7.4 Particle Detection—A “Nonideal” Measurement
179
observable determined by a POVM, which will be, however, the case of the following
example.
7.4 Particle Detection—A “Nonideal” Measurement
7.4.1 This model describes a compound system of a spin chain A with a particle
B; it is a completed version of the model presented originally in [38]. The model of
the spin chain is the half-infinite chain of the form described in the Sect. 7.3, and the
particle is a nonrelativistic scalar particle.
Let us use (essentially) the notation of Sect. 7.3. Hence, the algebra A of the
observables of the spin chain is now generated by the elements a
∗
n , a n , n ≥ 1. Let
the Hamiltonian of the chain be the operator (cf. (7.3.8))
H A :=
n≥1
a
∗
n a n (a
∗
n+1 + a n+1 )a n+2 a
∗
n+2
(7.4.1)
acting in the Hilbert space H vac of the GNS-representation of A with the cyclic
vector 0 corresponding to the state
ω
A
↓ (a
∗
j a j ) = 0, for all j ≥ 1.
(7.4.2)
The particle B is moving in the 3-dimensional Euclidean space and is described
as in elementary QM by operators acting in the space H B := L
2
(R
3
, d
3 x), so that its
states are described by vectors (resp. the corresponding unit rays) ψ ∈ H B . The free
particle’s Hamiltonian will be just the kinetic energy (in conveniently chosen units
and in the “x-representation”)
H B := ˆ
p
2
= −
3
j=1
∂
2
∂x
2
j
.
(7.4.3)
The interaction Hamiltonian will be V ϕ , with
V ϕ := (a
∗
1 + a 1 )a 2 a
∗
2 ⊗ |ϕϕ| ∈ L(H vac ⊗ H B ),
(7.4.4)
where ϕ ≡ |ϕ ∈ H B is a conveniently chosen normalized vector, hence |ϕϕ| ≡ P ϕ
is a one-dimensional projector in H B .
The total Hamiltonian H of the compound system {spin chain & particle} will
be
H := H A + H B + γV ϕ , γ ∈ R.
(7.4.5)
Some restrictions on the interaction constant γ and on the unit vector ϕ will be
specified later.
179
observable determined by a POVM, which will be, however, the case of the following
example.
7.4 Particle Detection—A “Nonideal” Measurement
7.4.1 This model describes a compound system of a spin chain A with a particle
B; it is a completed version of the model presented originally in [38]. The model of
the spin chain is the half-infinite chain of the form described in the Sect. 7.3, and the
particle is a nonrelativistic scalar particle.
Let us use (essentially) the notation of Sect. 7.3. Hence, the algebra A of the
observables of the spin chain is now generated by the elements a
∗
n , a n , n ≥ 1. Let
the Hamiltonian of the chain be the operator (cf. (7.3.8))
H A :=
n≥1
a
∗
n a n (a
∗
n+1 + a n+1 )a n+2 a
∗
n+2
(7.4.1)
acting in the Hilbert space H vac of the GNS-representation of A with the cyclic
vector 0 corresponding to the state
ω
A
↓ (a
∗
j a j ) = 0, for all j ≥ 1.
(7.4.2)
The particle B is moving in the 3-dimensional Euclidean space and is described
as in elementary QM by operators acting in the space H B := L
2
(R
3
, d
3 x), so that its
states are described by vectors (resp. the corresponding unit rays) ψ ∈ H B . The free
particle’s Hamiltonian will be just the kinetic energy (in conveniently chosen units
and in the “x-representation”)
H B := ˆ
p
2
= −
3
j=1
∂
2
∂x
2
j
.
(7.4.3)
The interaction Hamiltonian will be V ϕ , with
V ϕ := (a
∗
1 + a 1 )a 2 a
∗
2 ⊗ |ϕϕ| ∈ L(H vac ⊗ H B ),
(7.4.4)
where ϕ ≡ |ϕ ∈ H B is a conveniently chosen normalized vector, hence |ϕϕ| ≡ P ϕ
is a one-dimensional projector in H B .
The total Hamiltonian H of the compound system {spin chain & particle} will
be
H := H A + H B + γV ϕ , γ ∈ R.
(7.4.5)
Some restrictions on the interaction constant γ and on the unit vector ϕ will be
specified later.