Mathematical Foundations of Quantum Mechanics 792.1.5 A first look to the infinite dimensional case, CCR and
quantization procedures
All the described formalism, barring technicalities we shall examine in the rest
of the paper, holds also for quantum systems whose complex vector space of the
states isinfinitedimensional.
To extend the ideas treated in Sect. 2.1.2 to the general case, dropping the
hypothesis thatHis finite dimensional, it seems to be natural to assume thatHis
complete with respect to the norm associated to〈·,·〉. In particular, completeness
assures the existence of spectral decompositions, generalizing (2.4) for instance
when referring tocompactselfadjoint operators (e.g., see[ 5 ]). In other words,H
is acomplex Hilbert space.
The most elementary example of a quantum system described in an infinite di-
mensional Hilbert space is a quantum particle whose position is along the axisR.In
this case, as seen in the first part of this book, the Hilbert space isH:=L^2 (R,dx),
dxdenoting the standard Lebesgue measure onR. States are still represented by
elements ofPH, namely equivalence classes [ψ] of measurable functionsψ:R→C
with unit norm,||[ψ]||=
∫
R|ψ(x)|(^2) dx=1.
Remark 2.1.11.We therefore have heretwoquotient procedures.ψandψ′define
the same element[ψ]ofL^2 (R,dx)iffψ(x)−ψ′(x)=0on a zero Lebesgue measure
set. Two unit vectors[ψ]and[φ]define the same state if[ψ]=eia[φ]for some
a∈R.
Notation 2.1.12.In the rest of the paper, we adopt the standard convention of
many textbooks on functional analysis denoting byψ, instead of [ψ], the elements
of spacesL^2 and tacitly identifying pair of functions which are different on a zero
measure set.
The functionsψdefining (up to zero-measure set and phases) states, are called
wavefunctions. There is a pair of fundamental observables describing our quantum
particle moving inR. One is theposition observable. The corresponding selfadjoint
operator, denoted byX, is defined as follows
(Xψ)(x):=xψ(x),x∈R,ψ∈L^2 (R,dx).
The other observable is the one associated to the momentum and indicated byP.
Restoringfor the occasion, themomentum operatoris
(Pψ)(x):=−i
dψ(x)
dx
,x∈R,ψ∈L^2 (R,dx).
We immediately face several mathematical problems with these, actually quite
naive, definitions. Let us first focus onX. First of all, generallyXψ∈L^2 (R,dx)