TrickyDicky
- 3,507
- 28
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 Schrödinger 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 which depend on time but which 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.
"The evolution of a closed system is unitary (reversible). The evolution is given by the time-dependent Schrödinger 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 which depend on time but which 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: