# Decay of an electron not in eigenstate

Suppose the electron of a single electron atom is in the state $$a|{E_{1}}\rangle + b|{E_{2}}\rangle$$. If it decays to the ground state $$|E_{0}\rangle$$ which spectral lines do I expect to see? The ones that correspond to the transition from $$|E_{1}\rangle$$ to $$|E_{0}\rangle$$, the ones that correspond to the transition from $$|E_{2}\rangle$$ to $$|E_{0}\rangle$$ or neither? My guess is that "right before the decay" the electron's wave function collapses to $$|{E_{1}}\rangle$$ with a probability $$\lvert a \rvert ^{2}$$ or $$|{E_{2}}\rangle$$ with a probability of $$\lvert b \rvert ^{2}$$ and then it decays to $$|E_{0}\rangle$$, therefore I expect to see the spectral line of the transition $$|E_{1}\rangle$$ to $$|E_{0}\rangle$$ with a probability of $$\lvert a \rvert ^{2}$$ and the other one with a probability of $$\lvert b \rvert ^{2}$$. Is my reasoning correct?

You will observe transition $$E_1\rightarrow E_0$$ with probability $$|a|^2$$, and transition $$E_2\rightarrow E_0$$ with probability $$|b|^2$$. Of course, if you literally made one measurement on a single electron, you would observe only one transition. However, in practice you either observe an ensemble of identical systems or the same system multiple times.