182
7 Some Models of “Quantum Measurement”
In the case of higher dimensional arguments of the C-valued functions h ∈
L
1
(R
n
) the analogous formula applies:
ˆ
h(u) ≡ F(h)(u) ≡ F(h(t))(u) := (2π)
−
n
2
R n
e
−it·u h(t) d
n t, u ∈ R
n
.
(7.4.11b)
The inverse F
−1 of F defined on the image ˆ
h = F(h) has the similarly looking
form:
h(t) = F
−1
( ˆ
h)(t) = F( ˆ
h(−u))(t) = (2π)
−
n
2
R n
e
it·u ˆ
h(u) d
n u, t ∈ R
n
.
(7.4.11c)
Generalizations to various classes of functions h and also to tempered distributions is very useful in process of solution of various equations. Many important
properties of the Fourier transformation can be found, e.g. in [262, 324]. One
of the most useful properties of F is the possibility to extend it from L
1
(R
n
) to
a unitary transformation in the Hilbert space L
2
(R
n
)—the Plancherel theorem:
The scalar product ·|·· is invariant with respect to the transformation F; for
ϕ, ψ ∈ L
2 it means: ϕ|ψ = = ˆ
ϕ| ˆ
ψ. Moreover, the following important property
concerning the interconnection between the convolution and the Fourier transformation is valid:
F(h 1 ∗ h 2 ) = (2π)
n
2 F(h 1 )F(h 2 ) ≡ (2π)
n
2 ˆ
h 1 ˆ
h 2 ,
(7.4.12)
with the pointwise multiplication of functions.
For the proof of our main result formulated in Theorem 7.4.8, we shall also need
several following lemmas. The first one together with its proof can be deduced from
[151]:
7.4.4 Lemma. Let H be a lower-bounded selfadjoint operator on a Hilbert space
H v with its spectrum sp(H ) ≥ a. Then, for any two nonzero vectors ϕ, ψ ∈ H, it is:
(a) either ϕ|e
−it H
|ψ ≡ 0, ∀t ∈ R,
(b) or ϕ|e
−it H
|ψ = 0 for t in an open dense subset of R of total Lebesgue measure.
If the above chosen ϕ is fixed, then the set of all ψ ∈ H satisfying (a) forms a
closed linear subspace of H v , hence the open complement in H v of this set consists
of those ψ ∈ H which satisfy the point (b).
Proof. Let λ → E H (λ) be the projection measure of H . According to the functional
calculus (cf. e.g. [262]) it is
ϕ|e
−it H
|ψ =
∞
a
dλ e
−itλ
ϕ|E H (λ)|ψ.
(7.4.13)
This function of time t ∈ R can be analytically continued to the lower complex
half-plain of t, i.e. extended to t → t − iε =: z, ε ≥ 0:
7 Some Models of “Quantum Measurement”
In the case of higher dimensional arguments of the C-valued functions h ∈
L
1
(R
n
) the analogous formula applies:
ˆ
h(u) ≡ F(h)(u) ≡ F(h(t))(u) := (2π)
−
n
2
R n
e
−it·u h(t) d
n t, u ∈ R
n
.
(7.4.11b)
The inverse F
−1 of F defined on the image ˆ
h = F(h) has the similarly looking
form:
h(t) = F
−1
( ˆ
h)(t) = F( ˆ
h(−u))(t) = (2π)
−
n
2
R n
e
it·u ˆ
h(u) d
n u, t ∈ R
n
.
(7.4.11c)
Generalizations to various classes of functions h and also to tempered distributions is very useful in process of solution of various equations. Many important
properties of the Fourier transformation can be found, e.g. in [262, 324]. One
of the most useful properties of F is the possibility to extend it from L
1
(R
n
) to
a unitary transformation in the Hilbert space L
2
(R
n
)—the Plancherel theorem:
The scalar product ·|·· is invariant with respect to the transformation F; for
ϕ, ψ ∈ L
2 it means: ϕ|ψ = = ˆ
ϕ| ˆ
ψ. Moreover, the following important property
concerning the interconnection between the convolution and the Fourier transformation is valid:
F(h 1 ∗ h 2 ) = (2π)
n
2 F(h 1 )F(h 2 ) ≡ (2π)
n
2 ˆ
h 1 ˆ
h 2 ,
(7.4.12)
with the pointwise multiplication of functions.
For the proof of our main result formulated in Theorem 7.4.8, we shall also need
several following lemmas. The first one together with its proof can be deduced from
[151]:
7.4.4 Lemma. Let H be a lower-bounded selfadjoint operator on a Hilbert space
H v with its spectrum sp(H ) ≥ a. Then, for any two nonzero vectors ϕ, ψ ∈ H, it is:
(a) either ϕ|e
−it H
|ψ ≡ 0, ∀t ∈ R,
(b) or ϕ|e
−it H
|ψ = 0 for t in an open dense subset of R of total Lebesgue measure.
If the above chosen ϕ is fixed, then the set of all ψ ∈ H satisfying (a) forms a
closed linear subspace of H v , hence the open complement in H v of this set consists
of those ψ ∈ H which satisfy the point (b).
Proof. Let λ → E H (λ) be the projection measure of H . According to the functional
calculus (cf. e.g. [262]) it is
ϕ|e
−it H
|ψ =
∞
a
dλ e
−itλ
ϕ|E H (λ)|ψ.
(7.4.13)
This function of time t ∈ R can be analytically continued to the lower complex
half-plain of t, i.e. extended to t → t − iε =: z, ε ≥ 0: