188
7 Some Models of “Quantum Measurement”
If we take the restrictions of these functions to the values of the argument t ≥ 0
according to (7.4.9), we can rewrite (7.4.38a) as a convolution equation, cf. also
(7.4.10)
6 :
F + = F
0
+ − γ
2
g + ∗ f + ∗ F + .
(7.4.38b)
We shall express now the quantities ω
A&B
t
(P m ) in terms of the Sect. 7.4.3, with a
help of (7.4.33):
ω A&B
t
(P m ) = γ 2
t
0
dt
t
0
dt ¯
F(t )F(t )g(t − t ) ¯
f m (t − t ) f m (t − t ), m = 1, 2, . . . .
(7.4.39)
To obtain a similar expression for ω
A&B
t
(P 0 ) we shall use completeness of the set
of projections {P m : m ∈ Z + } in the subspace K, cf. (7.4.29). We can sum over m in
the argument of ω
A&B
t
(·) in (7.4.39) because of normality of the state ω
A&B
t
∈ S(C)
for finite t. After the summation we can perform also lim t→∞ . Summation over m
in (7.4.39) can be performed under the integral signs due to Lebesgue dominated
convergence theorem, cf. the definition of f m . The completeness of the orthonormal
basis { |m| m = 1, 2, . . . } in H A gives also:
∞
m=1
¯
f m (t − t
) f m (t − t
) = f 1 (t
− t
).
(7.4.40)
We then obtain:
ω
A&B
t
(P 0 ) = 1 − γ
2
t
0
dt
t
0
dt
¯
F(t
)F(t
) f (t
− t
).
(7.4.41)
To see the asymptotic properties of ω
A&B
t
(P m ) (t → +∞), we shall need some
properties of the solution F(t) of (7.4.38). We shall obtain them by expressing the
solution of the Volterra equation (7.4.38b) in the form of (Carl) Neumann series
F + =
∞
n=0
(−γ
2
g + ∗ f + ∗)
n F
0
+ ,
(7.4.42)
converging uniformly on any bounded interval for any γ and any continuous f, g.
7
Since the free particle Hamiltonian H B := ˆ
p
2 has an absolute continuous spectrum,
6 Note that, due to time-reflection symmetry of all the systems considered here, quite analogical
equations and the corresponding results could be obtained also for the function t → F(−t), t ≥ 0.
7 To see this, calculate
∞
n=0 (h + ∗) n (t) for h ≡ const.