It's fairly well known that "the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space", and that a Hilbert space is basically just the generalisation "from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces". However, the only examples I know of seem to be very low-dimensional: for example, in the DCQE, the state can be described using a basis consisting of just two vectors (eg. "slit A" and "slit B", or else some rotation thereof such as "in phase" and "anti phase", i.e. A+B and A-B). Looking at such examples, I can't see why the description needs to be a Hilbert space. Can someone explain a simple QM example for which an infinite-dimensional space is necessary?