328
7 Wave Functions in the Vicinity of Particle-Emission Threshold
Show that 2
1/2 verifies Eqs. (7.27), (7.28), (7.29) when κ → 0 + for
m = 1/2 (π = 1) and m = 1/2, 3/2 (π = −1).
Notice that 2
1/2 converges for κ → 0 + when m = 3/2, . . . , 7/2 (π =
1) and m = 5/2 (π = −1), in accordance with Eq. (7.30).
Explain the common features of 2
1/2 in the spherical and deformed
cases when κ → 0 + .
One will consider the asymptotic behavior of the root-mean-square radius of
Φ(k, r) for κ → 0. For this, let us calculate the expectation value of r 2 :
2
|Φ =
+∞
0
r
2 u j (k, r)
2 dr .
(7.26)
Using results of Sect. 7.2.1.1, one arrives at the conclusion that the integrals in
Eq. (7.26) either diverge if = 0, 1, or are finite if ≥ 2 when κ → 0. Hence,
one obtains for the root-mean-square radius at particle-emission threshold:
2
1/2 ∼
1
2
κ
−1 ,
|m| = 1/2 , π = 1
(7.27)
2
1/2 ∼
5
4
β
(0)
=1/2
2 +
β
(0)
=3/2
2
κ
−1/2 , |m| = 1/2 , π = −1
(7.28)
2
1/2 ∼
5
4
β
(0)
=3/2
κ
−1/2 ,
|m| = 3/2 , π = −1
(7.29)
2
1/2 → const otherwise
(7.30)
where β
(0)
j has been defined in Eq. (7.10). Consequently, the near-threshold properties of deformed neutron states are analogous to that of spherical neutron states,
as the = 0, 1 condition must be replaced by the π = ±1 and |m| = 1/2, 3/2
conditions. Numerical examples showing the validity of Eqs. (7.5), (7.12), (7.13)
and Eqs. (7.27–7.30) in practice, are done in Exercise I.
7.2.2 Effects of Nucleon–Nucleon Correlations on Halo States
The configuration mixing involving bound and unbound basis states may generate a
weakly bound many-body state with the density distribution extending by tunnelling
effect in the classically forbidden region. Indeed, even though halo states are bound,
they possess sizable components of unbound basis states and in that sense are the
states of open quantum systems. On the contrary, well-bound states whose density
7 Wave Functions in the Vicinity of Particle-Emission Threshold
Show that 2
1/2 verifies Eqs. (7.27), (7.28), (7.29) when κ → 0 + for
m = 1/2 (π = 1) and m = 1/2, 3/2 (π = −1).
Notice that 2
1/2 converges for κ → 0 + when m = 3/2, . . . , 7/2 (π =
1) and m = 5/2 (π = −1), in accordance with Eq. (7.30).
Explain the common features of 2
1/2 in the spherical and deformed
cases when κ → 0 + .
One will consider the asymptotic behavior of the root-mean-square radius of
Φ(k, r) for κ → 0. For this, let us calculate the expectation value of r 2 :
2
|Φ =
+∞
0
r
2 u j (k, r)
2 dr .
(7.26)
Using results of Sect. 7.2.1.1, one arrives at the conclusion that the integrals in
Eq. (7.26) either diverge if = 0, 1, or are finite if ≥ 2 when κ → 0. Hence,
one obtains for the root-mean-square radius at particle-emission threshold:
2
1/2 ∼
1
2
κ
−1 ,
|m| = 1/2 , π = 1
(7.27)
2
1/2 ∼
5
4
β
(0)
=1/2
2 +
β
(0)
=3/2
2
κ
−1/2 , |m| = 1/2 , π = −1
(7.28)
2
1/2 ∼
5
4
β
(0)
=3/2
κ
−1/2 ,
|m| = 3/2 , π = −1
(7.29)
2
1/2 → const otherwise
(7.30)
where β
(0)
j has been defined in Eq. (7.10). Consequently, the near-threshold properties of deformed neutron states are analogous to that of spherical neutron states,
as the = 0, 1 condition must be replaced by the π = ±1 and |m| = 1/2, 3/2
conditions. Numerical examples showing the validity of Eqs. (7.5), (7.12), (7.13)
and Eqs. (7.27–7.30) in practice, are done in Exercise I.
7.2.2 Effects of Nucleon–Nucleon Correlations on Halo States
The configuration mixing involving bound and unbound basis states may generate a
weakly bound many-body state with the density distribution extending by tunnelling
effect in the classically forbidden region. Indeed, even though halo states are bound,
they possess sizable components of unbound basis states and in that sense are the
states of open quantum systems. On the contrary, well-bound states whose density