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)