Journal of Physics: Conference Series PAPER OPEN ACCESS The solution of -dimensional Schrodinger equation for Scarf potential its partner potential constructed By SUSY QM To cite this article: Wahyulianti et al 7 J. Phys.: Conf. Ser. 99 View the article online for updates enhancements. Related content - Quantum Chemistry: The hydrogen atom A J Thakkar - The solution of -dimensional Schrodinger equation for polynomial inverse potential its partner potential by using ansatz wavefunction method A Suparmi, C Cari, Wahyulianti et al. - The construction of partner potential from the general potential Rosen-Morse Manning Rosen in dimensional Schrodinger system Kristiana Nathalia Wea, A. Suparmi, C. Cari et al. Recent citations - The solution of -dimensional Schrodinger equation for polynomial inverse potential its partner potential by using ansatz wavefunction method A Suparmi et al This content was downloaded from IP address 8.5..8 on //8 at 9:5
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// The solution of -dimensional Schrodinger equation for Scarf potential its partner potential constructed By SUSY QM Wahyulianti, A. Suparmi, C. Cari, Kristiana N. Wea Physics Department of Post Graduate Program Sebelas Maret University, Jl. Ir. Sutami 6A Kentingan Jebres Surakarta 576, INDONESIA E-mail: wahyuliantiyuli@yahoo.com Abstract. The angular part of -dimensional Schrodinger equation for Scarf potential was solved by using the Nikiforov-Uvarov method Supersymmetric Quantum Mechanic method. The determination of the ground state wave function has been used Nikiforov- Uvarov method by applying the parametric generalization of the hypergeometric type equation. By using manipulation of the properties operators of the Supersymmetric Quantum Mechanic method the partner potential was constructed. The ground state wave functions of original Scarf potential is different than the ground state wave functions of the construction result potential.. Introduction The D-dimensional Schrodinger equation is one of the wave equations for higher dimensions in quantum mechanics [-]. The D-dimensional Schrodinger equation for some potentials are exactly solved by some methods such as Nikiforov-Uvarov (NU) method [-6], Supersymmetric Quantum Mechanic (SUSY QM) [,8], Romanovski polynomial method [9], Asymptotic Iteration Method (AIM) [] Factorization methods []. The NU method is one of the methods with common application, NU differential equations have been formulated with general parameters of the general hypergeometric type equation []. By applying manipulating the operators of SUSY QM, the properties of SUSY QM the ground state wave functions of the original potential, the partner potential was constructed.[] The purpose of this research is to construct partner potential which is the new potential of the generalized Scarf potential by manipulating the partner potential equations with the ground state wave function of Scarf potential. In this paper, the solution of angular D-dimensional Schrodinger equation for Scarf potential was studied using Nikiforov-Uvarov method with hypergeometric type equations SUSY QM. The paper is organized as follows. The method is presented in Section. The results discussion are presented in Section a conclusion in Section.. Method In this section, we review the solution of D-dimensional Schrodinger equation with Nikiforov- Uvarov method Supersymmetry Quantum Mechanics method. The Generalized Schrodinger equation in D-dimensional shown as followed equation [,] given as D ( r, ) V ( r, ) ( r, ) E ( r, ) () Content from this work may be used under the terms of the Creative Commons Attribution. licence. Any further distribution of this work must maintain attribution to the author(s) the title of the work, journal citation DOI. Published under licence by Ltd
