Abstract
A general scheme of a symbolic-numeric approach for solving the eigenvalue problem for the one-dimensional Shrödinger equation is presented. The corresponding algorithm of the developed program EWA using a conventional pseudocode is described too. With the help of this program the energy spectra and the wave functions for some Schrödinger operators such as quartic, sextic, octic anharmonic oscillators including the quartic oscillator with double well are calculated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Landau, L.D., Lifshits, E.M.: Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, New York (1977)
Wilkinson, J.H., Reinsch, C.: Handbook for Automatic Computation. Linear Algebra, vol. 2. Springer, New York (1971)
Banerjee, K.: General anharmonic oscillators. Proc. Roy. Soc. London, A 364, 265–275 (1978)
Maslov, V.P., Fedoryuk, M.V.: Kvaziklassicheskie pribligeniya dlya uravnenii kvantovoi mekhaniki. Nayka, Moskva (1976)
Puzynin, I.V., Amirkhanov, I.V., et al.: Continuous Analogue of Newton’s Method for the Numerical Investigation of Some Nonlinear Quantum - Field Models. PEPAN 30, 210–265 (1999)
Flugge, S.: Practical Quantum Mechanics. Springer, Heidelberg (1971)
Birkhoff, G.D.: Dynamical Systems. A.M.S. Colloquium Publications. New York (1927)
Gustavson, F.G.: On construction of formal integral of a Hamiltonian system near an equilibrium point. Astronom. J. 71, 670–686 (1966)
Swimm, R.T., Delos: Semiclassical calculation of vibrational energy levels for nonseparable systems using Birkhoff–Gustavson normal form. J. Chem. Phys. 71, 1706 (1979)
Ali, M.K.: The quantum normal form and its equivalents. J. Math. Phys. 26, 25–65 (1985)
Chekanov, N.A.: Kvantovanie normalnoi formy Birkhoff–Gustavson. Jadernaya Fizika 50, 344–346 (1985)
Abrashkevich, A.G., Abrashkevich, D.G., Kaschiev, M.S., Puzynin, I.V.: FESSDE, a program for the finite-element solution of the coupled-channel Schroedinger equation using high-order accuracy approximations. Comp. Phys. Commun. 85, 65–74 (1995)
Jaffe, L.G.: Large N limits as classical mechanics. Rev. Mod. Phys. 54, 407–435 (1982)
Dineykhan, M., Efimov, G.V.: The Schroedinger equation for bound state systems in the oscillator representation. Reports of Math. Phys. 6, 287–308 (1995)
Jafarpour, M., Afshar, D.: Calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential. J. Phys. A: Math. Gen. 35, 87–92 (2002)
Ivanov, I.A.: Sextic and octic anharmonic oscillator: connection between strong-coupling and weak-coupling expansions. J. Phys. A: Math. Gen. 31, 5697–5704 (1998)
Ivanov, I.A.: Link between the strong-coupling and weak-coupling asymptotic perturbation expansions for the quartic anharmonic oscillator. J. Phys. A: Math. Gen. 31, 6995–7003 (1998)
Liu, X.S., Su, L.W., Ding, P.Z.: Intern. J. Quantum Chem. 87, 1–11 (2002)
Ince, E.L.: Ordinary Differential Equations. Dover Pubns, New York (1956)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1968)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Belyaeva, I.N., Chekanov, N.A., Gusev, A.A., Rostovtsev, V.A., Vinitsky, S.I. (2006). A Symbolic-Numeric Approach for Solving the Eigenvalue Problem for the One-Dimensional Schrödinger Equation. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2006. Lecture Notes in Computer Science, vol 4194. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11870814_2
Download citation
DOI: https://doi.org/10.1007/11870814_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-45182-2
Online ISBN: 978-3-540-45195-2
eBook Packages: Computer ScienceComputer Science (R0)