# 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.... Aug 20, 2017 at 15:38
• Yes, they are the "basic" points which generate my doubts
– steg
Aug 20, 2017 at 16:44

The basic issue with time-varying external forces is that they appear in the quantum theory as classical parameters, i.e. they have no Hamiltonian dynamics of their own and no quantum fluctuations. If one takes quantum theory as fundamental, then the appearance of such classical parameters can only be regarded as an approximation (albeit a very good one, in many cases). In general, this approximation consists of neglecting the back-action of the quantum system on whatever external system is producing the effective time-varying potential.

For example, if one considers a laser driving electronic transitions in an atom, the dynamics of the atom alone can be modelled to a very good approximation by considering a time-dependent classical electric field coupled to the electron dipole moment. This provides a great simplification since one does not need to explicitly model the dynamics of the laser field. However, this is only an approximation because spontaneous emission from the atom into the laser mode is neglected. Nevertheless, the photon flux in a typical laser beam is very high in comparison to the spontaneous emission rate. Therefore, the effect of the spontaneous emission on the light field is negligible, while its effect on the atom is relatively small and can be modelled using, for example, perturbation theory.

• Nice and clear example. Does this mean that we cannot predict (exactly) what happens whenever the quantum system "is open"? But, in principle, the Schrodinger equation is right, so it seems that the root of such "errors" is an incomplete knowledge of the Hamiltonian, is that right? I think that this reasoning is not straightforward for a noob in quantum mechanics and should be commented more on textbooks. Do you know some texts that give some insight behind such phenomena?
– steg
Aug 20, 2017 at 20:30
• I understand and agree with the above answer. A further point came to my mind? Is it not that "some approximation" is as well inherent with n-bodies, n>2 systems? That sentence "with some approximation" could be dropped almost everywhere. Aug 20, 2017 at 20:57
• Alchimista, I can't understand on which basis this statement is true, because it really seems that the Schrodinger equation "is wrong" in some sense. According to Mark's answer, it seems more like "we cannot have a good description of the time-varying Hamiltonian" (Implicity assuming that Schrodinger Equation is "right"). Am I wrong?
– steg
Aug 21, 2017 at 7:00
• @Stefano The Schroedinger equation is not wrong, but it strictly applies only to closed systems. If the system is open, then that means that there is an external environment with which the open system interacts and thus becomes correlated. In general, the Schroedinger equation must then be solved for the joint state describing system and environment together. However, for certain Hamiltonians and initial quantum states of the environment, system-environment correlations are negligible and the open system's dynamics can be well described by an effective (possibly time-dependent) Hamiltonian. Aug 21, 2017 at 12:19
• @MarkMitchison That's sounds good. Thank you so much, I think I've finally understood the point here. The interesting thing is that all textbook I've read does not make the point on such a question, and they leave the argument. But I think that the reasoning you made is not so straightforward for a "noob" in quantum mechanics.
– steg
Aug 21, 2017 at 12:57