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All of our theoretical understanding suggests that quantum transitions occur at a definite point in time. There is no experimental evidence to the contrary.
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The light in spectral line has quite well defined period of oscillation, which implies that the atom has to be in state of oscillation connected to pair of eigenfunctions for a time interval longer than this period.
But if atom was jumping instantaneously between the stationary states, all atoms would be either in one or the other stationary state.