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?
Best Answer
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.