From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 73

(1) Randomness:The eigenvalues of an observableAare physically interpreted
as the possible values of the outcomes of a measurement ofA.
Given a state, represented by the unit vectorψ∈H, the probability to obtain
a∈σ(A) as an outcome when measuringAis

μ(ψA)(a):=||Pa(A)ψ||^2.

Going along with this interpretation, the expectation value ofA,when the state
is represented byψ, turns out to be

〈A〉ψ:=

∑

a∈σ(A)

aμ(ψA)(a)=〈ψ,Aψ〉.

So that the identity holds

〈A〉ψ=〈ψ,Aψ〉. (2.5)

Finally, the standard deviation ΔAψresults to be

ΔA^2 ψ:=

∑

a∈σ(A)

(a−〈A〉ψ)^2 μ(ψA)(a)=〈ψ,A^2 ψ〉−〈ψ,Aψ〉^2. (2.6)

Remark 2.1.3.
(a)Notice that the arbitrary phase affecting the unit vectorψ∈H(eiaψand
ψrepresent the same quantum state for everya∈R)isarmlesshere.
(b)IfAis an observable andf:R→Ris given,f(A)is interpreted as an
observable whose values aref(a)ifa∈σ(a): Taking (2.4) into account,

f(A):=

∑

a∈σ(A)

f(a)Pa(A). (2.7)

For polynomialsf(x)=

∑n

k=0akx

k,itresultsf(A)=∑n

k=0akA

kas expected.

The selfadjoint operatorA^2 can naturally be interpreted this way as the natural
observable whose values area^2 whena∈σ(A). For this reason, looking at the last
term in (2.6) and taking (2.5) into account,

ΔA^2 ψ=〈A^2 〉ψ−〈A〉^2 ψ=〈(A−〈A〉ψI)^2 〉ψ=〈ψ,(A−〈A〉ψI)^2 ψ〉. (2.8)

(2) Collapse of the State.Ifais the outcome of the (idealized) measurement
ofAwhen the state is represented byψ, the new state immediately after the
measurement is represented by the unit vector

ψ′:= P

a(A)ψ

||Pa(A)ψ||

. (2.9)