This is soft question in the sense that I am looking to understand/motivate the naming behind the "infinitesimal generator" of a continuous semigroup. Namely, if we have a family of operators $\{U_t\}_{t \geq 0}$ s.t. $U_0 = I, U_{t+s} = U_t\circ U_s$ and the family is at least strongly continuous over a Hilbert/Banach space, why is the mapping $H\psi \equiv \lim_{t\to 0}\frac{i}{t}(U_t\psi – \psi)$ (in the quantum setting) called an infinitesimal generator? What, if anything, does it "generate"?
Understanding motivation/background for the name “infinitesimal generator” of a continuous semigroup
functional-analysisoperator-theorysemigroup-of-operatorssoft-questionterminology
Best Answer
My answer applies to this quantum mechanical setting as well as to basically all linear (autonomous) differential equations (so the $-i$ in front of $H$ does not really make a major difference from the heuristical point of view). Much of what I am writing is formal, and can be made rigorous depending on the context, but it is really not essential to focus on this to begin answering your question.
As from another answer, $H$ is the operator such that $U_t=e^{-itH}$. In other words, if you take $\psi(t)=U_t\psi_0$, that is supposed to be the evolution of the state in your physical system, then $\psi(t)$ is a solution to the differential equation $$ \left\{\begin{aligned} & \frac{d}{dt}\psi(t)=-iH\psi(t), \\ &\psi(0)=\psi_0 \end{aligned}\right. $$ In this setting, $H$ is the ‘vector field' that is telling you how $\psi$ is evolving in time. It tells you the law that governs the evolution of the state $\psi$ over time, and in principle uniquely determines all the future states $\psi(t)$, $t>0$ given the initial datum $\psi_0$. In this sense (to answer your question) it generates the flow $U_t$.
It is called "infinitesimal" because it only tells you what is the time derivative of your solution, and to recover the actual flow $U_t$ even for short times, you need to solve a differential equation. In the same way, for instance, the gravitational force between planets is the only thing that governs how planets move over time, but it is a long way from there to showing that planets' orbits are ellipses (to be fair, this specific example is actually a not very long exercise; but for a generic differential equation it is not possible in general to derive an explicit, exact formula for the solutions, and one simply proves that a solution exists and tries to study it qualitatively rather than quantitavely).
Heuristically, if you choose a very small time $h>0$, writing the first order Taylor expansion of your group, that is, $$ e^{-ihH}\approx \mathbb 1 -ihH, $$ you see that if your state at time $t=0$ is $\psi_0$, then $\psi(t)$ at time $t=h$ is approximately $$ \psi(h)\approx \psi_0-ihH\psi_0. $$ So that, if you approximate the evolution of $\psi$ with discrete time steps, the difference between one step and the next one is $$ \psi(t+h)-\psi(t)\approx-ihH\psi(t), $$ Which means that infinitesimally (that is, at leading order), the operator $-iH$ is what tells you the difference between the state in the near future and the present state. It tells you how the state changes from the present moment to the 'next' time instant... Except there is no 'next' instant, as $t$ is a continuous variable (unlike other examples of deterministic dynamical systems which evolve in discrete time steps; think for instance about how a computer works, with an internal clock that makes the state of the internal memory change over time with a very high, but still finite frequency). So, $-iH$ is nothing but the "velocity" operator, that tells you how fast and in which direction your state is moving depending on the current position (note that "velocity" and "direction" have an abstract sense, and are meant to refer to the vector space in which your state $\psi$ lies, not e.g. to the physical 3D space or the like).