From Classical Mechanics to Quantum Field Theory

116 From Classical Mechanics to Quantum Field Theory. A Tutorial

whereμ(ψ,ψP)(E)=〈ψ,PEψ〉=

∫

E×Rn−^1 |ψ(x)|

(^2) dnx. (2) in exercise 2.2.49 proves
thatX 1 =

∫

RλdP(λ) and thus (2.57) holds true.

(2)Considering that them-axis momentum operatorPminL^2 (Rn,dnx)intro-
duced in (2) of example 2.2.39, taking (2.36) into account whereFˆ(and thusFˆ†)
is unitary, in view of (i) in Proposition 2.2.66 we immediately have that the PVM
ofPmis

Q(EPm):=Fˆ†PE(Km)Fˆ.

Above,Kmis the operator andXmrepresented inL^2 (Rn,dnk) as in (1) of example
2.2.39.

(3)More complicated cases exist. Considering an operator of the form

H:=

1

2 m

P^2 +U

whereP is the momentum operator inL^2 (R,dx),m>0isaconstantandUis a
real valued function onRused as a multiplicative operator. IfU=U 1 +U 2 with
U 1 ∈L^2 (R,dx)andU 2 ∈L∞(R,dx) real valued, andD(H)=C∞(R;C),Hturns
out to be (trivially) symmetric but also essentially selfadjoint[ 5 ]as a consequence
of a well known result (Kato-Rellich’s theorem). The unique selfadjoint extension
H =(H†)†ofHphysically represent the Hamiltonian operator of a quantum
particle living alongRwith a potential energy described byU.Inthiscase,
generally speaking,σ(H) has both point and continuous part.

∫

σp(H)λdP

(H)(λ)

has a form like this ∫

σp(H)

λdP(H)(λ)=

∑

λ∈σp(H)

λPλ

wherePλis the orthogonal projector onto the eigenspace ofHwith eigenvalue
λ.Conversely,

∫

σc(H)λdP

(H)(λ) has an expression much more complicated and,

under a unitary transform, is similar to the integral decomposition ofX.

Remark 2.2.60.
(a)It is worth stressing that the notion (2.56) of a function of a selfadjoint
operator is just an extension of the analogous notion introduced for the finite di-
mensional case (2.7) and thus may be used in QM applications.
It is possible to prove that iff:σ(A)→Ris continuous, then
σ(f(A)) =f(σ(A)) (2.58)

where the bar denotes the closure and, iff:σ(A)→Ris measurable,

σp(f(A))⊃f(σp(A)). (2.59)