By using an ansatz for the eigenfunction, we have obtained the exact analytical solutions of the radial Schrödinger equation for the pseudoharmonic and the Kratzer potentials in two dimensions. The bound-state solutions are easily calculated from this eigenfunction ansatz. The corresponding normalized wavefunctions are also obtained.
[1] L.I. Schiff: Quantum Mechanics, 3rd ed., McGraw-Hill Book Co., New York, 1955.
[2] L.D. Landau and E.M. Lifshitz: Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, New York, 1977
[3] E.T. Whittaker and G.N. Watson: Modern Analysis, 4th ed., Cambridge University Press, London, 1927.
[4] R.L. Liboff: Introductory Quantum Mechanics, 4th ed., Addison Wesley, San Francisco, CA, 2003.
[5] M.M. Nieto: “Hydrogen atom and relativistic pi-mesic atom in N-space dimension”, Am. J. Phys., Vol. 47, (1979), pp. 1067–1072. http://dx.doi.org/10.1119/1.11976
[6] S.M. Ikhdair and R. Sever: “Exact polynomial eigensolutions of the Schrödinger equation for the pseudoharmonic potential”, J. Mol. Struc.-Theochem, Vol. 806, (2007), pp. 155–158. http://dx.doi.org/10.1016/j.theochem.2006.11.019
[7] S.M. Ikhdair and R. Sever: “Exact polynomial solutions of the Mie-type potential in the N-dimensional Schrödinger equation”, Preprint: arXiv:quant-ph/0611065.
[8] M. Sage and J. Goodisman: “Improving on the conventional presentation of molecular vibrations: Advantages of the pseudoharmonic potential and the direct construction of potential energy curves”, Am. J. Phys., Vol. 53, (1985), pp. 350–355. http://dx.doi.org/10.1119/1.14408
[9] F. Cooper, A. Khare and U. Sukhatme: “Supersymmetry and quantum mechanics and large-N expansions ”, Phys. Rep., Vol. 251, (1995), pp. 267–385 http://dx.doi.org/10.1016/0370-1573(94)00080-M
[10] T.D. Imbo and U.P. Sukhatme: “Supersymmetric quantum mechanics”, Phys. Rev. Lett., Vol. 54, (1985), pp. 2184–2187. http://dx.doi.org/10.1103/PhysRevLett.54.2184
[11] Z.-Q. Ma and B.-W. Xu: “Quantum correction in exact quantization rules”, Europhys. Lett., Vol. 69, (2005), pp. 685–691. http://dx.doi.org/10.1209/epl/i2004-10418-8
[12] S.-H. Dong, C.-Y. Chen and M. Lozada-Casson: “Generalized hypervirial and Balanchard’s recurrence relations for radial matrix elements”, J. Phys. B: At. Mol. Opt. Phys., Vol. 38, (2005), pp. 2211–2220. http://dx.doi.org/10.1088/0953-4075/38/13/013
[13] S.-H. Dong, D. Morales and J. Garc’ia-Ravelo: “Exact quantization rule and its applications to physical potentials”, Int. J. Mod. Phys. E, Vol. 16, (2007), pp. 189–198. http://dx.doi.org/10.1142/S0218301307005661
[14] W.-C. Qiang and S.-H. Dong: “Arbitrary l-state solutions of the rotating Morse potential through the exact quantization rule method”, Phys. Lett. A, Vol. 363, (2007), pp. 169–176. http://dx.doi.org/10.1016/j.physleta.2006.10.091
[15] A.F. Nikiforov and V.B. Uvarov: Special Functions of Mathematical Physics, Birkhauser, Basel, 1988.
[16] G. Sezgo: Orthogonal Polynomials, American Mathematical Society, New York, 1959.
[17] S.M. Ikhdair and R. Sever: “Exact polynomial solution of PT/non-PT-symmetric and non-Hermitian modified Woods-Saxon potential by the Nikiforov-Uvarov method”, Preprint: arXiv:quant-ph/0507272; S.M. Ikhdair and R. Sever: “Polynomial solution of non-central potentials”, Preprint: arXiv:quant-ph/0702186.
