Pseudospectral solution of the Schrödinger equation for the Rosen-Morse and Eckart potentials

  • Conor L. Morrison
  • Bernard ShizgalEmail author
Original Paper


Pseudospectral methods based on non-classical quadratures are used to numerically compute the eigenvalues and eigenfunctions of the Schrödinger equation for the Rosen-Morse and Eckart potentials. The method uses a basis set of non-classical polynomials, \(\{ P_n(x) \}\), orthonormal with respect to a weight function, \(w(x)>0\), to construct an \(N \times N\) matrix representative, \(\{ H_{nm} \}\), of the Hamiltonian, H. This matrix representative is transformed to an equivalent pseudospectral representative, \(\{H_{ij}\}\). The rate of convergence of the eigenvalues of \(\{ H_{ij} \}\) to the eigenvalues of H, versus the grid size N, is reported for non-classical basis functions in comparison with the use of Legendre and Laguerre polynomials as well as a Fourier basis. The use of non-classical polynomials is shown to provide the fastest convergence for the eigenvalues. The pseudospectral method based on nonclassical quadratures proposed in this paper should find wide applicability to other problems in quantum and statistical mechanics. A review is provided of the use of a tensor product of one dimensional basis functions to describe two and three dimensional problems.


Schroedinger equation Hamiltonian Eigenvalues Pseudospectral Nonclassical 



This research was supported by a grant to Bernard Shizgal from the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant Number (03190). Conor Morrison was supported in part by an Undergraduate Student Research Award (USRA) from NSERC, Reference Number (527568). We would like to thank Lucas Philipp for helpful discussions throughout the course of the research.


