6.5 Inclusion of Continuum Couplings in the Pairing Model
283
A useful measure of pairing correlations in a given eigenstate |Ψ (K) is the canonical
pairing gap:
Δ
(K)
= G
N
q
n
(K)
q
1 − n
(K)
q
,
(6.57)
where the sum runs over single-particle states, and n
(K)
q is the occupation probability
of the state q. Note that the canonical gap coincides with the BCS gap in the
BCS approximation where the occupation probability n
(K)
q
= |v q | 2 . The occupation
probability n
(K)
q
can be calculated exactly by diagonalizing the Hamiltonian using
the Gamow shell model.
Let us write the eigenstate |Ψ (K) of the pairing Hamiltonian (6.37) as an
expansion in the basis of Slater determinants |Φ α
|Ψ
(K)
=
α
C
(K)
α |Φ α .
(6.58)
The expectation value of the particle number operator ˆ
N in the eigenstate |Ψ (K)
reads
N = =Ψ
(K)
| ˆ
N |Ψ
(K)
=
α,α
C
(K)
α C
(K)
α Φ α | ˆ
N |Φ α =
q
2n
(K)
q .
(6.59)
The occupation probability can be evaluated numerically as:
n
(K)
q
=
α
g(α, q; K) (C
(K)
α )
2 .
(6.60)
In this expression, g(α, q; K) = 0 or 1 if the single-particle state q is occupied or
unoccupied in the Slater determinant α of an eigenstate K.
As one does not have access to the Slater determinant expansion of the manybody wave function when solving the Richardson equations, the particle numbers
must be calculated differently. In the generalized Richardson equations, the singleparticle occupation probabilities in an eigenstate K are determined using the
Hellmann-Feynman theorem [132, 133]:
n
(K)
q
=
∂ ˜
E (K)
∂∂ q
,
(6.61)
where ˜
E (K) is the total energy (6.38) of the eigenstate K.