An approximation to the Woods–Saxon potential based on a contact interaction

Abstract

We study a non-relativistic particle subject to a three-dimensional spherical potential consisting of a finite well and a radial δ − δ contact interaction at the well edge. This contact potential is defined by appropriate matching conditions for the radial functions, thereby fixing a self-adjoint extension of the non-singular Hamiltonian. Since this model admits exact solutions for the wave function, we are able to characterize and calculate the number of bound states. We also extend some well-known properties of certain spherically symmetric potentials and describe the resonances, defined as unstable quantum states. Based on the Woods-Saxon potential, this configuration is implemented as a first approximation for a mean-field nuclear model. The results derived are tested with experimental and numerical data in the double magic nuclei 132 Sn and 208 Pb with an extra neutron.

A solvable contact potential based on a nuclear model

“…The present work generalizes the model introduced in a recent publication in which the energy levels of neutrons in doubly magical nuclei were studied [1]. The analysis carried out there is taken a step further by introducing a Coulombic term so that we can study the neutron-and proton-level scheme.…”

“…In this section, we present the model that we intend to analyze using a notation adapted to possible applications in nuclear physics, as we did in [1]. The starting point is the following single-particle three-dimensional Hamiltonian, written with respect to the coordinates of the center of mass of the system,…”

“…The main advantage of these contact potentials is that they can often be resolved exactly, which gives a good insight of some quantum properties. This is why these interactions have a large number of applications in the modeling of real physical systems, some of which are listed in [1] and the references quoted therein [3][4][5][6][7][8][9][10][11][12][13]. In this regard, it is worth mentioning a recent work in which the vacuum interaction energy of a three-dimensional scalar field is calculated in the presence of two concentric spheres defined by the aforementioned contact interaction [14], so that the δ term leads to positive Casimir interaction energies, which are unattainable only with the Dirac δ.…”

“…These definitions are based on the construction of different self-adjoint extensions of the Hamiltonian without the singular term [17]. These extensions are initially defined for a one-dimensional Schrödinger equation but can be adapted to our problem with slightly modifications due to the spherical symmetry of the potential [1]. Here, we will use the local δ interaction, since it is compatible with the unambiguously defined δ potential when both interactions are supported on the same radius [17][18][19][20].…”

“…In fact, the Dirac δ interaction has been used for the calculation of resonant parameters and energy spectrum in [22], and it has also been employed to study the spectral function of the unbound nucleus 25 O [23]. Following the work [1], we now study the energy levels of protons and compare them with those of neutrons in the previous paper. As we will see, the general properties of the structure of the level scheme are maintained, although there are certain differences due to the Coulombic term.…”