Spin and Pseudo-Spin Symmetries in Radial Dirac Equation and Exceptional Hermite Polynomials (2107.13644v1)

Abstract: We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

• The paper derives solutions to the radial Dirac equation using exceptional Hermite polynomials to link energy eigenvalues with potential well depth.
• It transforms the Dirac equations into solvable forms, clarifying the behavior of spin and pseudospin symmetries in nuclear shell structures.
• The study proposes new energy-dependent potential models with significant implications for quantum mechanics and nuclear physics.

Spin and Pseudo-Spin Symmetries in Radial Dirac Equation and Exceptional Hermite Polynomials

Introduction

The paper investigates the solutions to the radial Dirac equation with a tensor potential, emphasizing spin and pseudospin symmetries. These solutions are extended to the exceptional Hermite polynomials family. This research develops new rational potential models that are generalizations of nonlinear isotonic potential families and are energy-dependent. The spin and pseudospin symmetries have historical roots in describing nuclear shell structures and have significant implications in addressing magic numbers in exotic nuclei.

Dirac Equation under Symmetry Limits

Spin Symmetry

In the spin-symmetric case, the scalar and vector potentials are equal, $S(r) = V(r)$. This leads to a pair of second-order differential equations for the radial functions $F(r)$ and $G(r)$. The symmetry results in simplifications that are crucial in nuclear physics, particularly regarding doublets and the splitting of orbitals in nuclear potentials. The paper constructs solutions to these equations by transforming them into forms solvable through exceptional orthogonal polynomials.

Pseudo-Spin Symmetry

For pseudospin symmetry, the vector potential is the negative of the scalar potential, $V(r) = -S(r)$. This transforms the Dirac equations into another form that can be related to exceptional Hermite polynomials through appropriate mappings and transformations. Pseudospin symmetry plays a critical role in heavy nuclei, and the solutions offered provide insights into the degeneracies observed in nuclear shells.

Transformations to Exceptional Hermite Polynomials

The paper utilizes a mathematical framework involving the Wronskian determinant and associated transformations to connect the solutions of Dirac equations under the symmetry limits to exceptional Hermite polynomials. Exceptional orthogonal polynomials, extending classical Hermite, Laguerre, and Jacobi polynomials, provide a robust theoretical tool for solving these quantum systems with potential functions that depend on energy. The methodology involves complex transformations and function mappings to handle the differential equations presented.

Analytical Solutions and Results

Spin Symmetric Solutions

The paper derives eigenvalues and solves the Dirac equation under spin symmetries using exceptional Hermite polynomials. The tensor potential in these cases is repulsive, and the energy eigenvalues are linked to polynomial properties. The results demonstrate how potential well depth increases with quantum number $n$. Graphical representations of probability densities and potentials further illustrate the dynamics.

Pseudo-Spin Symmetric Solutions

For pseudospin symmetric cases, the transformation framework applies similarly, leading to solutions that reveal the repulsive nature of the tensor potential. The potential models show a slow increase with model parameters, and the paper's methodology allows for a clear interpretation of the shifted energy landscapes in these models.

Implications and Future Directions

The extension of the radial Dirac equation solutions to include exceptional Hermite polynomials opens new avenues in studying quantum mechanics with potential functions not captured by traditional methods. These energy-dependent systems potentially influence the development of models in quantum field theory and nuclear physics, lending insight into heavy quark systems and semiconductors. Further exploration of exceptional orthogonal polynomials could yield rich theoretical and practical advancements in understanding quantum symmetry and interactions.

Conclusion

The paper presents a rigorous examination of the radial Dirac equation's solutions through the lens of modern mathematical frameworks, offering a nuanced view of spin and pseudospin symmetries substantiated by exceptional Hermite polynomials. By bridging complex symmetry considerations with practical, energy-dependent potential models, this paper advances the computational toolkit available to physicists in theoretical and applied domains.