7.4 Particle Detection—A “Nonideal” Measurement
181
2. Let us introduce also the symbol H A for the Hilbert (sub-)space of the chain
generated by the vectors {|m | m = 1, 2, . . . } introduced in (7.3.12). We shall
use also: U t := exp(−it H) with H from (7.4.5), and τ t c := e
it H c e
−it H for
c ∈ C. The vector 0 = |0 is defined in (7.3.2) and (7.3.3). We shall also use
χ
0 := 0 ⊗ χ, χ ∈ H B .
3. Let ϕ ∈ H B , ϕ 2 = 1, be the vector appearing in the interaction Hamiltonian
V ϕ in (7.4.4), and let ψ ∈ H B be the (also normalized) initial state-vector of the
particle. We shall introduce the symbols F
0
(t), g(t) , and F(t) as:
F
0
(t) ≡ F
0
ϕ (ψ)(t) := =ϕ|e
−it H B |ψ, g(t) := F
0
ϕ (ϕ)(t) ≡ ≡ϕ|e
−it H B |ϕ.
(7.4.6a)
F(t) ≡ F ϕ (ψ)(t) := =ϕ| ⊗ ⊗0|e
−it H
|0 ⊗ |ψ ≡ ≡
ϕ
0 |e
−it H
|
ψ
0 ,
(7.4.6b)
where H := H A + H B + γV ϕ is the total Hamiltonian of the compound system
(7.4.5).
The symbols f m (t), f (t) will be also useful abbreviations (cf. (7.3.24)):
f m (t) := =m|e
−it H A |1 = (−i)
m−1 m
t
J m (2t), m = 1, 2, . . . (7.4.7)
f (t) := g(t) f 1 (t) = =ϕ| ⊗ ⊗1|e
−it (H A +H B )
|1 ⊗ |ϕ.
(7.4.8)
4. To restrict a function t → h(t) defined on the whole real line t ∈ R to the positive (resp. negative) values of its argument t ∈ R + (resp. R − ), we shall use the
(Heaviside) θ(t)-function equal to zero for t < 0 and equal to one for t ≥ 0. We
shall denote these restrictions as h ± (t):
h + (t) := θ(t)h(t), resp. h − (t) := θ(−t)h(t), t ∈ R.
(7.4.9)
Such restrictions f → f + will be useful here, e.g., for rewriting certain equations
in the convolution form.
5. The convolution f ∗ h(t) of two complex-valued integrable functions is defined
by
f ∗ h(t) =
+∞
−∞
dτ f (t − τ )h(τ ) = h ∗ f (t).
(7.4.10)
For more details on existence conditions of convolutions see e.g. [262, IX.4].
The operation ∗ is not only commutative, but also associative. It can be trivially
extended to functions t → h(t) defined for t ∈ R
n , as well as to some other classes
of functions and of distributions, see e.g. [262, 324].
6. Let us define and denote, for purposes of the present section, to any integrable
function h ∈ L
1
(R), its Fourier transformed function F(h) ≡ ˆ
h:
ˆ
h(u) ≡ F(h)(u) :=
1
√
2π
+∞
−∞
e
−itu h(t) dt, u ∈ R.
(7.4.11a)
181
2. Let us introduce also the symbol H A for the Hilbert (sub-)space of the chain
generated by the vectors {|m | m = 1, 2, . . . } introduced in (7.3.12). We shall
use also: U t := exp(−it H) with H from (7.4.5), and τ t c := e
it H c e
−it H for
c ∈ C. The vector 0 = |0 is defined in (7.3.2) and (7.3.3). We shall also use
χ
0 := 0 ⊗ χ, χ ∈ H B .
3. Let ϕ ∈ H B , ϕ 2 = 1, be the vector appearing in the interaction Hamiltonian
V ϕ in (7.4.4), and let ψ ∈ H B be the (also normalized) initial state-vector of the
particle. We shall introduce the symbols F
0
(t), g(t) , and F(t) as:
F
0
(t) ≡ F
0
ϕ (ψ)(t) := =ϕ|e
−it H B |ψ, g(t) := F
0
ϕ (ϕ)(t) ≡ ≡ϕ|e
−it H B |ϕ.
(7.4.6a)
F(t) ≡ F ϕ (ψ)(t) := =ϕ| ⊗ ⊗0|e
−it H
|0 ⊗ |ψ ≡ ≡
ϕ
0 |e
−it H
|
ψ
0 ,
(7.4.6b)
where H := H A + H B + γV ϕ is the total Hamiltonian of the compound system
(7.4.5).
The symbols f m (t), f (t) will be also useful abbreviations (cf. (7.3.24)):
f m (t) := =m|e
−it H A |1 = (−i)
m−1 m
t
J m (2t), m = 1, 2, . . . (7.4.7)
f (t) := g(t) f 1 (t) = =ϕ| ⊗ ⊗1|e
−it (H A +H B )
|1 ⊗ |ϕ.
(7.4.8)
4. To restrict a function t → h(t) defined on the whole real line t ∈ R to the positive (resp. negative) values of its argument t ∈ R + (resp. R − ), we shall use the
(Heaviside) θ(t)-function equal to zero for t < 0 and equal to one for t ≥ 0. We
shall denote these restrictions as h ± (t):
h + (t) := θ(t)h(t), resp. h − (t) := θ(−t)h(t), t ∈ R.
(7.4.9)
Such restrictions f → f + will be useful here, e.g., for rewriting certain equations
in the convolution form.
5. The convolution f ∗ h(t) of two complex-valued integrable functions is defined
by
f ∗ h(t) =
+∞
−∞
dτ f (t − τ )h(τ ) = h ∗ f (t).
(7.4.10)
For more details on existence conditions of convolutions see e.g. [262, IX.4].
The operation ∗ is not only commutative, but also associative. It can be trivially
extended to functions t → h(t) defined for t ∈ R
n , as well as to some other classes
of functions and of distributions, see e.g. [262, 324].
6. Let us define and denote, for purposes of the present section, to any integrable
function h ∈ L
1
(R), its Fourier transformed function F(h) ≡ ˆ
h:
ˆ
h(u) ≡ F(h)(u) :=
1
√
2π
+∞
−∞
e
−itu h(t) dt, u ∈ R.
(7.4.11a)