7.4 Particle Detection—A “Nonideal” Measurement
187
with U
0
t := exp[−it (H A + H B )], will be used repeatedly in our work here. We shall
work in the Hilbert space K (for t < ∞). The restriction of the interaction Hamiltonian V ϕ to the subspace K has the form
P K V ϕ = (P 01 + P 10 )P ϕ .
(7.4.32)
Due to the commutativity of U
0
t with P 0 , we obtain for m = 0 after the insertion
from (7.4.32) into (7.4.31) :
P m U t P 0 = −iγ
t
0
dτ P m U
0
t−τ P 10 P ϕ P 0 U τ P 0 .
(7.4.33)
For m = 0, we obtain similarly:
P 0 U t P 0 = P 0 U
0
t − iγ
t
0
dτ U
0
t−τ P 01 P ϕ P 1 U τ P 0 .
(7.4.34)
Substitution of (7.4.33) with m = 1 to this equation leads, after a linear change of
integration variables, to an integral equation for P 0 U t P 0 :
P 0 U t P 0 = P 0 U
0
t − γ
2
t
0
dt
t−t
0
dτ U
0
t−t −τ P 01 P ϕ U
0
τ P ϕ P 10 P 0 U t P 0 . (7.4.35)
Also the commutativity of P ϕ with P mn was used here. Since U
A
t := e
−it H A leaves
the vector 0 invariant, it is also P 0 U
0
t = P 0 exp(−it H B ), and with (7.4.8) we have:
P 01 P ϕ U
0
τ P ϕ P 10 = =ϕ|e
−iτ H B |ϕ1|U
A
τ |1P ϕ P 0 ≡ f (τ )P ϕ P 0 .
(7.4.36)
The integral equation (7.4.35) can be rewritten now in the form:
P 0 U t P 0 = P 0 e
−it H B − γ
2
t
0
dt
t−t
0
dτ e
−i(t−t
−τ )H B P ϕ f (τ )P 0 U t P 0 . (7.4.37)
With the symbols from (7.4.6a) and (7.4.6b), by taking the matrix elements of
both sides of this equation as in (7.4.6b), we can write the equation for F(t), cf.
Notation 7.4.3:
F(t) = F
0
(t) − γ
2
t
0
dt
t−t
0
dτ g(t − t
− τ ) f (τ )F(t
).
(7.4.38a)