234
5 Formulation and Implementation of the Gamow Shell Model
Equation (5.51) is a direct consequence of Eq. (5.24) and of the binomial
theorem.
Conversely, (X core + i Y core ) k ∝ R k
core Y kk (Ω core ). Consequently, the core terms
in Eq. (5.51) are scalar operators only for k = 0. As the core wave function is
coupled to 0 + , all recoil terms, bearing k > 0, vanish identically in Eq. (5.51).
C. l lab and s in Eq. (5.58) are spherical tensors of order 1. As M L = L, one
necessarily has M L − 1 for the M-projection of Y L−1,M , and 1 for those of
l lab and s. Equations (5.52) and (5.53) then follow from Eqs. (5.58) and (5.59).
Equation (5.53) can be treated as Eq. (5.51) because spin is space-independent.
D. The orbital operator is handled from the equality:
l lab = r lab × p lab = (r + R core ) × p = l + R core × p ,
where one used Eqs. (5.24) and (5.31). Equation (5.54) is then straightforward
to deduce from Eqs. (5.24) and (5.31).
Equation (5.55) is obtained similarly to Eq. (5.51). l 1 , p x , p y , and p z are
operators involving valence nucleons only in Eq. (5.55). Consequently, the first
and second terms of Eq. (5.55) can be treated as that of Eq. (5.51). Z core is
proportional to R core Y 10 (Ω core ) because Z core = R core cos(θ core ). Therefore,
(X core + i Y core ) k Z core is proportional to R k+1
core Y kk (Ω core ) Y 10 (Ω core ).
The right-hand side of Eq. (5.56) is obtained by using a standard property of
spherical harmonics involving Y kk (Ω core ) Y 10 (Ω core ). One notices that the sum
in k of Eq. (5.56) possesses only non-scalar spherical tensors. Thus, they vanish
identically when acted on the core wave function. Consequently, all recoil terms
cancel out for magnetic transitions as well.
E. All the recoil terms induced by (5.51–5.53) vanish identically. As a consequence,
laboratory coordinates can be formally replaced by cluster orbital shell model
coordinates in Eqs. (5.57–5.59) without changing the results.
Exercise IV.
A. The ground state of 7 He mainly consists of the configuration having two
protons and two neutrons in the 0s 1/2 shells and three neutrons in the 0p 3/2
resonance shell. The p 1/2 partial wave is not occupied in the ground state of
7 He when considering a Hartree-Fock approximation of the many-body state.
Consequently, a model space built from a 4 He core and valence neutrons
occupying the neutron p 3/2 partial wave only recaptures the main features of
the ground state of 7 He.
B. The natural orbitals have been calculated from the scalar density matrix of
the exact ground state of 7 He. Consequently, the nucleon occupation of these
natural orbitals decreases very quickly when the number of natural orbitals
increases. About five natural orbitals per partial wave are typically needed to
obtain convergence. This number is much smaller than the tens of discretized
scattering states needed to numerically obtain completeness when discretizing
the Berggren basis contour.
5 Formulation and Implementation of the Gamow Shell Model
Equation (5.51) is a direct consequence of Eq. (5.24) and of the binomial
theorem.
Conversely, (X core + i Y core ) k ∝ R k
core Y kk (Ω core ). Consequently, the core terms
in Eq. (5.51) are scalar operators only for k = 0. As the core wave function is
coupled to 0 + , all recoil terms, bearing k > 0, vanish identically in Eq. (5.51).
C. l lab and s in Eq. (5.58) are spherical tensors of order 1. As M L = L, one
necessarily has M L − 1 for the M-projection of Y L−1,M , and 1 for those of
l lab and s. Equations (5.52) and (5.53) then follow from Eqs. (5.58) and (5.59).
Equation (5.53) can be treated as Eq. (5.51) because spin is space-independent.
D. The orbital operator is handled from the equality:
l lab = r lab × p lab = (r + R core ) × p = l + R core × p ,
where one used Eqs. (5.24) and (5.31). Equation (5.54) is then straightforward
to deduce from Eqs. (5.24) and (5.31).
Equation (5.55) is obtained similarly to Eq. (5.51). l 1 , p x , p y , and p z are
operators involving valence nucleons only in Eq. (5.55). Consequently, the first
and second terms of Eq. (5.55) can be treated as that of Eq. (5.51). Z core is
proportional to R core Y 10 (Ω core ) because Z core = R core cos(θ core ). Therefore,
(X core + i Y core ) k Z core is proportional to R k+1
core Y kk (Ω core ) Y 10 (Ω core ).
The right-hand side of Eq. (5.56) is obtained by using a standard property of
spherical harmonics involving Y kk (Ω core ) Y 10 (Ω core ). One notices that the sum
in k of Eq. (5.56) possesses only non-scalar spherical tensors. Thus, they vanish
identically when acted on the core wave function. Consequently, all recoil terms
cancel out for magnetic transitions as well.
E. All the recoil terms induced by (5.51–5.53) vanish identically. As a consequence,
laboratory coordinates can be formally replaced by cluster orbital shell model
coordinates in Eqs. (5.57–5.59) without changing the results.
Exercise IV.
A. The ground state of 7 He mainly consists of the configuration having two
protons and two neutrons in the 0s 1/2 shells and three neutrons in the 0p 3/2
resonance shell. The p 1/2 partial wave is not occupied in the ground state of
7 He when considering a Hartree-Fock approximation of the many-body state.
Consequently, a model space built from a 4 He core and valence neutrons
occupying the neutron p 3/2 partial wave only recaptures the main features of
the ground state of 7 He.
B. The natural orbitals have been calculated from the scalar density matrix of
the exact ground state of 7 He. Consequently, the nucleon occupation of these
natural orbitals decreases very quickly when the number of natural orbitals
increases. About five natural orbitals per partial wave are typically needed to
obtain convergence. This number is much smaller than the tens of discretized
scattering states needed to numerically obtain completeness when discretizing
the Berggren basis contour.