2.2 Three-Particle Hamiltonian, Channel Hamiltonian, Matrix-Elements …
15
where the reduced masses are given by
μ 1 =
m 2 m 3
m 2 + m 3
and
μ 1 =
m 1 (m 2 + m 3 )
m 1 + m 2 + m 3
(2.5)
The physical interpretation of the kinetic energy part of the Hamiltonian is that
the first term is the kinetic energy of the CM, the second term corresponds to the
relative kinetic energy of particles 2 and 3 in their center of mass system, while the
third term accounts for the relative kinetic energy of particle 1 and the composite
system (2, 3). Thus, in the CM system, the total Hamiltonian can be written as:
H =
q
2
i
2μ i
+
k
2
i
2
μ i
+
3
j=1
V j (i = 1, 2, or 3)
(2.6)
In the three-body problem, two-particle sub-systems have a crucial role for which
we introduce a channel Hamiltonian, H i which is defined as
H i =
q
2
i
2μ i
+
k
2
i
2
μ i
+ V i
(2.7)
and is related to the total Hamiltonian H as
H = H i + ¯
V i
where ¯
V i = V − V i =
3
j =i
V j
(2.8, 2.9)
It may be useful to include also the breakup channel (i.e., when all the three
particles are free) by extending the value of index i = 0 (in addition to 1, 2 or 3). In
that case V 0 = 0 and Eqs. (2.7) and (2.8) become also valid for i = 0. Along with
the channel Hamiltonian, we also introduce channel resolvent Green’s function G i
defined as:
G i (z) ≡ (z − H i )
−1
(2.10)
Thus, written explicitly, the matrix element of the channel resolvent would appear
as
q i ,
k i
Gi (z)
q
i ,
k
i
= δ(
k i −
k
i ) q i |G i (z − k
2
i /2
μ i )
q
i
(2.11)
These matrix elements are similar to the two-body matrix elements of two particle
Green’s functions introduced earlier [cf. Eq. (1.14)]. The Green’s function G i carries
the channel index ‘i’ because different potentials V i can act in different channels.
The δ− function accounts for the fact that G i contains only the interaction of the
15
where the reduced masses are given by
μ 1 =
m 2 m 3
m 2 + m 3
and
μ 1 =
m 1 (m 2 + m 3 )
m 1 + m 2 + m 3
(2.5)
The physical interpretation of the kinetic energy part of the Hamiltonian is that
the first term is the kinetic energy of the CM, the second term corresponds to the
relative kinetic energy of particles 2 and 3 in their center of mass system, while the
third term accounts for the relative kinetic energy of particle 1 and the composite
system (2, 3). Thus, in the CM system, the total Hamiltonian can be written as:
H =
q
2
i
2μ i
+
k
2
i
2
μ i
+
3
j=1
V j (i = 1, 2, or 3)
(2.6)
In the three-body problem, two-particle sub-systems have a crucial role for which
we introduce a channel Hamiltonian, H i which is defined as
H i =
q
2
i
2μ i
+
k
2
i
2
μ i
+ V i
(2.7)
and is related to the total Hamiltonian H as
H = H i + ¯
V i
where ¯
V i = V − V i =
3
j =i
V j
(2.8, 2.9)
It may be useful to include also the breakup channel (i.e., when all the three
particles are free) by extending the value of index i = 0 (in addition to 1, 2 or 3). In
that case V 0 = 0 and Eqs. (2.7) and (2.8) become also valid for i = 0. Along with
the channel Hamiltonian, we also introduce channel resolvent Green’s function G i
defined as:
G i (z) ≡ (z − H i )
−1
(2.10)
Thus, written explicitly, the matrix element of the channel resolvent would appear
as
q i ,
k i
Gi (z)
q
i ,
k
i
= δ(
k i −
k
i ) q i |G i (z − k
2
i /2
μ i )
q
i
(2.11)
These matrix elements are similar to the two-body matrix elements of two particle
Green’s functions introduced earlier [cf. Eq. (1.14)]. The Green’s function G i carries
the channel index ‘i’ because different potentials V i can act in different channels.
The δ− function accounts for the fact that G i contains only the interaction of the