4
Two-Particle Systems in the Berggren Basis
The theoretical and numerical developments of Chap. 3 can be directly applied to
the two-body problem which is solvable in terms of relative and center-of-mass
coordinates and reduces to a one-body problem therein. Consequently, it is possible
to study two-nucleon systems, consisting of diproton, dineutron, and deuteron, with
one-body Schrödinger equations. These systems are of fundamental importance in
nuclear physics as they provide most of the information gathered on the nucleonnucleon interaction. Moreover, diproton, dineutron, and deuteron ground states are
either loosely bound or unbound, so that their study with the Berggren basis is all
the more justified.
Many complicated many-body systems can also be conveniently represented as
two-body problems. This is particularly the case for weakly coupled subsystems,
consisting of weakly bound or unbound nucleon (electron) interacting with a
nuclear (molecular) core. Indeed, the action of the core on external particle can be
modeled by the so-called pseudo-potentials, i.e. coupling potentials which mimic
interactions in the many-body system. It will be shown in this chapter that in
many cases, pseudo-potentials with a collective rotation of the core reproduce
remarkably accurately the properties of many-body systems. As an example of such
an approach, the particle-rotor model will be formulated and applied to weakly
bound or unbound dipolar and quadrupolar anions, as well as to the one-neutron
halo nucleus 11 Be.
4.1
Exact Formulation of the Two-Particle Problem Using
Relative Coordinates
As in classical mechanics, the two-body problem in quantum mechanics is solvable,
as it reduces to two independent one-body problems. Let us introduce center-of© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_4
147
Two-Particle Systems in the Berggren Basis
The theoretical and numerical developments of Chap. 3 can be directly applied to
the two-body problem which is solvable in terms of relative and center-of-mass
coordinates and reduces to a one-body problem therein. Consequently, it is possible
to study two-nucleon systems, consisting of diproton, dineutron, and deuteron, with
one-body Schrödinger equations. These systems are of fundamental importance in
nuclear physics as they provide most of the information gathered on the nucleonnucleon interaction. Moreover, diproton, dineutron, and deuteron ground states are
either loosely bound or unbound, so that their study with the Berggren basis is all
the more justified.
Many complicated many-body systems can also be conveniently represented as
two-body problems. This is particularly the case for weakly coupled subsystems,
consisting of weakly bound or unbound nucleon (electron) interacting with a
nuclear (molecular) core. Indeed, the action of the core on external particle can be
modeled by the so-called pseudo-potentials, i.e. coupling potentials which mimic
interactions in the many-body system. It will be shown in this chapter that in
many cases, pseudo-potentials with a collective rotation of the core reproduce
remarkably accurately the properties of many-body systems. As an example of such
an approach, the particle-rotor model will be formulated and applied to weakly
bound or unbound dipolar and quadrupolar anions, as well as to the one-neutron
halo nucleus 11 Be.
4.1
Exact Formulation of the Two-Particle Problem Using
Relative Coordinates
As in classical mechanics, the two-body problem in quantum mechanics is solvable,
as it reduces to two independent one-body problems. Let us introduce center-of© Springer International Publishing AG 2021
N. Michel, M. Płoszajczak, Gamow Shell Model, Lecture Notes in Physics 983,
https://doi.org/10.1007/978-3-030-69356-5_4
147