From Classical Mechanics to Quantum Field Theory

74 From Classical Mechanics to Quantum Field Theory. A Tutorial

Remark 2.1.4. Obviously this formula does not make sense ifμ(ψA)(a)=0as
expected. Moreover the arbitrary phase affectingψdoes not lead to troubles, due
to the linearity ofPa(A).

(3) Compatible and Incompatible Observables. Two observables are com-
patible – i.e. they can be simultaneously measured – if and only if the associated
operatorscommute,thatis

AB−BA=0.

Using the fact thatHhas finite dimension, one easily proves that the observables
AandBare compatible if and only if the associated spectral projectors commute
as well

Pa(A)Pb(B)=Pb(B)Pa(A) a∈σ(A),b∈σ(B).

In this case,

||Pa(A)Pb(B)ψ||^2 =||Pb(B)Pa(A)ψ||^2

has the natural interpretation of the probability to obtain the outcomesaandb
for a simultaneous measurement ofAandB.IfinsteadAandBare incompatible,
it may happen that

||Pa(A)Pb(B)ψ||^2 =||Pb(B)Pa(A)ψ||^2.

Sticking to the case ofAandBincompatible, exploiting (2.9),

||Pa(A)Pb(B)ψ||^2 =

∣∣

∣∣

∣

∣∣

∣∣

∣P

(A)

a

Pb(B)ψ

||Pb(B)ψ||

∣∣

∣∣

∣

∣∣

∣∣

∣

2

||Pb(B)ψ||^2 (2.10)

has the natural meaning ofthe probability of obtaining firstband nextain a
subsequent measurement ofBandA.

Remark 2.1.5.
(a) Notice that, in general, we cannot interchange the rˆole ofA andBin

(2.10) because, in general, Pa(A)Pb(B)=Pb(B)Pa(A)ifAandBare incompatible.
The measurement procedures “disturb each other” as already said.
(b)The interpretation of (2.10) as a probability of subsequent measurements
can be given also ifAandBare compatible. In this case, the probability of ob-
taining firstband nextain a subsequent measurement ofBandAis identical
to the probability of measuringaandbsimultaneously and, in turn, it coincides
with the probability of obtaining firstaand nextbin a subsequent measurement of
AandB.