separable Hilbert space projective space complex projective space square-integrable spin nature momentum complex number
In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector 



ψ


{\displaystyle \psi }

 belonging to a (separable) complex Hilbert space 





H




{\displaystyle {\mathcal {H}}}

. This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys 



⟨
ψ
,
ψ
⟩
=
1


{\displaystyle \langle \psi ,\psi \rangle =1}

, and it is well-defined up to a complex number of modulus 1 (the global phase), that is, 



ψ


{\displaystyle \psi }

 and 




e

i
α


ψ


{\displaystyle e^{i\alpha }\psi }

 represent the same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions 




L

2


(

C

)


{\displaystyle L^{2}(\mathbb {C} )}

, while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors 





C


2




{\displaystyle \mathbb {C} ^{2}}

 with the usual inner product.