[18] S.M. Ikhdair and R. Sever: “Exact solution of the Klein-Gordon equation for the PTsymmetric generalized Woods-Saxon potential by the Nikiforov-Uvarov method”, Ann. Phys. (Leipzig), Vol. 16, (2007), pp. 218–232. http://dx.doi.org/10.1002/andp.200610232
[19] S.M. Ikhdair and R. Sever: “Approximate eigenvalue and eigenfunction solutions for the generalized Hulthén potential with any angular momentum”, Preprint: arXiv:quant-ph/0508009.
[20] S.M. Ikhdair and R. Sever: “A perturbative treatment for the energy levels of neutral atoms”, Int. J. Mod. Phys. A, Vol. 21, (2006), pp. 6465–6476. http://dx.doi.org/10.1142/S0217751X06034240
[21] S.M. Ikhdair and R. Sever: “Bound energy for the exponential-cosine-screened Coulomb potential”, Preprint: arXiv:quant-ph/0604073.
[22] S.M. Ikhdair and R. Sever: “Bound states of a more general exponential screened Coulomb potential”, Preprint: arXiv:quant-ph/0604078.
[23] S.M. Ikhdair and R. Sever: “A perturbative treatment for the bound states of the Hellmann potential”, J. Mol. Struc.-Theochem, Vol. 809, (2007), pp. 103–113. http://dx.doi.org/10.1016/j.theochem.2007.01.019
[24] O. Bayrak, I. Boztosun and H. Ciftci: “Exact analytical solutions to the Kratzer potential by the asymptotic iteration method”, Int. J. Quantum Chem., Vol. 107, (2007), pp. 540–544. http://dx.doi.org/10.1002/qua.21141
[25] R.L. Hall and N. Saad: “Smooth transformations of Kratzer’s potential in N dimensions”, J. Chem. Phys., Vol. 109, (1998), pp. 2983–2986. http://dx.doi.org/10.1063/1.476889
[26] M.R. Setare and E. Karimi: “Algebraic approach to the Kratzer potential”, Phys. Scr., Vol. 75, (2007), pp. 90–93. http://dx.doi.org/10.1088/0031-8949/75/1/015
[27] S.M. Ikhdair and R. Sever: “Heavy-quark bound states in potentials with the Bethe-Salpeter equation“, Z. Phys. C, Vol. 56, (1992), pp. 155–160 http://dx.doi.org/10.1007/BF01589718
[28] S.M. Ikhdair and R. Sever: “Bethe-Salpeter equation for non-self-conjugate mesons in a power-law potential”, Z. Phys. C, Vol. 58, (1993), pp. 153–157 http://dx.doi.org/10.1007/BF01554088
[29] S.M. Ikhdair and R. Sever: “Bound state enrgies for the exponential cosine screened Coulomb potential”, Z. Phys. D, Vol. 28, (1993), pp. 1–5
[30] S.M. Ikhdair and R. Sever: “Solution of the Bethe-Salpeter equation with the shifted 1/N expansion technique”, Hadronic J., Vol. 15, (1992), pp. 389–403
[31] S.M. Ikhdair and R. Sever: “Bc meson spectrum and hyperfine splittingsin the shifted large-N expansion technique”, Int. J. Mod. Phys. A, Vol. 18, (2003), pp. 4215–4231 http://dx.doi.org/10.1142/S0217751X03015088
[32] S.M. Ikhdair and R. Sever: “Spectroscopy of Bc meson in the semi-relativistic quark model using the shifted large-N expansion method”, Int. J. Mod. Phys. A, Vol. 19, (2004), pp. 1771–1791
[33] S.M. Ikhdair and R. Sever: “Bc and heavy meson spectroscopy in the local approximation of the Schrödinger equation with relativistic kinematics”, Int. J. Mod. Phys. A, Vol. 20, (2005), pp. 4035–4054 http://dx.doi.org/10.1142/S0217751X05022275
[34] S.M. Ikhdair and R. Sever: “Mass spectra of heavy quarkonia and Bc decay constant for static scalar-vector interactions with relativistic kinematics”, Int. J. Mod. Phys. A, Vol. 20, (2005), pp. 6509–6531 http://dx.doi.org/10.1142/S0217751X05021294
[35] S.M. Ikhdair and R. Sever: “Bound energy masses of mesons containing the fourth generation and iso-singlet quarks”, Int. J. Mod. Phys. A, Vol. 21, (2006), pp. 2191–2199 http://dx.doi.org/10.1142/S0217751X06031636
[36] S.M. Ikhdair and R. Sever: “A systematic study on non-relativistic quarkonium interaction”, Int. J. Mod. Phys. A, Vol. 21, (2006), pp. 3989–4002
[37] S.M. Ikhdair, O. Mustafa and R. Sever: “Light and heavy meson spectra in the shifted 1/N expansion method”, Tr. J. Phys., Vol. 16, (1992), pp. 510–518
[38] S.M. Ikhdair, O. Mustafa and R. Sever: “Solution of Dirac equation for vector and scalar potentials and some applications” Hadronic J., Vol. 16, (1993), pp. 57–74.