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// with ( r, ) U( r) Y ( xˆ) ; xˆ,, () D l ld l D r The -dimensional Schrodinger equation obtained by inserted Eq. () into Eq. () solve of using variable separation method with Y (,,..., ) P( ) P( ) P( ) is given as []: U U E V ( r) U r r P( ) V P( ) P( ) P( ) sin V (5) P( ) sin sin P( ) sin V (6) P( ) sin sin where, is variable separation constant. In this paper the potential energi of Scarf potential generally given as []: Ai Bi cos i Vi i, for i =,. (7) sin i sin i with Ai bi ai ai Bi bi ai.. Review of NU method The solution of Schrodinger equation is reduced to the hypergeometric type differential equation by using variable transformation. By using the NU method, the solution of the hypergeometric type differential equation is expressed as ( s) ( s) ( s) ( s) ( s) (8) s ( s) s ( s) where () s is first order polynomial, () s () s are mostly second order polynomials. By using variable separation method, substitution equation () with ( s) (s) y(s) (9) we obtain hypergeometric type as y( s) y( s) ( s) y( s) s s () where (s) is logarithmic derivative with (s),, are k () nn ( ), k n n, n,,,... () The solution of the part y n (s) by using Rodrigues relation, is given as () ()
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// n Cn d n yn ( s) ( s) w( s) n () w() s ds with C n is normalization constant, the weight function w(s) must satisfy the condition w ( s) w( s) () s The parametric generalization of the hypergeometric type equation in equation (8) is written as ( s) (c cs ) ( s) ( s s ) ( s) (5) s s( c s) s s ( c s) By comparing equation (8) equation (5), we obtained the energy value equation eigen function are respectively given as c n (n ) c (n ) c c c n( n )c c c c c c (6) where 5 9 8 7 8 8 9 c ( c / c ) ( c, ( c / c ) c ) ( s) N s c s P c s (7) c n n c ( c ), c5 (c c ), c6 c 5, c7 c c 5, c8 c c c c c c c, c c c c, c c c (8) 9 7 8 6 8 8 c c c c c c, c c c c c 5 9 8 5 9 8 By using equations (6) (7) for n = the ground state energy wave function are determined [-6].. Review of SUSY QM Witten defined two charge operators are commute with Hamiltonian (H ss ) of the supersymmetry quantum system [,8] is geven as d d( x) ( x) dx dx H H ss (9) d d( x) H ( x) dx dx with partner Hamiltonian H H. By setting the new SUSY operators for raising d operator A ( x) dx d A ( x dx ). The SUSY Hamiltonians of equation (9) are H () x H A A, H() x H AA () ( ) A A () The partners of potential in the SUSY QM method is V V V V are V ( x) V ( x) ( x) V ( x) V ( x) ( x) () where φ is superpotential. The partner potential V is determined as V ( x; a ) V ( x; a ) V ( x) E ef () with V ef is effective potential E is the ground state energy.
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// By manipulating equation () by applying equation (7) equation (), we construct the new partner potential V as d d V( x) Vef ( x) E dx () dx where is the ground state wave function [,8]. Then the new eigen function of new partner potential is also determined by using Nikiforov-Uvarov method.. Results Discussion In this section, the solution of the Schrodinger equation in -dimensional with Nikiforov-Uvarov method Supersymmetry Quantum Mechanics method. The solution of the Schrodinger equation in -dimensional for obtained by inserting Eq. (7) to Eq. () is given as P( ) A B cos ( P ) P( ) (5) sin sin By substituting Eq. (5) with cos s we obtain i s d P ( s) dp ( s) B A B s s P (s) ds s s ds s s (6) By comparing Eq. (6) Eq. (5) then by applying Eq. (6), Eq. (7) Eq. (8) we obtained the λ eigen function are respectively given as A n( n ) (n ) A B A B A B A B (7) A B A B A B, A cos B cos P( ) N P n cos (8) with B c, c c,,, A B (9) The solution of Eq. (7) Eq. (8) for the ground state n =, we obtained A A B A B A B A B A B A B cos cos P ( ) N The partner potential V as new potential constructed from Eq. () Eq. () is () ()
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// with A B cos c c V ( ) E sin cos sin () cos E, c A B c A B () The solution of the Schrodinger equation in -dimensional for partner potensial obtained by inserting Eq. () to Eq. () is given as P( ) A B cos c c E P( ) P( ) sin cos sin cos By substituting Eq. () with cos s d P( s) dp( s) we get i B s E s E c c s P ( s) ds s s ds s s A B c () (5) By comparing Eq. (5) Eq. (5) then by applying Eq. (6), Eq. (7) Eq. (8) we obtained eigen function are respectively given as E c A c A B c A B c 8 A B c A B c A B c A B c cos cos ( ) N P Secondly, the similarly with the steps of for solution of by applying Eq. (5) as the Schrodinger equation. The solution of for ground state n =, we obtained eigen function are respectively given as: A A B A B A B A B (8) (6) (7) 5
