2.6 Analytical Properties of the Wave Functions
65
k 0 in this equation is the linear momentum associated to E 0 . The integrand in
Eq. (2.194) is now O(ln(E ) E −5/2 ), so that the gain in precision is about one to
two orders of magnitude. As (S (k)) in Eq. (2.194) is a function of an integral
involving (S (k)), hence the knowledge of |S (k)| implies S (k).
The S-matrix is often considered for complex potentials as well. The addition
of an imaginary part to the real potential of Eq. (2.2) breaks the unitarity of the Smatrix, so that the particle flux is no longer conserved. This idea is behind the optical
model potential, where its absorptive, negative imaginary part accounts for the
flux lost to the compound nucleus component in the scattering process. Hence, the
complex optical potential arises from an attempted simplification of the description
of the reaction [45–47], and can be considered as an effective way to take into
account all of the physical processes which one does not want to consider explicitly.
In the exact formulation using a Hermitian Hamiltonian, a huge number of
reaction channels should be taken into account to generate particle absorption into
the compound nucleus configurations. This strategy in the description of direct
reactions is not only impractical but also impossible to realize in any practical
application. The phenomenological replacement of those reaction channels by the
(negative) imaginary part of optical potential can be justified by the fact that
the individual compound nucleus configurations cannot be studied in practice.
Therefore, their effects can be represented by simple functions, which take the form
of an imaginary part of the complex optical potential in practice.
2.6.7 Radial Equation and Shooting Method
The one-body bound, antibound, and resonances are calculated numerically by
integrating Eq. (2.2) with the shooting method which is based on finding the zeros
of the Jost function J + (k) of Eq. (2.169). For this, one has to provide a starting
value k start for the linear momentum of the one-body state to calculate, which is
then refined using the Newton method on k:
1. One considers k start , close to the exact k value of the resonant state. It is typically
given by a diagonalization of the potential with a harmonic oscillator basis for
well-bound states, and by a bisection method in the real E-axis or complex kplane for loosely bound, antibound, and resonance states.
2. Equation (2.2) is integrated from zero to a matching radius R m and Eq. (2.6) is
used for the boundary condition in r = 0. This provides with u(R m ).
3. Equation (2.2) is integrated from R to the matching radius R m and Eq. (2.7) is
used for the boundary condition in r = R, along with C + = 1 and C − = 0. This
provides with u + (R m ).
4. J + (k) is then calculated from u(R m ), u + (R m ) and their derivatives. dJ + (k)/dk
is calculated from the standard difference formula (J + (k p ) − J + (k))/(k p − k),
where k p is the linear momentum issued from the previous iteration. The linear
65
k 0 in this equation is the linear momentum associated to E 0 . The integrand in
Eq. (2.194) is now O(ln(E ) E −5/2 ), so that the gain in precision is about one to
two orders of magnitude. As (S (k)) in Eq. (2.194) is a function of an integral
involving (S (k)), hence the knowledge of |S (k)| implies S (k).
The S-matrix is often considered for complex potentials as well. The addition
of an imaginary part to the real potential of Eq. (2.2) breaks the unitarity of the Smatrix, so that the particle flux is no longer conserved. This idea is behind the optical
model potential, where its absorptive, negative imaginary part accounts for the
flux lost to the compound nucleus component in the scattering process. Hence, the
complex optical potential arises from an attempted simplification of the description
of the reaction [45–47], and can be considered as an effective way to take into
account all of the physical processes which one does not want to consider explicitly.
In the exact formulation using a Hermitian Hamiltonian, a huge number of
reaction channels should be taken into account to generate particle absorption into
the compound nucleus configurations. This strategy in the description of direct
reactions is not only impractical but also impossible to realize in any practical
application. The phenomenological replacement of those reaction channels by the
(negative) imaginary part of optical potential can be justified by the fact that
the individual compound nucleus configurations cannot be studied in practice.
Therefore, their effects can be represented by simple functions, which take the form
of an imaginary part of the complex optical potential in practice.
2.6.7 Radial Equation and Shooting Method
The one-body bound, antibound, and resonances are calculated numerically by
integrating Eq. (2.2) with the shooting method which is based on finding the zeros
of the Jost function J + (k) of Eq. (2.169). For this, one has to provide a starting
value k start for the linear momentum of the one-body state to calculate, which is
then refined using the Newton method on k:
1. One considers k start , close to the exact k value of the resonant state. It is typically
given by a diagonalization of the potential with a harmonic oscillator basis for
well-bound states, and by a bisection method in the real E-axis or complex kplane for loosely bound, antibound, and resonance states.
2. Equation (2.2) is integrated from zero to a matching radius R m and Eq. (2.6) is
used for the boundary condition in r = 0. This provides with u(R m ).
3. Equation (2.2) is integrated from R to the matching radius R m and Eq. (2.7) is
used for the boundary condition in r = R, along with C + = 1 and C − = 0. This
provides with u + (R m ).
4. J + (k) is then calculated from u(R m ), u + (R m ) and their derivatives. dJ + (k)/dk
is calculated from the standard difference formula (J + (k p ) − J + (k))/(k p − k),
where k p is the linear momentum issued from the previous iteration. The linear