There is radioactivity which was (and still) one of the most important aspects in physics (nuclear physics) and it is mainly about describing decaying state. Th

There is radioactivity which was (and still) one of the most important aspects in physics (nuclear physics) and it is mainly about describing decaying state. This is using non-hermitian QM. Because of hermitian operators giving always real eigenvalues (This exercise is left to the reader) we can see (and prove) that non hermitian operators will give rise to complex eigenvalues whose imaginary eigenvalues mean that the probability of finding the particle decreases exponentially with time (decay). That is actually an approximation that will violate the QM postulate that evolution shouls be unitary (and probability is not conserved obviously). People who study that in a more rigorous way will go and define somehow bigger Hilbert space that not only include the particle (atom) but will also include the decay products and only when you solve the system with the states of mother plus daughters you will return to your ordinary simple/ish quantum mechanics. The idea is that the decrease of probability of finding the particle will be opposed by increasing probability of finding decay products. So the total probability will be conserved and we will have unitary time operator. Hint: It is not simple as ordinary QM when you sometimes have to worry about resonances, mixed states and modeling these things mathematically is much difficult that solving your ordinary hermitian hamiltonian.