FOUNDATIONS OF QUANTUM MECHANICS

III. 4. COMPOSITE SYSTEMS 59
◃ Remark
Leaving out the eigenvalues u i = 0, the eigenvectors |u i ⟩ with eigenvalue 0 do not occur in the
expansion of Tr II W , however, they do belong to the complete basis basis {|u i ⟩}. ▹
Let {|v j ⟩} be a basis in H II . Then {|u i ⟩ ⊗ |v j ⟩} is a basis in H, and |ψ n ⟩ can be expanded as
where
|ψ n ⟩ =
|ϕ n i ⟩ :=
∑N I
∑N II
i=1 j=1
∑N II
j=1
ψ n
ij |u i ⟩ ⊗ |v j ⟩ =
∑N I
i=1
|u i ⟩ ⊗ |ϕ n i ⟩ (III. 79)
ψ n
ij |v j ⟩ ∈ H II . (III. 80)
These |ϕi n ⟩ are, in general, not orthogonal. Substituting (III. 79) in (III. 77) we have
W =
N∑
n=1
p n
∑N I
∑N I
i=1 k=1
|u i ⟩ ⟨u k | ⊗ |ϕ n i ⟩ ⟨ϕ n k |. (III. 81)
Subtitution of (III. 81) in (III. 69) yields
Tr II W =
∑N II
⟨β l | W | β l ⟩ =
l=1
N∑
∑N I
∑N I
n=1 i=1 k=1
∑N II
p n |u i ⟩ ⟨u k | ⟨β l | ϕi n ⟩ ⟨ϕk n | β l ⟩
l=1
=
N∑
∑N I
∑N I
n=1 i=1 k=1
p n ⟨ϕ n k | ϕ n i ⟩ |u i ⟩ ⟨u k |. (III. 82)
With {|ψ i ⟩} a basis, the coefficients in the expansion of an operator of the form ∑ ij c ij|ψ i ⟩⟨ψ j |
are unique, and comparison of (III. 82) with (III. 78) gives
therefore,
N∑
p n ⟨ϕk n | ϕi n ⟩ = u i δ ik , (III. 83)
n=1
Tr II W =
∑N I
∑N I
i=1 k=1
u i δ ik |u i ⟩ ⟨u k | =
∑N I
i=1
u i |u i ⟩ ⟨u i |. (III. 84)
◃ Remark
In (III. 83) it follows for i = k, due to the positivity of the p n , that if u i = 0 for certain i,
then |ϕ n i ⟩ = 0 for all n and we see that in (III. 79) only the terms appear for which u i ≠ 0.
Consequently, the same terms occur in (III. 79) as in the expansion (III. 78) of Tr II W . ▹