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// A B A B cos cos P ( ) N sin (9) By inserting Eq. (9) to Eq. () We get the partner potential θ as A B cos c c V ( ) E () sin cos sin sin cos with E, c A B c A B () By substituting Eq. () to Eq. (5) then by applying Eq. (5) - Eq. (8) we obtain c c E A A B c A B c A B c A B c () A B c A B c cos cos ( ) N P sin Finally, the similarly with the steps of for solution of by applying Eq. (6) as the Schrodinger equation. Then eigen function for ground state n = are respectively given as: 5 A A B A B 8 A B A B A B A B cos cos P ( ) N sin By Substituting Eq. (5) to Eq. () we obtain the partner potential θ as A B cos c c V ( ) E sin cos sin sin cos with E, c A B c A B (7) () () (5) (6) 6
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// By substituting Eq. (6) to Eq. (6) then by applying Eq. (5) - Eq. (8) we get: B E c c A B c A B c A B c A B c (8) A B c A B c cos cos ( ) N sin P The generalized wave functions are A B A B A B, A cos B cos n n P( ) N P cos A B A B cos cos, n n A B A B P( ) N P cos sin P( ) A B A B cos cos A B, A B N P cos n n sin A B c A B c A B c, A B c cos cos n n P ( ) N P cos A B c A B c cos cos Nn P( ) sin A B c, A B c n cos A B c A B c A B c cos cos, A B c n n sin P ( ) N P By using the same value of potential parameters, the wave functions for the original Scarf potential the partner potential as potential construction result are different. For details of the wave function of Equation (5) Equation (5) can be shown Table. P cos (9) (5) (5) 7
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// Table Three-dimensional representations of the angular wave function for variation of n l, a b. a b n l P(θ ) P (θ ) (a) (b) (c) (d) 6 6 (e) (f) 8 8 (g) (h) 8
International Conference on Science Applied Science 7 IOP Conf. Series: Journal of Physics: Conf. Series 56789 99 (7) doi :.88/7-6596/99// From Table, we have wave function for variations of a, b, n l. The increases of n l value is given the effect in form of wave functions the motion range of particle. The wave functions for the original Scarf potential its the partner potential are different. There is a decrease in the amplitude value at P (θ ) beside P (θ ). An increase in the value of n l causes a decrease in the amplitude value. The increase of a b values causes a decrease in the amplitude value.. Conclusion In this paper, we have presented the solution of -dimensional Schrodinger equation of angular part for Scarf potential partner potential using Nikiforov-Uvarov method Supersymmetric quantum mechanics method. We obtained the wave functions from angular part solution, which the wave functions depend on the parameters of all components of the composed potential. The partner potential have different wave functions compared to the ground state wave functions of the original potential. This partner potential was considered to be new potential. Acknowledgement This research was partly supported by PUT-MRG of Sebelas Maret University References [] Shi-Hai Dong Wave Equations in Higher Dimensions Springer New York P97 [] Wahyulianti, Suparmi, Cari, Fuad Anwar 7 IOP Conf. Series: Journal of Physics: Conf. Series 795 [] B H Yazarloo, L Lu, G Liu, S Zarrinkamar, H Hassanabadi Advances in High Energy Physics 5 [] A. Suparmi, C. Cari, Beta Nur Pratiwi, Dewanta Arya Nugraha 7 IOP Conf. Series: Journal of Physics: Conf. Series 8 [5] A. N. Ikot Communications in Theoretical Physics 59 68-7 [6] Benedict Iserom Ita, Alexer Immaanyikwa Ikeuba Applied Mathematics -6 [7] D. Agboola 9 Phys. Scr. 8 65 [8] Suparmi dan Cari J. Math. Fund. Sci. 6 5 [9] A Suparmi, C Cari, U A Deta Chin. Phys. B 9 [] R A Sari, A Suparmi, C Cari 6 Chin. Phys. B 5 [] Ituen B Okon, Eno E Ituen, Oyebola Popoola, Akaninyene D Antia International Journal of Recent Advances in Physics (IJRAP) [] Akpan N Ikot, O A Awoga, A D Antia Chin. Phys. B [] A Suparmi, C Cari, Beta Nur Pratiwi, Dewanta Arya Nugraha 7 IOP Conf. Series: Journal of Physics: Conf. Series 795 [] Luqman H, Suparmi, Cari, U.A. Deta IOP Conf. Series: Journal of Physics: Conference Series 59 9 9