[39] S. Özçelik and M. Şimşek: “Exact solutions of the radial Schr"odinger equation for inverse-power potentials”, Phys. Lett. A, Vol. 152, (1991), pp. 145–150. http://dx.doi.org/10.1016/0375-9601(91)91081-N
[40] S.-H. Dong: “Schrödinger equation with the potential V (r) = Ar −4+Br −3+Cr −2+Dr −1”, Phys. Scr., Vol. 64, (2001), pp. 273–276; S.-H. Dong: “Exact solutions of the two-dimensional Schrödinger equation with certain central potentials”, Preprint: arXiv:quant-ph/0003100. http://dx.doi.org/10.1238/Physica.Regular.064a00273
[41] S.-H. Dong: “On the solutions of the Schrödinger equation with some anharmonic potentials”, Phys. Scr., Vol. 65, (2002), pp. 289–295. http://dx.doi.org/10.1238/Physica.Regular.065a00289
[42] S.M. Ikhdair and R. Sever: “On the solutions of the Schrödinger equation with some molecular potentials: Wave function ansatz”, Preprint: arXiv:quant-ph/0702052.
[43] R.J. Le Roy and R.B. Bernstein: “Dissociation energy and long-range potential of diatomic molecules from vibration spacings of higher levels”, J. Chem. Phys., Vol. 52, (1970), pp. 3869–3879. http://dx.doi.org/10.1063/1.1673585
[44] C. Berkdemir, A. Berkdemir and J. Han: “Bound state solutions of the Schrödinger equation for modified Kratzer’s molecular potential”, Chem. Phys. Lett., Vol. 417, (2006), pp. 326–329. http://dx.doi.org/10.1016/j.cplett.2005.10.039
[45] A. Chatterjee: “Large N-expansion in Quantum mechanics, atomic physics and some O(N) invariant systems”, Phys. Rep., Vol. 186, (1990), pp. 249–370. http://dx.doi.org/10.1016/0370-1573(90)90048-7
[46] G. Esposito: “Complex parameters in quantum mechanics”, Found. Phys. Lett., Vol. 11, (1998), pp. 636–547.
[47] G.A. Natanzon: “General properties of potentials for which the Schr"odinger equation can be solved by means of hypergeometric functions”, Theor. Math. Phys., Vol. 38, (1979), pp. 146–153. http://dx.doi.org/10.1007/BF01016836
[48] G. Lévai: “A search for shape invariant solvable potentials”, J. Phys. A: Math. Gen., Vol. 22, (1989), pp. 689–702 http://dx.doi.org/10.1088/0305-4470/22/6/020
[49] G. Lévai: “A class of exactly solvable potentials related to the Jacobi polynomials”; J. Phys. A: Math. Gen., Vol. 24, (1991), pp. 131–146. http://dx.doi.org/10.1088/0305-4470/24/1/022
[50] K.J. Oyewumi and E.A. Bangudu: “Isotropic harmonic oscillator plus inverse quadratic potential in N-dimensional spaces”, Arab. J. Sci. Eng., Vol. 28, (2003), pp. 173–182.
[51] P.M. Morse: “Diatomic molecules according to the wave mechanics. II. Vibrational levels”, Phys. Rev., Vol. 34, (1929), pp. 57–64 http://dx.doi.org/10.1103/PhysRev.34.57
[52] N. Rosen and P.M. Morse: “On the vibrations of polyatomic molecules”, Phys. Rev., Vol. 42, (1932), pp 210–217. http://dx.doi.org/10.1103/PhysRev.42.210
[53] K.J. Oyewumi: “Analytical solutions of the Kratzer-Fues potential in an arbitrary number of dimensions”, Found. Phys. Lett., Vol. 18, (2005), pp. 75–84. http://dx.doi.org/10.1007/s10702-005-2481-9