Can anyone give an example of when infinite-dimensional Hilbert spaces are required to describe a physical system? The standard answer to this question is yes, and I'm sure some of you will be quick to point out several clear examples:
Ex. 1: Photon number states $\left\{ |0\rangle,|1\rangle,...,|n\rangle \right\}$
Ex. 2: Harmonic oscillator number states (same as above).
Ex. 3: Continuous-variable basis of a single free particle $\left\{ |x\rangle \right\} \forall x \in \mathbb{R}$
From the infinite number of basis vectors in these examples, it is generally concluded that the dimension of the Hilbert space is also infinite.
However, it is clear that a photon number state such as $|\psi\rangle=|100\rangle$ is unphysical (to be clear, this is the Fock number state with $n=100$, not a tripartite state with one photon in one channel). If you dispute this, I would suggest you try preparing such a state in a single-mode fiber in an optics lab. For this reason, we can apply a (somewhat arbitrary) cutoff value of $n$ and recover a finite-dimensional Hilbert space.
Similarly, the cardinality of the basis set for a free particle is actually $\mathbb{N}$, not $\mathbb{R}$ (due to the properties of the function space), and although this set of orthogonal functions is infinite, we can also apply a similar cutoff for any reasonably behaved wavefunction.
Can anyone give an example of a system described by an infinite-dimensional vector space for which such a cutoff cannot be applied?
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