A Unified Scheme of Central Symmetric Shape-Invariant Potentials
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.