186
7 Some Models of “Quantum Measurement”
The probability of the detection w(ψ) is here positive: w(ψ) > 0, and, moreover,
it depends on the initial state ψ of the particle as:
ψ → →ψ|W |ψ ≡ ≡ψ|W γ |ψ ≡ w(ψ),
(7.4.27)
where W ≡ W γ ∈ L(H B ) is a positive operator 0 < W γ < I H B , independent of ψ.
Moreover, for sufficiently small nonzero interaction constants γ ∈ [−γ 0 , γ 0 ] ⊂ R it
is W
2
γ = W γ , hence W γ is not a projector.
Proof. Let us use the notation introduced in 7.4.3. We want to prove the existence of
the limit (7.4.26) first. Let the state-vectors of the chain |m, m = 0, 1, 2, . . . ∞ be
defined as in (7.3.12) with |0 := 0 . The Hilbert subspace K ≡ (H A ⊕ H {∅} ) ⊗
H B of the state-space of the (initial-state representation of the) compound
system “the spin half-chain & the particle” generated by vectors |m⊗|ψ (m =
0, 1, . . .), ψ ∈ H B , is H -invariant, hence also invariant with respect to the time
evolution U t ≡ exp[−it (H A + H B + γV ϕ )]. Let P K be the orthogonal projector onto
K. Let us define the partial isometries P nm in K by
P nm |k ⊗ |ψ = δ mk |n ⊗ |ψ, forall ψ ∈ H B , n, m, k = 0, 1, 2, . . . (7.4.28)
Let P n := P nn , ∀n. Denote also by P ψ , ψ ∈ H B (ψ 2 = 1), the one dimensional
projector |ψψ| in H B . Clearly P 0
ψ
0 =
ψ
0 ≡ |0 ⊗ |ψ, and for all k, l, m, n ∈ Z +
it is
P
∗
nm = P mn , P nk P lm = δ kl P nm ,
∞
m=0
P m = P K .
(7.4.29)
We shall write elements x = a ⊗ b ∈ A ⊗ B as x = ab (hence also a ≡ a ⊗
I H B , b ≡ I A ⊗ b), if a confusion would be improbable. So, we are looking for limits
ω(x) := lim
t→∞
ω
A&B
t
(x), x ∈ A ⊗ B = C.
(7.4.30)
We shall see that the limits (7.4.30) for x ∈ A ⊂ C are expressible in terms of
ω(P mn ). The very well known Dyson equation (7.4.31) expressing the unitary evolution group U t = exp[−it (H A + H B + γV ϕ )] of a system with the interaction γV ϕ in
terms of this interaction and of the free system (without interaction) evolution group
U
0
t (t ∈ R):
U t = U
0
t − iγ
t
0
dτU
0
t−τ V ϕ U τ ,
(7.4.31)
7 Some Models of “Quantum Measurement”
The probability of the detection w(ψ) is here positive: w(ψ) > 0, and, moreover,
it depends on the initial state ψ of the particle as:
ψ → →ψ|W |ψ ≡ ≡ψ|W γ |ψ ≡ w(ψ),
(7.4.27)
where W ≡ W γ ∈ L(H B ) is a positive operator 0 < W γ < I H B , independent of ψ.
Moreover, for sufficiently small nonzero interaction constants γ ∈ [−γ 0 , γ 0 ] ⊂ R it
is W
2
γ = W γ , hence W γ is not a projector.
Proof. Let us use the notation introduced in 7.4.3. We want to prove the existence of
the limit (7.4.26) first. Let the state-vectors of the chain |m, m = 0, 1, 2, . . . ∞ be
defined as in (7.3.12) with |0 := 0 . The Hilbert subspace K ≡ (H A ⊕ H {∅} ) ⊗
H B of the state-space of the (initial-state representation of the) compound
system “the spin half-chain & the particle” generated by vectors |m⊗|ψ (m =
0, 1, . . .), ψ ∈ H B , is H -invariant, hence also invariant with respect to the time
evolution U t ≡ exp[−it (H A + H B + γV ϕ )]. Let P K be the orthogonal projector onto
K. Let us define the partial isometries P nm in K by
P nm |k ⊗ |ψ = δ mk |n ⊗ |ψ, forall ψ ∈ H B , n, m, k = 0, 1, 2, . . . (7.4.28)
Let P n := P nn , ∀n. Denote also by P ψ , ψ ∈ H B (ψ 2 = 1), the one dimensional
projector |ψψ| in H B . Clearly P 0
ψ
0 =
ψ
0 ≡ |0 ⊗ |ψ, and for all k, l, m, n ∈ Z +
it is
P
∗
nm = P mn , P nk P lm = δ kl P nm ,
∞
m=0
P m = P K .
(7.4.29)
We shall write elements x = a ⊗ b ∈ A ⊗ B as x = ab (hence also a ≡ a ⊗
I H B , b ≡ I A ⊗ b), if a confusion would be improbable. So, we are looking for limits
ω(x) := lim
t→∞
ω
A&B
t
(x), x ∈ A ⊗ B = C.
(7.4.30)
We shall see that the limits (7.4.30) for x ∈ A ⊂ C are expressible in terms of
ω(P mn ). The very well known Dyson equation (7.4.31) expressing the unitary evolution group U t = exp[−it (H A + H B + γV ϕ )] of a system with the interaction γV ϕ in
terms of this interaction and of the free system (without interaction) evolution group
U
0
t (t ∈ R):
U t = U
0
t − iγ
t
0
dτU
0
t−τ V ϕ U τ ,
(7.4.31)