QuantumPhysics.dvi

compatible observables, [px,py] = 0, while the spin componentsSxandSyare not, [Sx,Sy] =

i ̄hSz. The role played by compatibility of observables during measurementis expressed by

the following,

Theorem 3

(i) Two self-adjoint operatorsAandB, which satisfy[A,B] = 0, may be diagonalized in the

same basis, i.e. with common eigenspaces;

(ii) Two compatible observables may be observed simultaneously.

Proof (i) SinceAis self-adjoint, we may decompose it in a sum of projection operators,

A=

∑

i

aiPi

∑

i

Pi=IH (4.9)

where by assumption, the eigenvalues satisfyai 6 =aj wheni 6 =j. By multiplyingBto the

left and to the right by the identity operatorIH, we also have

B=

∑

i,j

PiBPj (4.10)

The commutator relation is easily computed in this basis,

0 = [A,B] =

∑

i,j

(ai−aj)PiBPj (4.11)

which implies thatPiBPj= 0 wheneveri 6 =j, and as a result

B=

∑

i

Bi Bi=PiBPi (4.12)

Inside the eigenspaceEi, the projection operatorPireduces to the identity operatorIi(which

manifestly commutes withBi). SinceBiis self-adjoint, it may be written as a direct sum of

projection operators weighted by its eigenvalues,

Bi=

∑

mi

bi,miPi,mi

∑

mi

Pi,mi=Ii (4.13)

wheremiis an index for matrixBiwhich labels all the distinct eigenvaluesbimiofBi. The

decomposition of both operators is then given by

A=

∑

i

∑

mi

aiPi,mi B=

∑

i

∑

mi

bi,miPi,mi (4.14)

(ii) It is manifest that an observation ofAwill produce an eigenstate ofA|φi〉with eigenvalue

ai, which with probability 1 will produce one of the eigenstates ofB in sectori, so that

simultaneous measurements ofAandBcan indeed be made.