194
5 Formulation and Implementation of the Gamow Shell Model
and therefore:
δ
|H |Ψ
|Ψ
= 0 ,
(5.22)
where E(Ψ ) is the expectation value of the Hamiltonian in |Ψ , which can always be
nonzero, and where one has used the fact that δE(Ψ ) and δE ∗ (Ψ ) can be considered
as independent.
As Eq. (5.22) is formally the same as for real-energy eigenstates, one can derive
the equation verified by |Ψ using the same technique as for real-energy ground
states:
δ
|H |Ψ
|Ψ
= 0
⇔ ⇔δΨ |H |Ψ − E δΨ |Ψ = 0
⇔ ⇔δΨ |H − E|Ψ = 0 ,
(5.23)
where E = E(Ψ ).
Exercise II
One will illustrate now in a numerical example involving a nucleus with two
valence nucleons that the generalized variational principle holds in practical
calculations.
A. Run the two-body Gamow shell model code using the modified surface
Gaussian interaction (see Eq. (9.174)) for 18 O to calculate the low energy
observables: eigenstates, densities, and electromagnetic transitions. Notice
how the generalized variational principle acts on the convergence of energies
and widths of unbound states. Explain why energies do not converge in the
same way as for bound states in the standard shell model, i.e., by increasing
binding energy when the model space is enlarged by adding more Lanczos or
Jacobi-Davidson vectors.
B. Notice in numerical studies that observables involving only bound states are
always real, and that those involving at least one resonance state are complex.
Explain from completeness arguments why the observables involving only
bound states are not complex, even though basis states are complex in general.
C. Notice that narrow resonance states have observables whose imaginary part
is very small, whereas broad resonance states bear observables with sizable
imaginary parts. Explain qualitatively these values in terms of the Berggren
interpretation of complex observables of Sect. 5.1.
Equation (5.23) is equivalent to the Hamiltonian matrix eigenproblem, as it must
be fulfilled for all variations of |Ψ , denoted as |δΨ . Consequently, even though E
5 Formulation and Implementation of the Gamow Shell Model
and therefore:
δ
|H |Ψ
|Ψ
= 0 ,
(5.22)
where E(Ψ ) is the expectation value of the Hamiltonian in |Ψ , which can always be
nonzero, and where one has used the fact that δE(Ψ ) and δE ∗ (Ψ ) can be considered
as independent.
As Eq. (5.22) is formally the same as for real-energy eigenstates, one can derive
the equation verified by |Ψ using the same technique as for real-energy ground
states:
δ
|H |Ψ
|Ψ
= 0
⇔ ⇔δΨ |H |Ψ − E δΨ |Ψ = 0
⇔ ⇔δΨ |H − E|Ψ = 0 ,
(5.23)
where E = E(Ψ ).
Exercise II
One will illustrate now in a numerical example involving a nucleus with two
valence nucleons that the generalized variational principle holds in practical
calculations.
A. Run the two-body Gamow shell model code using the modified surface
Gaussian interaction (see Eq. (9.174)) for 18 O to calculate the low energy
observables: eigenstates, densities, and electromagnetic transitions. Notice
how the generalized variational principle acts on the convergence of energies
and widths of unbound states. Explain why energies do not converge in the
same way as for bound states in the standard shell model, i.e., by increasing
binding energy when the model space is enlarged by adding more Lanczos or
Jacobi-Davidson vectors.
B. Notice in numerical studies that observables involving only bound states are
always real, and that those involving at least one resonance state are complex.
Explain from completeness arguments why the observables involving only
bound states are not complex, even though basis states are complex in general.
C. Notice that narrow resonance states have observables whose imaginary part
is very small, whereas broad resonance states bear observables with sizable
imaginary parts. Explain qualitatively these values in terms of the Berggren
interpretation of complex observables of Sect. 5.1.
Equation (5.23) is equivalent to the Hamiltonian matrix eigenproblem, as it must
be fulfilled for all variations of |Ψ , denoted as |δΨ . Consequently, even though E