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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: [email protected] 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 UNIFIED TREATMENT OF SCREENING COULOMB AND ANHARMONIC OSCILLATOR POTENTIALS IN ARBITRARY DIMENSIONS Abstract A mapping is obtained relating radial screened Coulomb systems with low screening parameters to radial anharmonic oscillators in N-dimensional space. Using the formalism of supersymmetric quantum mechanics, it is shown that exact solutions of these potentials exist when the parameters satisfy certain constraints. p.1 MAPPINGS BETWEEN THE TWO DISTINCT SYSTEMS 1 d 2 R N 1 dR ( N 2) R E V ( r )R 2 2 2 dr r dr 2r (1) Eq. (1) is transformed to d 2 ( M 1)( M 3) 2V (r ) 2 E 2 2 dr 4r where (2) (r ) r ( N 1) / 2 R(r ) and M N 2 . If it is substituted r 2 and R F ( ) / 1 d 2 F N 1 dF L( L N 2) F Eˆ Vˆ ( ) F (3) 2 d 2 d 2 2 where N 2 N 2 2 , L 2 p.2 And from Eq. (3), it is obtained that Eˆ Vˆ ( ) E 2 2 2 2V (2 / 2) (4) The static-screened Coulomb potential is given as VSC ( r ) e 2 e r r (5) Then, within the frame of low screening parameter, , it becomes as e2 e2 2 e2 3 2 e2 4 3 e2 5 4 2 VSC (r ) e r r r r r 2 6 24 120 A1 (6) A2 A3r A4r 2 A5r 3 A6r 4 r p.3 NOW, Eq. (6) is transformed to the anharmonic oscillator using the procedure as mentioned above (with the choice of 2 1 / En 0 ) A2 ˆ V ( ) 1 En 0 2 A3 A4 4 6 3/ 2 2 2 En 0 4 En 0 A5 A6 8 10 5/ 2 3 8 En 0 16 En 0 (7) with the eigenvalue Eˆ n 0 2 A1 En 0 1/ 2 Thus the system of Eq. (5) is reduced to another system defined by Eq. (7) !!! (8) p.4 Supersymmetric treatment for the ground state Using the SUSYQM, we set the superpotential term as a W (r ) 1 a2 a3r a4r 2 , a4 0 r (9) for the potential given in Eq. (6). Then the SUSY-partner potential is found as V (r ) W 2 (r ) W (r ) 2a1a2 a22 a3 (2a1 1) 2(a1a4 a4 a2a3 )r r a (a 1) 2(a2a4 a32 )r 2 2a3a4r 3 a42r 4 1 1 r2 2 A1 2 A2 2 A3r 2 A4r 2 2 A5r 3 2 A6r 4 r ( M 1)( M 3) 2 En 0 (10) 4r 2 p.5 where a1 M 1 2 A1 , a2 , a3 2 M 1 A5 , a4 2 A6 2 A6 (11) The physically observables for the interested potential under the constraints 2 8 A6 A4 2 A5 A1 ( M 1) , 16 A6 2 A6 ( M 1) A1 A5 A3 2 A6 2 ( M 1 ) A 6 (12) are found as a a n 0 (r ) N0 r a1 exp a2r 3 r 2 4 r 3 2 3 1 4 A12 En 0 A2 2 2 ( M 1) A5 M 2 A6 (13) p.6 For the anharmonic oscillator potential, we set W ( ) a b 5 3 c d , a0,d 0 (14) b d a n 0 ( ) C0 c exp 6 4 2 4 2 6 (15) which leads to and leads to physically meaningful eigenvalue 2 d 8 A A 2 A M 6 4 5 ˆ En 0 (2c 1) 2 16 A6 2 A6 En 0 1 / 2 where M N 2 L . (16) 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 M A1 ˆ En 0 1/ 2 M 1 E n 0 To be consistent with Eq. (8), it is imposed that such that 0 1 M 2( N 1 ) 2(2 ) 2 M 1 N 2 1 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. p.8 Table 1. The first four eigenvalues of the screening Coulomb potential as a function of the screening parameter in atomic units. p.9 Table 2. Ground-state eigenvalues of the anharmonic potential 8 p.10 CONCLUDING REMARKS As the objective of this presentation we have highlighted a different facet of these studies and established a very general connection between the screened Coulomb and anharmonic oscillator potentials in higher dimensional space through the application of a suitable transformation. The purpose being the emphasize the pedagogical value residing in this interrelationship between two of the most practical applications of quantum mechanics. The exact ground state solutions for the potentials considered are obtained analytically within the framework of supersymmetric quantum mechanics.