  1. 1.
    B.D. Shizgal, H. Chen, The quadrature discretization method (QDM) in the solution of the Schrödinger equation with nonclassical basis functions. J. Chem. Phys. 104, 4137–4150 (1996)CrossRefGoogle Scholar
  2. 2.
    B.D. Shizgal, H. Chen, The quadrature discretization method in the solution of the Fokker–Planck equation with nonclassical basis functions. J. Chem. Phys. 107, 8051–8063 (1997)CrossRefGoogle Scholar
  3. 3.
    D. Baye, P.H. Heenen, Generalized meshes for quantum-mechanical problems. J. Phys. A Math. Gen. 19, 2041–2059 (1986)CrossRefGoogle Scholar
  4. 4.
    D. Baye, M. Hesse, M. Vincke, The unexplained accuracy of the Lagrange-mesh method. Phys. Rev. E 65, 026701 (2002)CrossRefGoogle Scholar
  5. 5.
    D.T. Colbert, W.H. Miller, A novel discrete variable representation for quantum-mechanical reactive scattering via the S-Matrix Kohn method. J. Chem. Phys. 96, 1982–1991 (1992)CrossRefGoogle Scholar
  6. 6.
    J.C. Light, T. Carrington Jr., Discrete variable representations and their utilization. Adv. Chem. Phys. 114, 263–310 (2000)Google Scholar
  7. 7.
    V. Szalay, Optimal grids for generalized finite basis and discrete variable representations: definition and method of calculation. J. Chem. Phys. 125, 154115 (2006)CrossRefGoogle Scholar
  8. 8.
    V. Szalay, G. Czako, A. Nagy, T. Furtenbacher, A.G. Csaszar, On one-dimensional discrete variable representations with general basis functions. J. Chem. Phys. 119, 10512–10518 (2003)CrossRefGoogle Scholar
  9. 9.
    D. Kosloff, R. Kosloff, A Fourier method of solution for the time-dependent Schrödinger-equation as a tool in molecular dynamics. J. Comput. Phys. 52, 35–53 (1983)CrossRefGoogle Scholar
  10. 10.
    R. Kosloff, The Fourier method, in Numerical Grid Methods and their Application to Schrödinger’s Equation, ed. by C. Cerjan (Kluwer Academic, Dordrecht, 1993), pp. 175–194CrossRefGoogle Scholar
  11. 11.
    C.C. Marston, G.G. Balint-Kurti, The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions. J. Chem. Phys. 91, 3571–3576 (1989)CrossRefGoogle Scholar
  12. 12.
    J. Stare, G.G. Balint-Kurti, The Fourier grid Hamiltonian method for solving the vibrational Schrödinger equation in internal coordinates: theory and test applications. J. Phys. Chem. A 107, 7204–7214 (2003)CrossRefGoogle Scholar
  13. 13.
    A. Derevianko, E. Luc-Koenig, F. Masnou-Seeuws, Application of B-splines in determining the eigenspectrum of diatomic molecules: robust numerical description of halo-state and Feshbach molecules. Can. J. Phys. 87, 67–74 (2009)CrossRefGoogle Scholar
  14. 14.
    B.W. Shore, Solving the radial Schrödinger equation by using cubic-spline basis functions. J. Chem. Phys. 58, 3855–3866 (1973)CrossRefGoogle Scholar
  15. 15.
    F. Cooper, A. Kharem, U. Sukhatme, Supersymmetry and quantum mechanics. Phys. Rep. 251, 267–385 (1995)CrossRefGoogle Scholar
  16. 16.
    F. Cooper, J.N. Ginocchio, A. Khare, Relationship between supersymmetry and solvable potentials. Phys. Rev. D 36, 2458–2473 (1987)CrossRefGoogle Scholar
  17. 17.
    C.-L. Ho, Simple unified derivation and solution of Coulomb, Eckart and Rosen-Morse potentials in prepotential approach. Ann. Phys. 324, 1095–1104 (2009)CrossRefGoogle Scholar
  18. 18.
    S. Dominguez-Hernandez, D.J. Fernandez, C. Rosen-Morse, Potential and its supersummetric partners. Int. J. Theor. Phys. 50, 1993–2001 (2011)CrossRefGoogle Scholar
  19. 19.
    G.-H. Sun, S.H. Dong, Quantum information entropies of the eigenstates for a symmetrically trigonometric Rosen-Morse potential. Phys. Scr. 87, 045003 (2013)CrossRefGoogle Scholar
  20. 20.
    G.-H. Sun, S.H. Dong, Quantum information entropies for an asymmetric trigonometric Rosen-Morse potential. Ann. Phys. 525, 934–943 (2013)CrossRefGoogle Scholar
  21. 21.
    S.A. Najafizade, H. Hassanabadi, D. Zarrinkamar, Information theoretic global measures of Dirac Equation with Morse and trigonometric Rosen-Morse potentials. Few Body Syst. 68, 149–163 (2017)CrossRefGoogle Scholar
  22. 22.
    R. Dutt, A. Khare, U.P. Sukhatme, Supersymmetry, shape invariance, and exactly solvabe potentials. Am. J. Phys. 56, 163–168 (1987)CrossRefGoogle Scholar
  23. 23.
