# Hilbert space of smallest dimension

I have read in Quantum mechanics that Hilbert space has infinite dimensions. But is it possible to find smallest dimension of Hilbert space.

• Duplicate on Math.SE: math.stackexchange.com/q/868681 Jul 25, 2021 at 14:21
• The Hilbert space of a qubit has just two complex dimensions. Theoretically you can have a one dimensional Hilbert space, but it would not be physically interesting. Jul 25, 2021 at 15:05

The full Hilbert space can usually be broken up in smaller pieces. For instance in the case of the hydrogen atom, we can break up the system in subspace where states have the same energy, so there is one subspace for each $$n$$, and the size of this subspace is $$n^2$$: the number of states with energy $$-13.6/n^2$$ eV. This subspace can be further broken up into subspaces where all the states have the same $$\ell$$ value. Thus the $$n=2$$ subspace itself contains a subspace of states with $$\ell=0$$ (a single state) and $$\ell=1$$ (3 states).
Depending on the specific calculation at hand, it might be possible to restrict the full Hilbert space to even a subspace of dimension $$1$$.
If you are only interested in specific observables, such as angular momentum or spin, then the full Hilbert space need not be infinite-dimensional. For instance, the full Hilbert space for two particles with spin-1/2 is $$4$$ dimensional, and can be broken into a $$3$$-dimensional subspace and a $$1$$-dimensional subspace.