Many questions in StackExchange and papers that I see consider the following von Neumann equation for describing evolution of density matrix $\rho$ in time $t$ given a Hamiltonian $H$:

$$ \frac{d\rho}{dt} = i\hbar [H (t), \rho] \;.$$

Here, both state $\rho$ and observable $H$ are assumed to depend on parameter which we call time.

I am confused here in the following way: the material that I used to learn quantum mechanics talks about two pictures (Schrodinger and Heisenberg) where one allows evolution on states only and the only one allows evolution on observables only. Then, there seems to be contradiction in the previously defined evolution.

I give the definition I use that can be found here (https://www.ithaca.edu/hs/depts/math/docs/theses/mastroenithesis.pdf) and is basically taken from L. Takhtajan "Quantum Mechanics for Mathematicians" by making minor changes of language.

Axiom on evolution: There exists a strongly continuous one parameter unitary group $\{U(t)\}$ on Hilbert space $\mathscr{H}$ and a pair of bijections for each $t \in \mathbb{R}$ both denoted by $U_t$ such that :

$$ U_t : \text{Space Of Observables} \to \text{Space Of Observables}, U_t(A) = A(t) = A $$ and $$ U_t : \text{Space Of States} \to \text{Space Of States}, U_t(M) = M(t) = U(t) M U(t)^{-1} \;.$$

For the infinitesimal generator $H$ of $\{U(t)\}$:

$$ \frac{dM}{dt} = i\hbar[H,M] \;.$$

Therefore, by this axiom, it is defined that $H$ cannot depend on parameter $t$.

Question 1: Maybe problem is with axiom? If yes, then what is it?

Question 2: I assume that solution for my question could be related that this axiom (even though it is not mentioned explicitly) assumes that this kind of evolution happens only for closed systems.

When, for example, I want to consider how my system interacts with time-dependent field then of course by constructing such model I assume that there is some interaction with outside world and, therefore, the system is not closed. Then, the evolution given by axiom does not hold and one can somehow argue that the one given at the beginning of the question when Hamiltonian depends on time is some "effective" time evolution.

If this is the case then can someone give derivation of situation where one gets time-dependent Hamiltonian dynamics represented by the first equation from axiom and some assumptions on system + environment? I assume that it can be done using partial trace and some conditions on being weakly interacting (just like it is in the case of Hamiltonian classical mechanics).

Unfortunately, I have not found such derivation or some argument. The only thing I have found is the Markov approximation and Lindblad operator that is related to the fact that system is open. But there, Lindblad operator is added to already equation which is the first equation in this question.

I would appreciate any sort of comments and opinions!

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    $\begingroup$ I've take the liberty of introducing some minor "improvements" to your markup, though I decided against removing the camel-case when setting the text as text. Feel free to roll them back if you feel they don't constitute an improvement. $\endgroup$ – dmckee --- ex-moderator kitten May 6 '18 at 23:25
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    $\begingroup$ It sounds like your main confusion is related to a time-dependent version of the Heisenberg picture. This question might help: physics.stackexchange.com/questions/333024/… $\endgroup$ – probably_someone May 7 '18 at 0:18

Your axioms only work for time-independent Hamiltonians. If the Hamiltonian depends on time, the time-evolution operator will no longer be a one-parameter group $U(t)$, but rather a two-parameter family of unitary operators $U(t,s)$ that satisfies $U(t,s)U(s,r)=U(t,r)$. You can find more on this, for example, in Reed Simon "Methods of Mathematical Physics", vol. II chap. X.12 "time-dependent Hamiltonians".


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