192
7 Some Models of “Quantum Measurement”
Let us take Fourier transform of the (7.4.38b) for F(t) = F ϕ (ψ)(t). We shall use
the notation
8 :
ˆ
F + (u) ≡ F(F ϕ (ψ) + (•))(u) ≡ F(θ · F ϕ (ψ))(u),
(7.4.57a)
and similarly for other functions g + → ˆ
g + , f + → ˆ
f + , or also
F(F
0
+ ) ≡ (F
0
+ )ˆ ≡ ˆ
F
0
+ ≡ F(F
0
ϕ (ψ) + (•)) ≡ F(θ · F
0
ϕ (ψ)).
(7.4.57b)
We obtain then from (7.4.38b) the transformed equation:
ˆ
F + = ˆ
F
0
+ − 2π γ
2
ˆ
g + ˆ
f + ˆ
F + ,
(7.4.58)
which can be solved immediately:
ˆ
F + (u) =
ˆ
F
0
+ (u)
1 + 2π γ 2 ˆ
g + (u) ˆ
f + (u)
, u ∈ R,
(7.4.59a)
or in another form
F(F + )(u) =
F(F
0
+ )(u)
1 + 2π γ 2 F(g + )(u)F( f + )(u)
.
(7.4.59b)
This is the Fourier transform of the explicit expression (7.4.42) of the solution of
(7.4.38) obtained with the help of Carl Neumann series.
Let us rewrite the expression (7.4.56c) for the probability w(ψ) ≡ ≡ψ|W γ |ψ with
the help of (7.4.59) (remember the notation (7.4.6b)):
ψ|W γ |ψ = γ
2
( ˆ
F + , ˆ
F + · ˆ
f ) = γ
2
√
2π
R
du ˆ
f (u)
| ˆ
F
0
+ (u)|
2
|1 + 2π γ 2 ˆ
g + (u) ˆ
f + (u)| 2
.
(7.4.60)
Let us investigate properties of the above integrand in some details. Let us express first
the function ˆ
f + (u) = F( f 1 · g · θ)(u) =
√
2π ˆ
f 1 ∗ ˆ
g + (u) =
√
2πF( f 1 · θ) ∗ ˆ
g(u).
The Fourier image ˆ
f 1 (u) of f 1 (t) ≡
1
t
J 1 (2t) can be obtained with a help of its
integral representation taken from [129, 3.752-2]:
f 1 (t) =
1
t
J 1 (2t) =
4
π
1
0
cos(2t x)
1 − x 2 dx.
(7.4.61)
8 This notation should not be confused with F (F) + := θ · F (F) ≡ ( ˆ
F) + , differing by the place
where the sign “+” occurs.