Infinite-dimensional Hilbert spaces in physical systems

Question

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?

Answer 1

Isn't your Ex 3 such an example?

In any event, you might have trouble producing a pure photon state $\left|\,001\right>$, but you should have no trouble creating a coherent state. Just turn on your laser pointer. The coherent state is a superposition of single photon states, summing over all single photon number states. You can't impose a cutoff.

See this Wikipedia article, in particular search for the phrase "This can be easily seen" and look at the equation that follows.

Answer 2

They are much more physical systems than you can imagine that have solutions in a Hilbert space. Most common are physical systems which are classically described by a periodic unknown function. The unknown function is developed in a Fourier series whose members are trigonometric functions that form an orthonormal complete base in a Hilbert space.

Another example is a classical distribution of particles $u(r,\varphi)$ in phase space that has circular symmetry. The dynamics of such a distribution is described by the Vlasov equation. The $\varphi$-dependence can be described by the functions $e^{im\varphi}$ with $m=-\infty,\ldots-1,0,1,\ldots\infty$. The $r$-dependence is non-periodic and a priori unknown, but can be developed in a series of Laguerre-polynomials. A scalar product can be defined. The Laguerre-polynomials form a orthonormal complete infinite basis on a vector space, which has with the scalar product and completeness all characteristics of a Hilbert space. Such type of problems appear in physics everywhere where there is a differential equation whose solution is not right obvious to be guessed, so that the solution has to be searched in a larger function space, with a scalar product in a Hilbert space. (and the Schroedinger equation is just a example as many other classical examples.)

And in order to have complete and correct description all found modes have to be considered.