Exact solution of Schrodinger equation for the pseudoharmonic potential is
obtained for an arbitrary angular momentum. The energy eigenvalues and
corresponding eigenfunctions are calculated by Nikiforov-Uvarov method.
Wavefunctions are expressed in terms of Jacobi polynomials. The energy
eigenvalues are calculated numerically for some values of l and n with n<5 for
some diatomic molecules.Comment: 10 page

A. Khare

A. Rajaki

A. Turbiner

A. Voros

A.F. Nikiforov

A.J. Sous

A.O. Barut

B.P. Mahapatra

C. Berkdemir

Cevdet Tezcan

D. Chelkak

F.L. Lu

G. Chen

G. Esposito

G. Herzberg

G.C. Maitland

H.A. Alhendi

J. Plieva

J.A. Yu

M. Jafapour

M. Karplus

M. Znojil

M.C. Zhang

M.L. Klein

M.S. Child

Metin Aktaş

N. Saad

R.S. Kaushal

Ramazan Sever

S. Bose

S.H. Dong

S.K. Maayedi

T. Barakat

Özlem Yeşiltaş

English

arXiv

WOS: 000252563900026Exact solution of Schrodinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of l and n with n <= 5 for some diatomic molecules

Sever, Ramazan

Tezcan, Cevdet

Aktas, Metin

Yesiltas, Oezlem

Kırıkkale Üniversitesi Akademik Arşiv Sistemi

Journal of Mathematical Chemistry

Exact solution of Schrodinger equation for Pseudoharmonic potential

Crossref

Exact solution of Schrödinger equation for Pseudoharmonic potential

Exact solution of Schrodinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of l and n with n <= 5 for some diatomic molecules

TEZCAN, CEVDET

OpenMETU (Middle East Technical University)

