NUCLEAR PHYSICS GROUP DEPARTMENT OF ENGINEERING PHYSICS

1. NUCLEAR PHYSICS GROUP • DEPARTMENT OF • ENGINEERING PHYSICS Unıversity of Gaziantep 27310, Gaziantep, Türkiye NPG Web Page : http://www1.gantep.edu.tr/~ozer E-Mail Addresses: ozer@gantep.edu.tr Unified treatment of screening Coulomb and anharmonic oscillator potentials in arbitrary dimensions Okan Özer, Bülent Gönül Department of Engineering Physics, University of Gaziantep, 27310, Gaziantep, Türkiye

2. UNIFIED TREATMENT OF SCREENING COULOMBAND ANHARMONIC OSCILLATOR POTENTIALS IN ARBITRARY DIMENSIONS • Abstract • A mapping is obtained relating radial screened Coulomb systemswith low screening parameters to radial anharmonic oscillatorsin N-dimensional space. Using the formalism ofsupersymmetric quantum mechanics, it isshown that exact solutions of these potentials exist when theparameters satisfy certain constraints.

3. p.1 MAPPINGS BETWEEN THE TWO DISTINCT SYSTEMS (1) Eq. (1) is transformed to (2) where and . If it is substituted (3) where

4. p.2 And from Eq. (3), it is obtained that (4) The static-screened Coulomb potential is given as (5) Then, within the frame of low screening parameter, , it becomes as (6)

5. p.3 NOW, Eq. (6) is transformed to the anharmonic oscillator using the procedure as mentioned above (with the choice of ) (7) with the eigenvalue (8) Thus the system of Eq. (5) is reduced to another system defined by Eq. (7) !!!

6. p.4 Supersymmetric treatment for the ground state Using the SUSYQM, we set the superpotential term as (9) for the potential given in Eq. (6). Then the SUSY-partner potential is found as (10)

7. p.5 where (11) The physically observables for the interested potential under the constraints (12) are found as (13)

8. p.6 For the anharmonic oscillator potential, we set (14) which leads to (15) and leads to physically meaningful eigenvalue (16) where .

9. p.7 Significance of mapping parameter To make clear the significance of the mapping parameter, , we consider Eq. (13) and Eq. (16) together with and arrive at To be consistent with Eq. (8), it is imposed that such that Numerical results for the interested potentials are tabulated for different values of screening parameter, angular momentum quantum number in arbitrary dimensions in Table 1 and 2.

10. p.8 Table 1. The first four eigenvalues of the screening Coulomb potential as a function of the screening parameter in atomic units.

11. p.9 8 Table 2.Ground-state eigenvalues of the anharmonic potential

12. p.10 CONCLUDING REMARKS As theobjective of this presentation we have highlighted a differentfacet of these studies and established a very general connectionbetween the screened Coulomb and anharmonic oscillator potentialsin higher dimensional space through the application of a suitabletransformation. The purpose being the emphasize the pedagogicalvalue residing in this interrelationship between two of the mostpractical applications of quantum mechanics. The exact groundstate solutions for the potentials considered are obtainedanalytically within the framework of supersymmetric quantummechanics.