194
7 Some Models of “Quantum Measurement”
This proves, also due to the condition γ g 1 < 2 in (7.4.25), that the denominator
of the integrand in (7.4.60) is everywhere different from zero and finite.
Another part of the integrand in (7.4.60) is the function ˆ
f = F( f 1 · g) =
1
2π
ˆ
f 1 ∗
ˆ
g. The Fourier image of g(t) ≡ ≡ϕ| exp(−it H B )ϕ is
ˆ
g(u) =
1
√
2π
+∞
−∞
e
−itu
ϕ|e
−it H B ϕ dt
=
1
√
2π
+∞
−∞
dt e
−itu
+∞
0
dλ e
−itλ
ϕ| E H B (λ)|ϕ, (7.4.68a)
where E H B (λ) ≡ E H B ((−∞, λ]) is the projection-measure of the selfadjoint operator
H B . Because the spectrum of H B is positive (and absolutely continuous with respect
to Lebesgue measure on R), and the function g(t) is proportional to the Fourier
image of λ → →ϕ|E H B (λ)|ϕ, one has
ˆ
g(u) = θ(−u)F(g)(u) =
√
2πϕ|E H B (−u)|ϕ.
(7.4.68b)
This can be rewritten in the “p-representation”, which allows us to see better the
dependence on the specific functions ϕ. We shall write the element of the solid angle
φ in terms of the Euler angles θ, ϕ in R
3 as dφ := sin θ dθ dϕ, and the function
ˆ
ϕ(p) ≡ ˆ
ϕ( p, φ) (p := |p|). It is
g(t) = =ϕ|e
−it H B ϕ =
R 3
d
3 p ˆ
ϕ(p)e
−it p
2 ˆ
ϕ(p)
=
+∞
0
d p p
2 e
−it p
2
4π
dφ | ˆ
ϕ( p, φ)|
2
, (7.4.68c)
which, after the change of variables λ := p
2 , leads to
g(t) =
1
2
+∞
0
dλ
√ λe
−itλ
4π
dφ | ˆ
ϕ(
√ λ, φ)|
2
;
(7.4.68d)
this has the form of the Fourier image of
F
−1
(g)(λ) := θ(λ)
π
2
√ λ
4π
dφ | ˆ
ϕ(
√
λ, φ)|
2
,
(7.4.68e)
and the Fourier image ˆ
g has now the form
ˆ
g(u) = ˆ
g(u) θ(−u) = F
−1
(g)(−u) = θ(−u)
π
2
√
−u
4π
dφ | ˆ
ϕ(
√
−u, φ)|
2
.
(7.4.68f)