Your question seems to presume that the old Bohr attempt at a quantum theory is correct, which is very far from true. In Bohrís picture, an electron would sit in a state with definite energy (e.g. a íshellí state) then suddenly jump to a state with a different definite energy.
In actual quantum mechanics, the time-dependent state will change in a gradual, continuous manner, at least as far as it is described by the SchrŲdinger equation. If ímeasurementí processes occur, what happens on the way to a definite perceived outcome is controversial. The predicted distributions of observed measured outcomes if the experiment is repeated many times is not controversial, however.
Anyway, letís get to your last question: whether itís the same electron at the beginning and end. Thereís a definite sense in which it is, because the quantum state of the beginning electron has changed in a continuous way to that of the final one. However, it is a little misleading to think of electrons as having identities. Consider a helium atom with two electrons, in the same íshellí. They have opposite spins. You might think that this constitutes two states: either up-down or down-up, depending on which electron is which. However, nature tells us which states it considers different via statistical mechanics, and the He atom has just one ground state. The electrons donít have identities.
(published on 10/22/2007)