122 From Classical Mechanics to Quantum Field Theory. A Tutorial
Let us show how the mathematical assumptions (1)-(3) permit us to set the phys-
ical properties of quantum systems (1)-(3) of Section 2.1.1.2 into mathematically
nice form in the general case of an infinite dimesional Hilbert spaceH.
(1) Randomness. The Borel subsetE ⊂σ(A), represents the outcomes of
measurement procedures of the observable associated with the selfadjoint operator
A. (In case of continuous spectrum the outcome of a measurement is at least an
interval in view of the experimental errors.) Given a state represented by the unit
vectorψ∈H, the probability to obtainE⊂σ(A) as an outcome when measuring
Ais
μ(P(A))
ψ,ψ (E):=||P(A)
E ψ||(^2) ,
where we have used the PVMP(A)of the operatorA.
Going along with this interpretation, theexpectation value,〈A〉ψ,ofAwhen
the state is represented by the unit vectorψ∈H, turns out to be
〈A〉ψ:=
∫
σ(A)λdμ(P(A))
ψ,ψ (λ). (2.64)This identity makes sense provided id : σ(A) λ → λ belongs to
L^1 (σ(A),μ(P
(A))
ψ,ψ ) (which is equivalent to say thatψ∈Δ|id|^1 /^2 and, in turn, that
ψ∈D(|A|^1 /^2 )), otherwise the expectation value is not defined.
Since
L^2 (σ(A),μ(P
(A))
ψ,ψ )⊂L(^1) (σ(A),μ(P(A))
ψ,ψ )
becauseμ(P
(A))
ψ,ψ is finite, we have the popular identity arising from (2.48),
〈A〉ψ=〈ψ,Aψ〉 ifψ∈D(A). (2.65)
The associatedstandard deviation,ΔAψ, results to be
ΔA^2 ψ:=
∫
σ(A)(λ−〈A〉ψ)^2 dμ(P(A))
ψ,ψ (λ). (2.66)This definition makes sense providedid∈L^2 (σ(A),μ(P
(A))
ψ,ψ ) (which is equivalent
to say thatψ∈Δidand, in turn, thatψ∈D(A)).
As before, the functional calculus permits us to write the other popular identity
ΔA^2 ψ=〈ψ,A^2 ψ〉−〈ψ,Aψ〉^2 ifψ∈D(A^2 )⊂D(A). (2.67)
We stress that now, Heisenberg inequalities, as established in exercise 2.1.14, are
now completely justified as the reader can easily check.
(3) Collapse of the state. If the Borel setE ⊂σ(A) is the outcome of the
(idealized) measurement ofA, when the state is represented by the unit vector