I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems.
However, one of the axioms in quantum mechanics postulates that the hamiltonian as the generator of the one-parameter unitary group $U(t)$ is not time-dependent, and the evolution of a quantum state should obey the Schroedinger equation with time-independent hamiltonian.
So why can Schroedinger equation with time-dependent hamiltonian be used without hesitation?
Where does the time-dependent hamiltonian come from, except for the case of interaction picture? And should there be an additional axiom for it?
The key is to recognize which are from axiom and which are from model. I foolishly thought that every Shroedinger equation with a time-dependent Hamiltonian can be derived by starting from that strong-continuity axiom in quantum mechanics. Also, I do not think the above comments answered my question.
Best Answer
From the given information it's a little hard to deduce what you want. But say you have a time dependent magnetic field interacting with a system of particles. Then the spin state of the system of the particles will be time dependent. Therefore, the Hamiltonian of the system of particles will be time dependent. Usually a time dependent Hamiltonian comes from a non-conserved system (here we observe only the system of particles interacting with the magnetic field, rather than the whole picture, which would also include what is generating the magnetic field). Since we ignore what is generating the magnetic field, then the system is not conserved therefore the Hamiltonian is often time dependent.
That's why it is used without hesistation. It's all about the context of the problem.