2.4 Pöschl-Teller-Ginocchio Potential
41
The wave function u(k, r) is provided by Eq. (2.96) for 0 < r < r as , while one
must use Eq. (2.7) to obtain u(k, r) when r > r as . As V PTG–mod (r) is continuous for
all radii, Eq. (2.2) implies that u(k, r), u (k, r), and u (k, r) are all continuous.
As V PTG–mod (r) is not differentiable in r = r as , u (k, r) is not differentiable
therein as well, so that the third and higher derivatives of u(k, r) are all discontinuous in r = r as . The Jost functions read:
J
± (k) = u
PTG (r as )H
±
,η (kr as ) − u PTG (r as )kH
±
,η
(kr as ) .
(2.125)
Its second-order Taylor expansion is:
J
± (k + δk) J
± (k) + δk J
± (k) +
δk 2
2
J
± (k) ,
(2.126)
so that the equation J ± (k + δk) = 0 leads to the second-order value of δk:
δk = −
J ± (k)
J ± (k)
1 −
1 − 2
J ± (k)J ± (k)
J ± (k)
2
.
(2.127)
The choice of k associated to E = E PTG + E b corresponds to the eigenstate
of V PTG (r) + E b , which is the closest analytical potential to V PTG–mod (r). If
one considers a narrow resonance, one has E < E b so that the zeroth-order
approximation of the exact eigenstate is bound, hence without a width. But, as
E > 0, one has k > 0 and thus H
+
,η (kr as ) is complex. Consequently, δk is complex
in Eq. (2.127) and provides a nonzero approximation for the width.
The second-order approximation of energies works well except very close or
very far from the particle-emission threshold. Indeed, in this latter case the width
becomes very large and the width arising from the second-order expansion can no
longer be correct. Even though the second-order approximation of energies close to
zero energy is usually good, the width might become imprecise as it is very small.
Here, it is better to calculate energies numerically but to evaluate widths with the
current formula (2.196). The eigenenergies and phase shifts of the modified PöschlTeller-Ginocchio potential are studied in numerical examples in Exercise X.
Exercise X
In this exercise, we will concentrate on the ability of the modified PöschlTeller-Ginocchio potential to provide with physical values of eigenenergies and
phase shifts.
A. Calculate analytically the phase shifts of the scattering states generated by the
V PTG (r) (see Eq. (2.92)).
B. Run the one-particle code of radial wave functions to calculate phase shifts
numerically for the exact V PTG (r) and modified V PTG–mod (r) Pöschl-Teller-
41
The wave function u(k, r) is provided by Eq. (2.96) for 0 < r < r as , while one
must use Eq. (2.7) to obtain u(k, r) when r > r as . As V PTG–mod (r) is continuous for
all radii, Eq. (2.2) implies that u(k, r), u (k, r), and u (k, r) are all continuous.
As V PTG–mod (r) is not differentiable in r = r as , u (k, r) is not differentiable
therein as well, so that the third and higher derivatives of u(k, r) are all discontinuous in r = r as . The Jost functions read:
J
± (k) = u
PTG (r as )H
±
,η (kr as ) − u PTG (r as )kH
±
,η
(kr as ) .
(2.125)
Its second-order Taylor expansion is:
J
± (k + δk) J
± (k) + δk J
± (k) +
δk 2
2
J
± (k) ,
(2.126)
so that the equation J ± (k + δk) = 0 leads to the second-order value of δk:
δk = −
J ± (k)
J ± (k)
1 −
1 − 2
J ± (k)J ± (k)
J ± (k)
2
.
(2.127)
The choice of k associated to E = E PTG + E b corresponds to the eigenstate
of V PTG (r) + E b , which is the closest analytical potential to V PTG–mod (r). If
one considers a narrow resonance, one has E < E b so that the zeroth-order
approximation of the exact eigenstate is bound, hence without a width. But, as
E > 0, one has k > 0 and thus H
+
,η (kr as ) is complex. Consequently, δk is complex
in Eq. (2.127) and provides a nonzero approximation for the width.
The second-order approximation of energies works well except very close or
very far from the particle-emission threshold. Indeed, in this latter case the width
becomes very large and the width arising from the second-order expansion can no
longer be correct. Even though the second-order approximation of energies close to
zero energy is usually good, the width might become imprecise as it is very small.
Here, it is better to calculate energies numerically but to evaluate widths with the
current formula (2.196). The eigenenergies and phase shifts of the modified PöschlTeller-Ginocchio potential are studied in numerical examples in Exercise X.
Exercise X
In this exercise, we will concentrate on the ability of the modified PöschlTeller-Ginocchio potential to provide with physical values of eigenenergies and
phase shifts.
A. Calculate analytically the phase shifts of the scattering states generated by the
V PTG (r) (see Eq. (2.92)).
B. Run the one-particle code of radial wave functions to calculate phase shifts
numerically for the exact V PTG (r) and modified V PTG–mod (r) Pöschl-Teller-