I read some notes saying,
$$i\hbar \frac{dC_{i}(t)}{dt} = \sum_{j}^{} H_{ij}(t)C_{j}(t)\tag{1}$$
where $C_{i}(t) = \langle i|\psi(t)\rangle$ and $H_{ij}$ is hamiltonian matrix.
However, what is obscure to me is the way to deduce
$$H^*_{ij} = H_{ji}\tag{2}$$
in the notes.
They explain (2) as
(2) follows from the condition that the total probability that the system is in some state does not change. If you start with a particle then you have still got it as time goes on. The total probability of finding it somewhere is
$$\sum_{i}^{}\left|C_{i}^2 \right|$$
which must not vary with time. If this is to be true for any starting condition $\phi$, then (2) must also be true.
I cannot understand the relationship between the total amplitude and (2).
Well.. though I omit other contents/derivative of (1) in the notes to simplify my question, it will be very helpful for me if you figure out the explanation of the notes above.
