Electronic tunneling between two states I am reading Steering Electrons on Moving Pathways (Beratan et al. 2009) which is  about electron tunneling in biomolecules, and specifically the processes via which an electron can move from an initial state $|D\rangle$ (on the donor side) to a final state $|A\rangle$ (on the acceptor side). In my research, I just came across the following equation:

The time-dependent transition amplitude from an initial complex (momentum-carrying) state $D$ to a real final state $A$ in a system with Hamiltonian $H$ is given by$$\langle A|e^{-iHt/\hbar}|D\rangle=\sum_m\langle A|\Psi_m\rangle\langle\Psi_m|D\rangle e^{-iE_mt/\hbar}.$$

Here, $\Psi_m$ and $E_m$ are the corresponding eigenvalues and eigenfunctions of the Hamiltonian. The reason this confuses me is that I am used to think of transition probabilities as being proportional to $\langle A|H|D\rangle$, per the Schrödinger equation. I had always thought that the time evolution operator $e^{-iHt/\hbar}$ introduces a harmless phase factor which does not in and of itself determine quantum dynamics.
So what am I not understanding here? Is this a matter of terminology, or is there a deeper meaning to the time evolution operator in this context that I had not previously been aware of?
 A: If your system is in the state $|\psi(0)\rangle$ at time $t=0$, it will be in the state $|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle$ at time $t$, where $H$ is the Hamiltonian of the system. With $|\psi(0) \rangle = | D \rangle$, $\langle A | e^{-iHt/\hbar} |D \rangle$ is thus the transition amplitude to find the system in the state $|A\rangle$ at time $t$ if it had been in the state $|D\rangle$ at $t=0$. The corresponding transition probability is then given by $|\langle A | e^{-iHt/\hbar} |D \rangle|^2$, being time dependent, in general.
Only in the special case $|D\rangle=|\psi_n\rangle$, where $|\psi_n \rangle$  is (a normalized) eigenvector of the Hamiltonian ($H |\psi_n\rangle = E_n | \psi_n\rangle$), the transition amplitude is just given by $\langle A | e^{-iHt/\hbar} |\psi_n\rangle=\langle A | \psi_n \rangle e^{-iE_n t /\hbar}$ resulting in a time independent transition probability, a rather boring case.
The matrix element $\langle A | H | D \rangle$ (in general a complex number) is certainly NOT a transition probability!
