exotic nuclei structure and reaction noyaux exotiques ... - IPN - IN2P3

The rotational spectrum and the attractive delta-shell potential.
IPNO Participation: M. Lassaut and R.J. Lombard
Collaboration : R. Yekken, Faculté de Physique, USTHB, Bab Ezzouar, Alger, Algeria.
Le spectre d’un potentiel « delta-shell » attractif est calculé dans l’espace à D dimensions, D ≥ 2. Une
formule compacte est dérivée dans la limite d’une forte constante d’interaction pour un numbre impair de
dimensions. Par rapport à l’état fondamental, le spectre a une allure rotationnelle ; il est proportionnel à
L(L + D — 2) , où L est le grand moment orbital. L’extension aux valeurs paires de D s’obtient par
application du théorème de Hellmann-Feynman. En simulant le potentiel delta-shell par une gaussienne,
nous discutons les effets de portée finie.
In the D=3 dimensional space, it is well known that
the L(L + 1) spectrum is generated by the L 2 operator
acting on the spherical harmonics. When use is
made of the Schrödinger equation, the mark of this
L dependence in the eigenvalues depends very
much on the radial shape of the potential (here
assumed local and spherically symmetric). For instance,
this mark is absent in the Coulomb
and harmonic oscillator cases, whereas a L (L + 1)
component appears explicitly in the eigenvalues
with the Kratzer and Morse potentials.
On the other hand, it has recently been found that
a L(L + 1) spectrum arises also from a finite range
attractive potential having a two parameter Fermi
function shape with a deep well at the surface.
A similar conclusion was reached by applying the
techniques of the inverse problem, in the case of
discrete states to this spectrum. In this work, it was
conjectured in that indeed the L(L + 1) spectrum is
characteristic of an attractive delta-shell potential
with a large coupling constant.Our purpose is to
provide a proof of this conjecture by solving directly
the Schrödinger equation for the delta-shell
potential. We started by studying the D=3 dimensional
space. We found afterwards that the results
can be easily extended to any dimension D ≥ 2.
The eigenvalues are obtained by matching the free
particle solutions at the shell radius. For D odd,
the matching conditions yields equations, which
can be written explicitly for each L. In the asymptotic
domain, namely for large potential strength,
these equations take a compact form The results
are extended to even D on the ground of the Hellmann-Feynman
theorem.
In the limit of large potential strength, the spectra
with respect to the ground state follow the rotational
formula in L( L + D – 2 ).
Finally, finite range effects are studied by simulating
the delta-shell potential by a normalized Gaussian.
References:
M. Barranco, M. Pi, S.M. Gatica, E.S. Hernandes
and J.Navarro, Phys. Rev. B 56 (1997) 8997.
J. Navarro, A. Poves, M. Barranco and M. Pi,
Phys. Rev. A 69 (2004) 023202.
R. Yekken, F.-Z. Ighezou and R.J. Lombard, Ann.
Phys. (NY) 323 (2008) 61.
K. Gottfried, Quantum Mechanics I, W.A. Benjamin,
Inc, New York Amsterdam (1966).
The solutions of the Schrödinger equation for the
delta-shell potential are known since a long time. A
comprehensive treatment is given, for instance, in
the textbook by Gottfried.
However, to our knowledge the relationship with
the L(L + 1) spectrum has never been studied.
Actually, in his textbook, Gottfried underlines the
fact that as L is increasing, the solution of the
Schrödinger equation becomes rapidly messy. We
show that in the case of a large coupling constant,
the eigenvalue are given by a compact expression,
which is solved easily.
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