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I'm learning infinite-dimensional systems in mathematical viewpoint and trying to understand it from physical perspective.

I would like to understand if infinite-dimensional systems make sense in physics, especially when it becomes necessary in quantum control theory. Are there any simple and intuitive examples?

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I do not know much about quantum control theory, but I can give you a simple example from regular quantum mechanics: that of a particle in a box. This is one of the simplest systems one can study in QM, but even here an infinite dimensional space shows up.

Indexing the energy eigenstates by $n$ so that $$H|nrangle=E_n|nrangle$$ there is an infinite number of possible states, one for every integer. Every state with its own energy: $$E_n=fracn^2pi^2hbar^22mL^2.$$ Thus if you want to describe a general quantum state in this system you would write it down as $$|psirangle=sum_n=1^infty c_n|nrangle,$$ where $c_n$ is a complex number. $|psirangle$ is then an example of a vector in an infinite dimensional space where every possible state is a basis vector, and the $c_n$ are the expansion coefficients in that basis.

You can of course imagine the $n$ indexing some other collection of states of some other system. Indeed, in most cases the dimension of the space of all possible states of a quantum system will be infinite dimensional.

"Infinite" for me does not make much sense in physics. It is a nice mathematical tools, you need it to make calculations; but for real things I prefer "very large". I have seen many "very large" things; I have never seen anything infinite.

Anyway, more seriously, we use all the time infinite dimensional spaces, they are useful. From the example of "particle in a box" by >JSorngard:

You want to keep your particle in one eigenstate, against external disturbance. It can excape going to other eigenstates. Those, in theory, are infinite, so to calculate the probability for your particle to fall off from your favourite eigenstate, you sum the transition probability to each of all those (infinite) states. And it works!!

In practice there are not really infinite eigenstates; the particle is confined by some actual physical trap that is finite in size and can hold only finite energy. But incredibly often you can disregard this finiteness, as the infinite sum is almost identical to the (very large) sum of the actual eigenstate.

Another argument in favour of usefulness/reality of infinites is about notation: we use infinite things to define & manipulate a normal object.
Example is the Taylor expansion of $e^x$:
It is an infinite sum, its useful, don't give rise to anything nonsensical.