Naively I assumed that an electron would sit in an excited state (in the shell of an atom) having some stochastic lifetime, so nothing changes and suddenly the electron would jump back into the ground state (emitting a photon), and the process of this jump is instantaneous...
However, from what I learned in this thread Do electrons really perform instantaneous quantum leaps? and this Does the new finding on "reversing a quantum jump mid-flight" rule out any interpretations of QM?, this view is wrong.
My current understanding: Quantum mechanically, both the ground state $|g\rangle$ as well as the excited state $|e\rangle$ are eigenvectors of the Hamilton operator $H$ of the atom, meaning the corresponding eigenvalues are observables $E_g$ and $E_e$ (which can be measured or observed). As $|g\rangle$ and $|e\rangle$ are base vectors, we can write linear combinations $|\Psi\rangle$ (not corresponding to an obervable eigenvalue of $H$, i.e. if we measure, the state would collapse into an eigenvector). However, temporal evolution of quantum states is deterministic, and as soon as the electron is in the excited state $|e\rangle$, the transition or quantum jump starts and the electron's state can be described as a linear combination $|\Psi\rangle = a(t) |e\rangle + b(t) |g\rangle$, which develops continuously into $|g\rangle$ when $a(t$) changes with time from 1 to 0 while $b(t$) changes from 0 to 1. And during this, the electromagnetic wave (the photon) is emitted meaning the longer it takes (the more stable the excited level is) the longer the emitted wave train (more coherent light is emitted).
Question: Is that correct, meaning is the lifetime of an excited state the transition time (or quantum jump time)? If so, where is the randomness of the lifetime, i.e. what determines $a(t)$ and $b(t)$ in the equation above?