# Physical meaning of time-varying Hamiltonian in Quantum Mechanics

I'm a self-taught in Quantum Mechanics, with the aim to understand Quantum Information theory. I have the following doubt which I cannot solve:

Assuming as a postulate that the evolution of a quantum system is governed by a unitary operator: $|\psi(t)\rangle = U(t,t_0)|\psi(t_0)\rangle$

The Schrodinger equation could be derived: $i\hbar\frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle$. Where $H(t$) is the Hamiltonian operator since it could be seen as the "total energy of the system". When it is time-independent, the justification is physically reasonable since it is a conserved quantity and the system is closed (no problem here). If it is not the case, according to Dirac textbook (pg.110), the system is "open":

If the energy depends on t, it means the system is acted on by external forces.

In my opinion, this assumption is also reasonable, according to the energy conservation principle.

My doubts arise from the fact that different textbooks (e.g. Nielsen-Chuang) states that:

[If the Hamiltonian is time-variant] The system is not, therefore, closed, but it does evolve according to Schrodinger’s equation with a time-varying Hamiltonian, to some good approximation.

Or they make the assumption that the "evolution postulate" is true iff the system is closed.

I can't really take the physical insight behind that. According to such a version, it seems that the Schrodinger equation is not universal or, in some sense, imprecise. This raises some questions to me: What is the correct version of the "evolution postulate"? Does it predict the evolution of any quantum system or only of the closed ones? Why a time-varying Hamiltonian does not describe the real evolution of the system?

• Basically your doubt is generated by the bold text? Just for me to understand.... – Alchimista Aug 20 '17 at 15:38
• Yes, they are the "basic" points which generate my doubts – steg Aug 20 '17 at 16:44