16
2 Scattering Theory of Three-Particle System
sub-system ‘i’ and particle ‘i’ is free. The argument of G i is the energy of the subsystem ‘i’, obtained by subtracting the kinetic energy of the particle ‘i’ from the free
energy z. For the Green’s function G 0 for the three particles, we have
q i ,
k i
G0(z)
q
i ,
k
i
=
δ(
k i −
k
i ) δ( q i − −
q
i )
z − k
2
i /2M i − q
2
i /2μ i
(2.12)
The expression for the T-matrix T i for the two-particle sub-system in three-particle
Hilbert space can also be written similar to Eq. (2.11) as
q i ,
k i
Ti (z)
q
i ,
k
i
= δ(
k i −
k
i ) q i |T i (z − k
2
i /2
μ i )
q
i
(2.13)
2.3 Boundary Conditions and Moller Operators
The discussion of boundary conditions presented in the case of two-particle scattering
can be easily extended to the three-particle case. Defining the time development of
a three-particle wave packet by
ψ
(+)
i j (t) = exp(−i Ht) ψ
(+)
i j
(2.14)
and the reference wave packets which develop according to
φ i j (t) = exp(−i H i t) φ i j
(2.15)
The wave packet φ i j describes the free motion of the particle ‘i’ (i = 1, 2, 3)
relative to the other two particles which are in their jth bound state. We demand that
the norm of the difference vanishes in the distant past leading to the representation
of scattering states
ψ
(+)
i j = s − lim
t→ ∓ ∞
exp(i Ht) exp(−i H i t)φ i j ≡
(±)
i φ i j
where Moller operators are defined as
(±)
i
= s − lim
t→∓
exp(−i Ht) exp(−i H i t), i = 1, 2, 3
(2.16)
Here we also include the t → +∞ which is required for the construction of
S-matrix.
The fact that a separate Moller operator is needed for every partition ‘i’ itself
shows that the three-particle problem is much more difficult to handle. The domain
of Moller operators defined above is the space of channel states
2 Scattering Theory of Three-Particle System
sub-system ‘i’ and particle ‘i’ is free. The argument of G i is the energy of the subsystem ‘i’, obtained by subtracting the kinetic energy of the particle ‘i’ from the free
energy z. For the Green’s function G 0 for the three particles, we have
q i ,
k i
G0(z)
q
i ,
k
i
=
δ(
k i −
k
i ) δ( q i − −
q
i )
z − k
2
i /2M i − q
2
i /2μ i
(2.12)
The expression for the T-matrix T i for the two-particle sub-system in three-particle
Hilbert space can also be written similar to Eq. (2.11) as
q i ,
k i
Ti (z)
q
i ,
k
i
= δ(
k i −
k
i ) q i |T i (z − k
2
i /2
μ i )
q
i
(2.13)
2.3 Boundary Conditions and Moller Operators
The discussion of boundary conditions presented in the case of two-particle scattering
can be easily extended to the three-particle case. Defining the time development of
a three-particle wave packet by
ψ
(+)
i j (t) = exp(−i Ht) ψ
(+)
i j
(2.14)
and the reference wave packets which develop according to
φ i j (t) = exp(−i H i t) φ i j
(2.15)
The wave packet φ i j describes the free motion of the particle ‘i’ (i = 1, 2, 3)
relative to the other two particles which are in their jth bound state. We demand that
the norm of the difference vanishes in the distant past leading to the representation
of scattering states
ψ
(+)
i j = s − lim
t→ ∓ ∞
exp(i Ht) exp(−i H i t)φ i j ≡
(±)
i φ i j
where Moller operators are defined as
(±)
i
= s − lim
t→∓
exp(−i Ht) exp(−i H i t), i = 1, 2, 3
(2.16)
Here we also include the t → +∞ which is required for the construction of
S-matrix.
The fact that a separate Moller operator is needed for every partition ‘i’ itself
shows that the three-particle problem is much more difficult to handle. The domain
of Moller operators defined above is the space of channel states