Abstract
Symbolic-numerical algorithms for solving a boundary value problem (BVP) for the 2D Schrödinger equation with homogeneous third type boundary conditions to study the quantum tunneling model of a coupled pair of nonidentical ions are described. The Kantorovich reduction of the above problem with non-symmetric long-range potentials to the BVPs for sets of the second order ordinary differential equations (ODEs) is given by expanding solution over the one-parametric set of basis functions. Symbolic algorithms for evaluation of asymptotics of the basis functions, effective potentials, and linear independent solutions of the ODEs in the form of inverse power series of independent variable at large values are given by using appropriate etalon equations. Benchmark calculation of quantum tunneling problem of coupled pair of identical ions through Coulomb-like barrier is presented.
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
Hofmann, H.: Quantum mechanical treatment of the penetration through a two-dimensional fission barrier. Nucl. Phys. A 224, 116–139 (1974)
Hagino, K., Rowley, N., Kruppa, A.T.: A program for coupled-channel calculations with all order couplings for heavy-ion fusion reactions. Comput. Phys. Commun. 123, 143–152 (1999)
Pen’kov, F.M.: Metastable states of a coupled pair on a repulsive barrier. Phys. Rev. A 62, 044701-1-4 (2000)
Pen’kov, F.M.: Quantum Transmittance of Barriers for Composite Particles. JETP 91, 698–705 (2000)
Kantorovich, L.V., Krylov, V.I.: Approximate Methods of Higher Analysis. Wiley, New York (1964)
Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: ODPEVP: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm–Liouville problem. Comput. Phys. Commun. 180, 1358–1375 (2009)
Chuluunbaatar, O., Gusev, A.A., Vinitsky, S.I., Abrashkevich, A.G.: KANTBP 2. 0: New version of a program for computing energy levels, reaction matrix and radial wave functions in the coupled-channel hyperspherical adiabatic approach. Comput. Phys. Commun. 179, 685–693 (2008)
Chuluunbaatar, O., Gusev, A., Gerdt, V., Kaschiev, M., Rostovtsev, V., Samoylov, V., Tupikova, T., Vinitsky, S.: A Symbolic-numerical algorithm for solving the eigenvalue problem for a hydrogen atom in the magnetic field: cylindrical coordinates. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 118–133. Springer, Heidelberg (2007)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)
Barnett, A.R., Feng, D.H., Steed, J.W., Goldfarb, L.J.B.: Coulomb wave functions for all real η and ρ. Comput. Phys. Comm. 8, 377–395 (1974)
Goodvin, G.L., Shegelski, M.R.A.: Three-dimensional tunneling of a diatomic molecule incident upon a potential barrier. Phys. Rev. A 72, 042713-1-7 (2005)
Giannakeas, P., Melezhik, V.S., Schmelcher, P.: D-wave confinement-induced resonances in harmonic waveguides. arXiv:1102.5686v1 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gusev, A.A., Vinitsky, S.I., Chuluunbaatar, O., Gerdt, V.P., Rostovtsev, V.A. (2011). Symbolic-Numerical Algorithms to Solve the Quantum Tunneling Problem for a Coupled Pair of Ions. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-23568-9_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23567-2
Online ISBN: 978-3-642-23568-9
eBook Packages: Computer ScienceComputer Science (R0)