196
5 Formulation and Implementation of the Gamow Shell Model
• For i ∈ core:
∂
∂x i,lab
=
j
∂x j
∂x i,lab
∂
∂x j
=
∂
∂x i
−
j ∈val
m i
M core
∂
∂x j
(5.28)
• For i ∈ val:
∂
∂x i,lab
=
j
∂x j
∂x i,lab
∂
∂x j
=
∂
∂x i
(5.29)
Equations for ∂/∂y i,lab and ∂/∂z i,lab are obtained by substituting x by y and z. As a
consequence, the expression of p i in cluster orbital shell model reads:
• For i ∈ core:
p i,lab = p i −
j ∈val
m i
M core
p j
(5.30)
• For i ∈ val:
p i,lab = p i
(5.31)
One can then calculate the center-of-mass linear momentum P lab in cluster orbital
shell model coordinates from Eq. (5.31):
P lab =
i∈core
p i,lab +
i∈val
p i,lab =
i∈core
p i −
i∈core
j ∈val
m i
M core
p j +
i∈val
p i
=
i∈core
p i −
j ∈val
p j +
i∈val
p i =
i∈core
p i
(5.32)
Interestingly, P lab is a function of core linear momenta only.
In order to deal with the kinetic energy operator, one will derive the Laplacian in
cluster orbital shell model coordinates using Eqs. (5.30) and (5.31):
• For i ∈ core:
p
2
i,lab = p
2
i +
j,j ∈val
m i
M core
2
p j · p j − 2
j ∈val
m i
M core
p i · p j
(5.33)
• For i ∈ val:
p
2
i,lab = p
2
i
(5.34)