FOUNDATIONS OF QUANTUM MECHANICS

28 CHAPTER II. THE FORMALISM
for which, with (II. 64), we have
χ ak (A) : =
M∑
χ ak (a i ) P ai = P ak , (II. 66)
i=1
and we see that the projectors from the spectral decomposition of A, (II. 63), are functions of A.
We use the spectral decompositions in the proof of the following theorem.
THEOREM:
If two self - adjoint operators A and B commute, there is a maximal, self - adjoint operator
C of which both A and B are a function.
To prove this theorem we first prove a useful lemma.
LEMMA:
If [A, B] = 0, a basis {|γ i ⟩} exists in which A and B are simultaneously diagonal.
Proof
Let {|a i , j⟩} be an orthonormal eigenbasis of operator A, where j = 1, . . . , n i is the degeneracy
of eigenvalue a i , and we have
⟨a p , q | a i , j⟩ = δ pi δ qj . (II. 67)
Analogously, let there be an orthonormal eigenbasis {|b k , l⟩} for operator B. From [A, B] = 0
and (II. 63) it follows that
A ( B |a i , j⟩ ) = B A |a i , j⟩ = a i B |a i , j⟩, (II. 68)
and B |a i , j⟩ is, apparently, an eigenvector of A with the eigenvalue a i , i.e., B |a i , j⟩ is in the
eigenspace spanned by |a i , 1⟩, . . . , |a i , n i ⟩. Or, equivalently,
B |a i , j⟩ =
∑n i
k=1
holds for certain numbers Λ [i]
j,k ∈ C.
Λ [i]
j,k |a i, k⟩ (II. 69)
By assmuptionion, B is self - adjoint and therefore the matrix Λ [i] must be Hermitian,
and we see that
⟨a k , l | B | a i , j⟩ = Λ [i]
l,j δ ki = Λ [i]
l,j
, (II. 70)
⟨a k , l | B | a i , j⟩ ∗ = Λ [i] ∗
l,j = ⟨ai , j | B † | a k , l⟩ = Λ [k]
j,l δ ik = Λ [i]
j,l
, (II. 71)
Λ [i]
l,j
∗ [i] = Λ
j,l
. (II. 72)