arXiv:quant-ph/0701206v1  29 Jan 2007Exact solution of Schrödinger equation forPseudoharmonic potentialRamazan Sever1∗, Cevdet Tezcan2, Metin Aktaş3,Özlem Yeşiltaş41Department of Physics, Middle East Technical University06531 Ankara, Turkey2Faculty of Engineering, Başkent University, Bag̃lıca Campus,Ankara, Turkey3Department of Physics, Faculty of Arts and Sciences, Kırıkkale University,Kırıkkale, Turkey,4Turkish Atomic Energy Authority, Istanbul Road 30km, Kazan, Ankara, TurkiyeOctober 2, 2018AbstractExact solution of Schrödinger equation for the pseudoharmonic potential is ob-tained for an arbitrary angular momentum. The energy eigenvalues and correspond-ing eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions areexpressed in terms of Jacobi polynomials. The energy eigenvalues are calculated nu-merically for some values of ℓ and n with n ≤ 5 for some diatomic molecules.PACS numbers: 03.65.-w; 03.65.Ge; 12.39.FdKeywords: Schrödinger equation, pseudoharmonic potential, diatomic molecules, Nikiforov-Uvarov methodOctober 2, 2018∗Corresponding Author: sever@metu.edu.tr11 IntroductionThe study of anharmonic oscillators in three dimensions has much attention in chemicalphysics phenomena [1-45]. They have wide applications in molecular physics [1]. The solu-tions of Schrödinger equation for any ℓ-state for such potentials have also a special interest.The Morse potential is commonly used for an anharmonic oscillator. However, its wave-function does not vanish at the origin, but the Mie type and the pseudoharmonic potentialsdo. The Mie type potential has the general features of the true interaction energy [1], in-teratomic and inter-molecular and dynamical properties of solids [2]. The pseudoharmonicpotential may be used for the energy spectrum of linear and non-linear systems [3]. TheMie type and pseudoharmonic are two exactly solvable potentials other than the Coulombicand harmonic oscillator. This potential may be considered as an intermediate potential be-tween the harmonic oscillator potential and anharmonic potentials, as the Morse oscillator(MO) potential, which is a more realistic potential, in good agreement with the experimen-tal spectroscopical data [47]. A comparative analysis of potentials HO-3D (3-dimensionalharmonic oscillator potential) and PHO is performed in [2].To solve the Schrödinger equation for the pseudoharmonic potential, the Nikiforov-Uvarov method [46] which is introduced for the solution of the hypergeometric type secondorder differential equations appeared in the time-independent problems. The method isbased on the determination of the solution in terms of special orthogonal functions for anygeneral second-order differential equations [47].By taking an appropriate s = s(r) coordinate transformation, the Schrödinger equationin spherical coordinates is reduced to a generalized equation of hypergeometric type asΨ′′(s) +τ̃(s)σ(s)Ψ′(s) +σ̃(s)σ2(s)Ψ(s) = 0, (1)where, σ(s) and σ̃(s) are polynomials, at most second-degree, and τ̃ (s) is in a first-degreepolynomial.The contents of this work is as follows. In section 2, solution of Schrödinger equationfor any ℓ−state is introduced. In section 3, solutions for Mie-type and pseudoharmonicpotentials are given. As an example, some numerical values of the energy levels of N2, CO,NO and NH molecules are computed. Section 4 is devoted for the calculations.2 Method and SolutionsThe pseudoharmonic potential has the form [3]V (r) = V0(rr0− r0r)2, (2)where V0 is the dissociation energy and r0 is the equilibrium intermolecular separation. Theradial part of the Schrödinger equation for a diatomic molecule potential is(− h̄22µ1r2ddr(r2ddr)+ℓ(ℓ+ 1)h̄22µr2+ V (r))Rnℓ(r) = EnℓRnℓ(r), (3)2where µ is the reduced mass of the diatomic molecules. n denotes the radial quantum num-ber (n and ℓ are named as the vibration-rotation quantum numbers in molecular chemistry),r is the internuclear separation. Substituting the explicit form of V (r), we getd2Rnℓ(r)dr2+2rdRnℓ(r)dr+2µh̄2(Enℓ − V0(rr0− r0r)2− ℓ(ℓ+ 1)h̄22µr2)Rnℓ(r) = 0. (4)By defining a coordinate transformationr2 = s, (5)and the following dimensionless parametersα2 =V0r20µ2h̄2(6)ε = (Enℓ + 2V0)µ2h̄2(7)β =(V0r20 +ℓ(ℓ+ 1)h̄22µ)µ2h̄2, (8)we obtaind2Rds2+3/2sdRds+1s2(−α2s2 + εs− β)R(s) = 0. (9)Thus, comparing the Eqs. (1) and (8), we obtain the following polynomialsτ̃ (s) =32, σ(s) = s, and σ̃(s) = −α2s2 + εs− β. (10)To find a particular solution of Eq. (9), we apply separation variables as Rnℓ(s) = φ(s)y(s),Schrödinger equation, Eq. (1) is reduced to an equation of hypergeometric typeσ(s)y′′ + τ(s)y′ + λy = 0, (11)where φ(s) is a solution of φ′(s) 6= φ(s) = π(s)/σ(s) and y(s) is a hypergeometric typefunction whose polynomial solutions are given by the well-known Rodrigues relation [46]y(s) = yn(s) =Bnρ(s)dndsn[σn(s)ρ(s)] . (12)Here, σ(s) is a polynomial of degree at most one, ρ(s) is a weight function and Bn is anormalization constant. It satisfy the condition(σρ)′ = τρ. (13)The function π and parameter λ are defined as3π =(σ′ − τ̃2)±√(σ′ − τ̃2)2− σ̃ + kσ, (14)andλ = k + π′. (15)Here π(s) is a polynomial with the parameter s. Determination of k is the essential pointin the calculation of π(s). k is determined by setting the square root as the square of apolynomial. Thus a new eigenvalue equation for the Schrödinger equation becomesλ = λn = −nτ ′ −n(n− 1)2σ′′; n = 0, 1, 2 . . . , (16)whereτ(s) = τ̃ (s) + 2π(s), (17)and it will have a negative derivative. Therefore the polynomial of π(s) is found in fourpositive valuesπ(s) = −14±(αs+√β + 116), k+ = ε+ γ= −14±(αs−√β + 116), k− = ε− γ, (18)where γ = 2α√β + 116. By following Eq. (17), we getτ(s) =32+ 2−14±(αs+√β + 116), for k+ = ε+ γ−14±(αs−√β + 116), for k− = ε− γ.(19)To have a negative derivative of τ(s) and a physical eigenfunction, we take the followingsubstitution: for k− = ε− γ, we haveπ(s) = −14−αs−√β +116 , (20)andτ(s) = 1− 2αs−√β +116 , (21)Thus, using Eq.(16) we obtainλ = λn = 2nα, (22)and using Eqs.(22) and (15) we obtain the following energy eigenvalues,4ε =2n + 1 + 2√β +116α. (23)Substituting Eq.(23) into Eq. (7), the energy eigenvalue becomesEnℓ = −2V0 +h̄r0√2V0µ(2n+ 1) + 2√√√√µ2h̄2(V0r20 +ℓ(ℓ+ 1)h̄22µ)+116 . (24)In order to calculate the wave function, the weight function ρ can be obtained by usingEq.(13) asρ(s) = s2√β+ 116 e−2αs, (25)and using Eq.(12) which is Rodrigues relation, we can get the function y,yn(s) = s1+2√β+ 116 e−2αs (26)Finally the wave function is obtained asRnℓ = s−14+√β+ 116 e−αs L2√β+ 116n (2αs). (27)3 ConclusionsWe have studied analytical solution of Schrödinger equation for a Pseudoharmonic potential.Energy eigenvalues and the corresponding wave functions are calculated for a diatomicsystem with any angular momentum L . The NU method is used in the computations.Numerical values of energy for N2, CO, NO and NH molecules are calculated for differentvalues of principal and angular quantum numbers n and ℓ. Results are listed in Table I.The potential parameters are obtained from Ref. [10].4 AcknowledgementsThis research was partially supported by the Scientific and Technological Research Councilof Turkey.5References[1] G.C. Maitland, M. Rigby, E. B. Smith and W. A. 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Exact solution of Schrodinger equation for Pseudoharmonic potential

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