The Problem of Irreversible Change in Quantum Mechanics

I prove that, if a change happens in a closed quantum system so that its state is perfectly distinguishable from all past or future states, the Hamiltonian is $\widehat{H}=-i\hbar\frac{\partial\ }{\partial\tau}$. A time operator $\widehat{\tau}$ can be defined as its canonical conjugate. This Hamiltonian is usually rejected because it has no ground state, but I show that even a weaker form of irreversibility is inconsistent with a ground state. What is the right choice, that the world's Hamiltonian is $-i\hbar\frac{\partial\ }{\partial\tau}$, or that changes are reversible?