From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 75

(c)Ais always compatible with itself. MoreoverPa(A)Pa(A)=Pa(A)just due
to the definition of projector. This fact has the immediate consequence that if
we obtainameasuringAso that the state immediately after the measurement is
represented byψa=||Pa(A)ψ||−^1 ψ,itwillremainψaeven after other subsequent
measurements ofAand the outcome will result to be alwaysa. Versions of this
phenomenon, especially regarding the decay of unstable particles, are experimen-
tally confirmed and it is called thequantum Zeno effect.

Example 2.1.6. An electron admits a triple of observables,Sx,Sy,Sz,known
as the components of thespin. Very roughly speaking, the spin can be viewed
as the angular momentum of the particle referred to a reference frame always at
rest with the centre of the particle and carrying its axes parallel to the ones of the
reference frame of the laboratory, where the electron moves. In view of its peculiar
properties, the spin cannot actually have a complete classical corresponding and
thus that interpretation is untenable. For instance, one cannot “stop” the spin of a
particle or change the constant value ofS^2 =Sx^2 +Sy^2 +S^2 z: It is a given property
oftheparticlelikethemass. Theelectron spin is described within aninternal
Hilbert spaceHs, which has dimension 2. IdentifyingHswithC^2 , the three spin
observables are defined in terms of the three Hermitian matrices (occasionally
re-introducing the constant)

Sx=



2

σx,Sy=



2

σy,Sz=



2

σz, (2.11)

where we have introduced the well knownPauli matrices,

σx=

[

01

10

]

,σy=

[

0 −i

i 0

]

,σz=

[

10

0 − 1

]

. (2.12)

Notice that [Sa,Sb]=0ifa=bso that the components of the spin are incompatible
observables. In fact one has

[Sx,Sy]=iSz

and this identity holds also cyclically permuting the three indices. These com-
mutation relations are the same as for the observablesLx,Ly,Lzdescribing the
angular momentum referred to the laboratory system which have classical corre-
sponding (we shall return on these observables in example 2.3.77). So, differently
from CM, the observables describing the components of the angular momentum
are incompatible, they cannot be measured simultaneously. However the failure of
the compatibility is related to the appearance ofon the right-hand side of

[Lx,Ly]=iLz.