    R.K. Yadav, A. Khare, B.P. Mandal, The scattering amplitude for rationally extended shape invariant Eckart potentials. Phys. Lett. A 379, 67–70 (2015)CrossRefGoogle Scholar
  24. 24.
    C. Quesne, Novel enlarged shape invariance property and eckartly solvable rational extensions of the Rosen-Morse II and Eckart potentials. SIGMA 8, 80–99 (2012)Google Scholar
  25. 25.
    H. Hassanabadi, B.H. Yazarloo, A.N. Ikot, N. Salehi, S. Zarrinkamr, Exact analytical versus numerical solutions of the Schrödinger equation for Hua plus modified Eckart potential. Indian J. Phys. 87(12), 1219–1223 (2013)CrossRefGoogle Scholar
  26. 26.
    B.D. Shizgal, Pseudospectral method of solution of the Schrödinger equation with non-classical polynomials; the Morse and Poschl-Teller (SUSY) potentials. Comput. Theor. Chem. 1084, 51–58 (2016)CrossRefGoogle Scholar
  27. 27.
    B.D. Shizgal, Pseudospectral solution of the Fokker–Planck equation with equilibrium bistable states: the eigenvalue spectrum and the approach to equilibrium. J. Stat. Phys. 164, 1379–1393 (2016)CrossRefGoogle Scholar
  28. 28.
    B.D. Shizgal, A comparison of pseudospectral methods for the solution of the Schrödinger equation: the Lennard-Jones \((n,6)\) potential. Comput. Theor. Chem. 114, 25–32 (2017)CrossRefGoogle Scholar
  29. 29.
    B. Shizgal, Spectral Methods in Chemistry and Physics: Applications to Kinetic Theory and Quantum Mechanics (Springer, New York, 2015)CrossRefGoogle Scholar
  30. 30.
    W. Gautschi, On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput. 3, 289–317 (1982)CrossRefGoogle Scholar
  31. 31.
    W. Gautschi, Algorithm 726: ORTHOPOL—a package of routines for generating orthogonal polynomials with Gauss-type quadrature rules. ACM Trans. Math. Softw. 20, 21–82 (1994)CrossRefGoogle Scholar
  32. 32.
    W. Gautschi, Orthogonal Polynomials in MATLAB: Excercies and Solutions (SIAM, Philadelphia, 2016)Google Scholar
  33. 33.
    J.Q.W. Lo, B.D. Shizgal, Pseudospectral methods of solution of the Schrödinger equation. J. Math. Chem. 44, 787–801 (2008)CrossRefGoogle Scholar
  34. 34.
    B.D. Shizgal, N. Ho, X. Yang, The computation of radial integrals with nonclassical quadratures for quantum chemistry and other applications. J. Math. Chem. 55, 413–422 (2017)CrossRefGoogle Scholar
  35. 35.
    K. Leung, B.D. Shizgal, H. Chen, The quadrature discretization method (QDM) in comparison with other numerical methods of solution of the Fokker–Planck equation for electron thermalization. J. Math. Chem. 24, 291–319 (1998)CrossRefGoogle Scholar
  36. 36.
    C.-I. Gheorghiu, Laguerre collocation solutions to boundary layer type problems. Numer. Algorithm 64, 385–401 (2013)CrossRefGoogle Scholar
  37. 37.
    B.D. Shizgal, H. Chen, The quadrature discretization method (QDM) in the solution of the Schrödinger equation with nonclassical basis functions. J. Chem. Phys. 104, 4137–4150 (1996)CrossRefGoogle Scholar
  38. 38.
    H. Chen, Y. Su, B.D. Shizgal, A direct spectral collocation poisson solver in polar and cylindrical coordinates. J. Comput. Phys. 160, 453–469 (2000)CrossRefGoogle Scholar
  39. 39.
    H.H. Yang, B.D. Shizgal, Chebyshev pseudospectral multi-domain technique for viscous flow calculation. Comput. Methods Appl. Mech. Eng. 118, 47–61 (1994)CrossRefGoogle Scholar
  40. 40.
    L. Gibelli, B.D. Shizgal, A.W. Yau, Ion energization by wave–particle interactions: comparison of spectral and particle simulation solutions of the Vlasov equation. Comput. Math. Appl. 59, 2566–2581 (2010)CrossRefGoogle Scholar
  41. 41.
    D. Bǵué, N. Gohaud, C. Pouchan, P. Cassam-Chenaï, J. Lie\(\acute{\text{v}}\)in, A comparison of two methods for selecting vibrational configuration interaction spaces on a heptatomic system: ethylene oxide. J. Chem. Phys. 127, 164115 (2007)Google Scholar
  42. 42.
    G. Avila, T. Carrington Jr., Reducing the cost of using collocation to compute vibrational energy levels: results for \(\text{ CH }_2\)NH. J. Chem. Phys. 147, 064103 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of ChemistryUniversity of British ColumbiaVancouverCanada
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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