4.3 The Particle-Rotor Model in the Berggren Basis
157
wave functions, as the asymptote of two-body systems is built from scattering basis
states, which are unbound, and one loosely bound basis state. Additional numerical
illustrations of the two-body problem in the complex plane are done in Exercise III.
Exercise III
Using several sets of Hamiltonian parameters, we will study in a numerical
example the physical observables associated to two-body systems presented in this
section.
A. To calculate the observable quantities presented in this section, run the code
dedicated to the two-body systems in intrinsic coordinates using default
parameters for the three cases shown above, that is dineutron, diproton, and
deuteron. Modify parameters in the Hamiltonian to check the precision of a
numerical method, based on the diagonalization of two-body Hamiltonian in
the Berggren basis. The set of parameters consists of the parameters of WoodsSaxon basis, the k values defining Berggren basis contours, and the number of
discretized scattering states. Modify each of these parameters by 10%. Notice
that results are stable against these changes.
B. Explain why the use of a Fermi function is independent of the precision of the
Berggren basis expansion.
4.3
The Particle-Rotor Model in the Berggren Basis
The main objective in physics of nonrelativistic complex systems is to solve the
many-body Schrödinger equation as precisely as possible. While this is feasible
when one has few nucleons or electrons, the numerical cost of calculations augments
very rapidly with the number of active particles, so that approximation schemes
become quickly necessary.
The simplest approach in this case is to separate particles of nucleus or molecules
into inactive particles of the core and active particles of the valence part. The
core has to possess the global features of the whole many-body system. It is
modeled by a potential, whose parameters are typically fitted from experimental
data. While in most applications the core can be spherical, there are situations
where this assumption is no longer sufficient. Indeed, the multiple interactions
existing at microscopic level may generate collective excitations, of vibrational and
rotational nature, which implies deformation of the core. Even though this could
be simply solved by the use of a nonspherical potential, a direct implementation of
deformation in the core potential breaks rotational invariance of the Hamiltonian. In
fact, the many-body wave function of a deformed nucleus is the linear combination
of intrinsic wave functions of fixed deformation along a given axis, so that the full
wave function fulfills the rotational invariance.
157
wave functions, as the asymptote of two-body systems is built from scattering basis
states, which are unbound, and one loosely bound basis state. Additional numerical
illustrations of the two-body problem in the complex plane are done in Exercise III.
Exercise III
Using several sets of Hamiltonian parameters, we will study in a numerical
example the physical observables associated to two-body systems presented in this
section.
A. To calculate the observable quantities presented in this section, run the code
dedicated to the two-body systems in intrinsic coordinates using default
parameters for the three cases shown above, that is dineutron, diproton, and
deuteron. Modify parameters in the Hamiltonian to check the precision of a
numerical method, based on the diagonalization of two-body Hamiltonian in
the Berggren basis. The set of parameters consists of the parameters of WoodsSaxon basis, the k values defining Berggren basis contours, and the number of
discretized scattering states. Modify each of these parameters by 10%. Notice
that results are stable against these changes.
B. Explain why the use of a Fermi function is independent of the precision of the
Berggren basis expansion.
4.3
The Particle-Rotor Model in the Berggren Basis
The main objective in physics of nonrelativistic complex systems is to solve the
many-body Schrödinger equation as precisely as possible. While this is feasible
when one has few nucleons or electrons, the numerical cost of calculations augments
very rapidly with the number of active particles, so that approximation schemes
become quickly necessary.
The simplest approach in this case is to separate particles of nucleus or molecules
into inactive particles of the core and active particles of the valence part. The
core has to possess the global features of the whole many-body system. It is
modeled by a potential, whose parameters are typically fitted from experimental
data. While in most applications the core can be spherical, there are situations
where this assumption is no longer sufficient. Indeed, the multiple interactions
existing at microscopic level may generate collective excitations, of vibrational and
rotational nature, which implies deformation of the core. Even though this could
be simply solved by the use of a nonspherical potential, a direct implementation of
deformation in the core potential breaks rotational invariance of the Hamiltonian. In
fact, the many-body wave function of a deformed nucleus is the linear combination
of intrinsic wave functions of fixed deformation along a given axis, so that the full
wave function fulfills the rotational invariance.