Appendix A: Equation of motion for a superposition state Consider a state in a quantum superposition
$$\Large |\psi\rangle = \sum_k c_k e^{-i E_kt / \hbar}|k\rangle \normalsize \qquad \text{(1)} $$
the time-evolution of the coefficients $c_k$ can be obtained by plugging this definition into the Schrödinger equation
$$\Large i\hbar \frac{\partial |\psi\rangle}{\partial t} = \hat{H} |\psi\rangle \normalsize \qquad \text{(2)}$$
First, we evaluate the left-hand side of this equation by taking the derivative of (1)
$$\Large i\hbar \frac{\partial |\psi\rangle}{\partial t} = i\hbar\sum_k \dot{c_k}e^{-i E_kt / \hbar}|k\rangle + \sum_k E_k c_k e^{-i E_kt / \hbar}|k\rangle \normalsize \qquad \text{(3)}$$
Next, we partition the Hamiltonian into diagonal and off-diagonal parts
$$\Large \hat{H} = \hat{H}_0 + \hat{V}$$
such that
$$\Large \hat{H}_0 |k\rangle = E_k |k\rangle $$
then, the right-hand side of (2) becomes
$$\Large \hat{H}|\psi\rangle = \sum_k E_k c_k e^{-i E_kt / \hbar}|k\rangle + \sum_k c_k e^{-i E_kt/\hbar}\hat{V}|k \rangle \normalsize \qquad \text{(4)}$$
Plugging (3) and (4) back into (2) gives us
$$\Large i\hbar\sum_k \dot{c_k}e^{-i E_kt / \hbar}|k\rangle + \sum_k E_k c_k e^{-i E_kt / \hbar}|k\rangle = \sum_k E_k c_k e^{-i E_kt / \hbar}|k\rangle + \sum_k c_k e^{-i E_kt/\hbar}\hat{V}|k \rangle$$
we can cancel the common term out to get
$$\Large i\hbar\sum_k \dot{c_k}e^{-i E_kt / \hbar}|k\rangle = \sum_k c_k e^{-i E_kt/\hbar}\hat{V}|k \rangle $$
Projecting $\Large \frac{-i}{\hbar}\langle n | e^{iE_nt/\hbar}$ from the left side yields
$$\Large \dot{c_n} = -\frac{i}{\hbar}\sum_kc_k e^{-i \omega_{kn} t} V_{nk}$$
where
$$\Large V_{nk} = \langle n | \hat{V} | k \rangle$$
and
$$\Large \omega_{kn} = (E_k - En) / \hbar $$