# Descriptions of time evolution: closed vs open systems

1. May 8, 2015

### TrickyDicky

The equivalence between descriptions of time evolution in QM are rigorously defined and proved for conservative systems as explained for instance among many other sources in Jauch's "Foundations of quantum mechanics" in the chapter 10. However, and an exception is the cited reference, it is not usually stressed how this rigorous definition of equivalence refers to closed systems, probably because it is obvious from the postulates of QM that the systems usually described are closed physical systems for instance:
"The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schrodinger equation: $i \hbar \frac{d |\psi \rangle}{d t} = \hat H|\psi \rangle$ "
On the other hand in the nonconservative case with explicitly time dependent Hamiltonian $i \hbar \frac{d |\psi \rangle}{d t} = \hat H(t)|\psi \rangle$ and quoting Jauch: "For such systems it is no longer possible to give a simple expression for the integrated form of the dynamical law, although states at different times are still connected by unitary transformations wich depend on time but wich no longer have the group property. Thus while we can still write $\Psi_t=U_t\Psi$, we must admit that $U_{t1}U_{t2}≠U_{t1}+U_{t2}$."
If we define the equivalence of descriptions of the time evolution in the same way it is done for the conservative case by the presence of the one-parameter group it follows there is no longer rigorous equivalence. This has been obvious for many years. If one decides the one-parameter group property is not relevant to define equivalence then it will follow a different conclusion, as simple as that.
Just for reference here's a couple of peer reviewed references dealing with a different aspect of the equivalence of descriptions of time evolution in quantum theory more centered on QFT, they discuss discrepant results applying different pictures.
A.J. Faria, H.M. Fanca, C. P. Malta, R. C. Sponchiado, Physics Letters A, 305 (2002) 322-328.
P. A. M. Dirac. Physical Review, Vol. 139 (1965) B684 – B690.

I think such an obvious distinction between closed and open systems is often overlooked.

Last edited: May 8, 2015
2. May 13, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. May 14, 2015

### Demystifier

4. May 14, 2015

### TrickyDicky

Yes, in general it isn't. I was dealing with the asymptotic case. As soon as we want to obtain a finite term approximation of the time evolution operator we have a non-unitary perturbative operator in general.
My point is that it is the closed systems that are always in fact some sort of approximation, due to external forces not being constant as seen above or/and back-reaction inherent to any formulation with a cut either system-environment (decoherence) or system-apparatus(Copenhagen).Although in many situations the closed system approximation is very good.
Considering the open or interacting systems as the fundamental ones is much more physically adequate, don't you think?