Here we consider the role of [Hilbert space dimension](Hilbert%20space%20dimension.md) in the formulation of quantum theory. While we may model many quantum phenomena in [finite dimensional Hilbert spaces](Finite%20dimensional%20Hilbert%20spaces.md) descriptions of [quantum systems](Quantum%20systems.md) that account for all degrees of freedom for any real world system must be [infinite dimensional.](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md#Infinite%20dimensional%20Hilbert%20spaces%20in%20quantum%20mechanics) ^2c3a00 %%One way of showing this is with the commutation relation between position and momenta operators which give rise to the Heisenberg algebra. but this is universally extendable to all quantum systems?%% # Finite dimensional Hilbert spaces in quantum mechanics ## 2 Dimensional Hilbert spaces in quantum mechanics The lowest dimensional [Hilbert space](Hilbert%20Spaces%20in%20Quantum%20Mechanics.md) that describes any [quantum mechanical system](Quantum%20systems.md) that may form a [superposition](Quantum%20superposition.md) is a [2 dimensional Hilbert space.](2%20dimensional%20Hilbert%20space.md) Such a space would be used to model [two-level systems.](Two-Level%20Systems.md) ^be30c4 # Infinite dimensional Hilbert spaces in quantum mechanics #QuantumMechanics/MathematicalFoundations