problem with the so-called "spontaneous emission" in atoms, which occurs when an electron undergoes a transition between two stationary states without any perturbation or external agent (that's why it's called "spontaneous").

How is that possible?! According to $(1)$, there would be no transition at all because its probability is zero. But we know it's not and yet this probability is given in terms of the Einstein coefficient $A$.

Question : Is this a contradiction between theory and experiment? Or is it a flaw in the frame work of the Schrodinger picture of QM? If so, how do we fix it ?!

I really need a satisfactory explanation. Thanks!

Samà
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The flaw in your analysis is the assumption that spontaneous emission occurs between stationary states in the complete absence of perturbations. In the real world there are always perturbations. If these perturbations are weak and not time varying, then we have what we call spontaneous emission. In actuality the upper level state is a mixed state with a small admixture of the lower state and the always present perturbations lead to random decays.

Lewis Miller
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