198
7 Some Models of “Quantum Measurement”
where the c-number coefficients c
(m)
j ( p) (m ∈ Z + , j, p ∈ Z) satisfy following recurrent relations:
c
(m+1)
( p) = −
κ
2
(c
(m)
( p − 1) + c
(m)
( p + 1)),
(7.5.11)
where c
(0)
( p) = δ 0 p , c
(m)
( j − p) ≡ c
(m)
j ( p).
It is seen that the coefficients c
(m)
j ( p) depend on p − j only: they are expressible
as linear combinations of the Kronecker deltas δ j, p+c . Notice also that c
(m)
(− p) =
c
(m)
( p), ∀ p ∈ Z. Note moreover that for each m ≥ 0 only finite number of the
coefficients c
(m)
( j − p) is nonzero.
From (7.5.5) and (7.5.10) we have:
τ t (b j ) =
k∈Z
C t ( j − k) b k ,
(7.5.12a)
where
C t (r ) :=
∞
m=0
(it)
m
m!
c
(m)
(r ).
(7.5.12b)
The Bessel functions of the first kind J r (t), r ∈ Z + , t ∈ R, can be expressed by
the power series:
J r (t) =
∞
k=0
(−1)
k
t
2
2k+r
1
k!(r + k)!
.
(7.5.13)
By calculation of coefficients c
(m)
(r ) in (7.5.12b) with the help of (7.5.11) and by
comparison of coefficients at equal powers t
m of the variable t ∈ R in the expressions
(7.5.12b) for C t (r ) and in (7.5.13) for J r (t), we can see that for r ∈ Z + it is
C t (r ) ≡ (−i)
r J r (κt).
(7.5.14)
After inserting this into (7.5.12a) (keep in mind that C t (−r ) = C t (r ) =
(−i)
|r | J |r | (κt)) we obtain explicit expression for time evolution of elements b j ∈ A,
hence the time-automorphism group τ t , t ∈ R, of A in terms of standard special
functions J r , r ∈ Z + .
7.5.2 Interaction with a small system. Let us use the just described X-Y spin chain
to construction of an alternative “model of quantum measurement” now.
Let us represent the algebra A in a subspace of the CTPS = ⊗ j∈Z C
2
j (cf. 5.1.3)
corresponding to the product-vector 0 defined as follows: Let the spins on our chain
be well ordered and numbered by j ∈ Z. Let | ± j be the states of the j-th spin being
eigenvectors of the Pauli matrix σ
z
j corresponding to the up-, resp. down-orientations:
σ
z
j | ± j = ±| ± j. Let then
7 Some Models of “Quantum Measurement”
where the c-number coefficients c
(m)
j ( p) (m ∈ Z + , j, p ∈ Z) satisfy following recurrent relations:
c
(m+1)
( p) = −
κ
2
(c
(m)
( p − 1) + c
(m)
( p + 1)),
(7.5.11)
where c
(0)
( p) = δ 0 p , c
(m)
( j − p) ≡ c
(m)
j ( p).
It is seen that the coefficients c
(m)
j ( p) depend on p − j only: they are expressible
as linear combinations of the Kronecker deltas δ j, p+c . Notice also that c
(m)
(− p) =
c
(m)
( p), ∀ p ∈ Z. Note moreover that for each m ≥ 0 only finite number of the
coefficients c
(m)
( j − p) is nonzero.
From (7.5.5) and (7.5.10) we have:
τ t (b j ) =
k∈Z
C t ( j − k) b k ,
(7.5.12a)
where
C t (r ) :=
∞
m=0
(it)
m
m!
c
(m)
(r ).
(7.5.12b)
The Bessel functions of the first kind J r (t), r ∈ Z + , t ∈ R, can be expressed by
the power series:
J r (t) =
∞
k=0
(−1)
k
t
2
2k+r
1
k!(r + k)!
.
(7.5.13)
By calculation of coefficients c
(m)
(r ) in (7.5.12b) with the help of (7.5.11) and by
comparison of coefficients at equal powers t
m of the variable t ∈ R in the expressions
(7.5.12b) for C t (r ) and in (7.5.13) for J r (t), we can see that for r ∈ Z + it is
C t (r ) ≡ (−i)
r J r (κt).
(7.5.14)
After inserting this into (7.5.12a) (keep in mind that C t (−r ) = C t (r ) =
(−i)
|r | J |r | (κt)) we obtain explicit expression for time evolution of elements b j ∈ A,
hence the time-automorphism group τ t , t ∈ R, of A in terms of standard special
functions J r , r ∈ Z + .
7.5.2 Interaction with a small system. Let us use the just described X-Y spin chain
to construction of an alternative “model of quantum measurement” now.
Let us represent the algebra A in a subspace of the CTPS = ⊗ j∈Z C
2
j (cf. 5.1.3)
corresponding to the product-vector 0 defined as follows: Let the spins on our chain
be well ordered and numbered by j ∈ Z. Let | ± j be the states of the j-th spin being
eigenvectors of the Pauli matrix σ
z
j corresponding to the up-, resp. down-orientations:
σ
z
j | ± j = ±| ± j. Let then