A Unified Scheme of Central Symmetric Shape-Invariant Potentials
- URL: http://arxiv.org/abs/2001.02068v4
- Date: Thu, 4 Jan 2024 12:07:38 GMT
- Title: A Unified Scheme of Central Symmetric Shape-Invariant Potentials
- Authors: Taha Koohrokhi and Abdolmajid Izadpanah and Mitra Gerayloo
- Abstract summary: Angular momentum, denoted as $ell$, is a conserved quantity that appears in the centrifugal potential when a particle moves under a central force.
This study introduces a formalism in which $ell$ plays a unifying role, consolidating central potentials into a superpotential.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Most physical systems, whether classical or quantum mechanical, exhibit
spherical symmetry. Angular momentum, denoted as $\ell$, is a conserved
quantity that appears in the centrifugal potential when a particle moves under
the influence of a central force. This study introduces a formalism in which
$\ell$ plays a unifying role, consolidating solvable central potentials into a
superpotential. This framework illustrates that the Coulomb potential emerges
as a direct consequence of a homogenous ($r$-independent) isotropic
superpotential. Conversely, a $\ell$-independent central superpotential results
in the 3-Dimensional Harmonic Oscillator (3-DHO) potential. Moreover, a local
$\ell$-dependent central superpotential generates potentials applicable to
finite-range interactions such as molecular or nucleonic systems. Additionally,
we discuss generalizations to arbitrary $D$ dimensions and investigate the
properties of the superpotential to determine when supersymmetry is broken or
unbroken. This scheme also explains that the free particle wave function in
three dimensions is obtained from spontaneous breakdown of supersymmetry and
clarifies how a positive 3-DHO potential, as an upside-down potential, can have
a negative energy spectrum. We also present complex isospectral deformations of
the central superpotential and superpartners, which can have interesting
applications for open systems in dynamic equilibrium. Finally, as a practical
application, we apply this formalism to specify a new effective potential for
the deuteron.
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