From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 123

ψ∈H, the new state immediately after the measurement is represented by the
unit vector

ψ′:=

PE(A)ψ

||PE(A)ψ||

. (2.68)

Remark 2.2.67.Obviously this formula does not make sense ifμ(P

(A))
ψ,ψ (E)=0
as expected. Moreover the arbitrary phase affectingψdoes not give rise to troubles
due to the linearity ofPE(A).

(2) Compatible and Incompatible Observables.Two observablesA,Bare
compatible – i.e. they can be simultaneously measured – if and only if their
spectral measures commutewhich means

PE(A)PF(B)=PF(B)PE(A),E∈B(σ(A)),F∈B(σ(B)). (2.69)

In this case,

||PE(A)PF(B)ψ||^2 =||PF(B)PE(A)ψ||^2 =||PE(A,B×F)ψ||^2

whereP(A,B)is the joint spectral measure ofAandB, has the natural inter-
pretation of the probability to obtain the outcomesEandFfor a simultaneous
measurement ofAandB.IfinsteadAandBare incompatible it may happen
that

||PE(A)PF(B)ψ||^2 =||PF(B)PE(A)ψ||^2.

Sticking to the case ofAandBbeing incompatible, exploiting (2.68),

||PE(A)PF(B)ψ||^2 =

∣∣

∣∣

∣

∣∣

∣∣

∣

PE(A)

PF(B)ψ

||PF(B)ψ||

∣∣

∣∣

∣

∣∣

∣∣

∣

2

||PF(B)ψ||^2 (2.70)

has the natural meaning ofthe probability of obtaining firstF and nextEin a
subsequent measurement ofBandA.

Remark 2.2.68.
(a)It is worth stressing that the notion of probability we are using here cannot
be a classical notion because of the presence of incompatible observables. The
theory of conditional probability cannot follows the standard rules. The probability
μψ(EA|FB), that (in a state defined by a unit vectorψ) a certain observableA
takes the valueEAwhen the observableBhas the valueFB, cannot be computed
by the standard procedure

μψ(EA|FB)=

μψ(EAANDFB)

μψ(FB)