In full glory, a dynamical system is defined as a tuple $(T,M,\Phi)$ where $T$ is a monoid, written additively, $M$ is a set, and $\Phi$ is a function. $$ \Phi : U \subset T \times M \to M$$ with

$ I(x) = \{t \in T : (t,x) \in U \}$
$ \Phi(0,x) = x $
$ \Phi(t_2, \Phi(t_1(x)) = \Phi(t_1 + t_2, x)$ for $t_1, t_2,t_1+t_2 \in I(x)$

See this Math stack exchange post.

I am interested in quantum dynamics generated by a Hamiltonian that could be time-dependent, that is, a quantum state in a $d$-dimensional Hilbert space evolving via the Schrodinger equation \begin{align} i \partial_t |\psi(t)\rangle = H(t)|\psi(t)\rangle \end{align} with initial state $|\psi(0)\rangle$. The solution is $|\psi(t)\rangle = U(t) |\psi(0)\rangle$ where $U(t)$ is the unitary time-evolution operator which is formally given as a time-ordered exponential \begin{align} U(t) = \mathcal{T}e^{-i\int_0^t H(t') dt'}. \end{align}

My question is: does this constitute an example of a dynamical system?

My understanding is, if the Hamiltonian is time-independent $H(t) = H$, then yes. This is because $U(t) = e^{-iHt}$ so $U(t+s) = U(t)U(s)$. If the Hamiltonian has non-trivial time-dependence, then no, according to Is the initial value problem of an ODE considered as a dynamic system?. This is because $U(t)$ fails the semi-group property in general. Would my understanding be right?

The reason I am interested in asking this is because I am interested in ergodic theorems (e.g., Birkhoff's, point wise ergodic, Von Neumann ergodic theorems). But it seems to me they all deal with dynamical systems, in particular, with the flow map $\Phi$ satisfying the semigroup property. My next question is: are there any ergodic theorems in the case when the semigroup property does not hold?


1 Answer 1


For a time-dependent Hamiltonian you will need an evolution operator $U(t_2,t_1)= \mathcal{T}e^{-i\int_{t_1}^{t_2} H(t') dt'}$ that depends on initial and final times ($0<t_1<t_2$). With the composition property $U(t_2,t_1)U(t_1,0)=U(t_2,0)$ you can then consider this a non-stationary dynamical system.

The main difference between classical and quantum dynamical systems lies in the fact that the Hilbert space is typically infinite-dimensional. A formulation of quantum dynamical systems based on $C^\ast$ algebras is given by Claude Pillet (see section 4.4).

  • $\begingroup$ thanks. I am interested in the case of the Hilbert space being finite dimensional. Are there ergodic theorems for "non-stationary dynamical systems" then? $\endgroup$
    – nervxxx
    Mar 16 at 8:27
  • $\begingroup$ a typical definition of ergodicity implies stationarity, see for example the discussion at math.stackexchange.com/q/593746/87355 $\endgroup$ Mar 16 at 16:37

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