NUMERICAL MATRIX METHOD FOR SOLVING STATIONARY ONE-DIMENSIONAL SCHRÖDINGER EQUATION. AMSSE PROGRAM
Salejda Włodzimierz, Just Marcin, Tyc Michał H.
Institute of Physics, Wrocław University of Technology
Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
DOI: 10.12921/cmst.2000.06.01.73-100
OAI: oai:lib.psnc.pl:511
Abstract:
An implementation of numerical algebraic methods of solving a stationary one-dimensional Schrödinger equation (SODSE) is presented. In the framework of the proposed approach, SODSE is converted into an algebraic eigenvalue problem, which represents a discrete version of studied problem on an equally spaced grid. The AMSSE program written in Delphi calculates eigenvalues and corresponding eigenvectors by means of various methods and algorithms described here. It is an efficient and valuable computational environment, which can be used in science and nanotechnology. Arbitrary potentials can be introduced into AMSSE program in the form of analytic formulae or data tables, or with the mouse. The user-friendly graphical interface takes advantage of full capabilities of the Windows operating system. Main program features are described. Efficiency and accuracy of different numerical algorithms are comprehensively tested and compared. Factors influencing accuracy are discussed. Examples are widely presented. Matrix approach extension to the case of an effective-mass equation is mentioned.
References:
[1] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics. Vol. Ill Quantum Mechanics, 4th
ed,, Nauka, Moscow (1989) (in Russian).
[2] E. Merzbacher, Quantum Mechanics, 3rd ed., John Wiley & Sons, New York (1998).
[3] R. L. Liboff, Introductoiy Quantum Mechanics, 3rd ed., Addison-Wesley Longman, New York, (1998).
[4] W. Greiner, Quantum Mechanics. An Introduction, 2nd corrected ed., Springer-Verlag, Berlin
(1993).
[5] S. Flügge, Practical Quantum Mechanics, Springer-Verlag, Berlin (1999).
[6] R. Shankar, Principles of Quantum Mechanics, 2nd ed., Plenum Press, N.Y. and London (1994).
[7] F. Mandl, Quantum Mechanics, Wiley, Chichester (1992).
[8] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Reading (1994).
[9] F. Schwabl, Quantum Mechanics, 2nd revised ed., Springer-Verlag, Berlin (1995).
[10] J. F. Van der Maelen Uria, S. Garcia-Granda, A. Menendez-Veläzquez, Am. J. Phys. 64, 327
(1996).
[11] W. Salejda, M. H. Tyc, J. Andrzejewski, M. Kubisa, J. Misiewicz, M. Just, K. Ryczko, Acta
Phys. Pol., 95, 881 (1999).
[12] J. Killingbeck, Microcomputer Algorithms, Hilger, Bristol, 1991; J. Killingbeck and G. Jolicard, Phys. Lett. A172, 313 (1993).
[13] P. Harrison, Quantum Wells, Wires and Dots. Theoretical and Computational Physics, Wiley,
Chichester, Ch. 2 and 3 (2000).
[14] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. Art of
Scientific Computing. Cambridge University Press, Cambridge, Ch. 16 (1992).
[15] R. Guardiola, J. Ros, J. Comput. Phys. 45, 374 (1982).
[16] B. Lindberg, J. Chem. Phys. 88, 3805 (1988).
[17] G. C. Groenenboom and H. M. Buck, J. Chem. Phys. 92, 4374 (1990).
[18] Ch. 11 in [14],
[19] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford 1965; see also
Handbook for Automatic Computations, vol. 2, Linear Algebra, J. H. Wilkinson and C. Reinsch
(Eds.), Springer-Verlag, Heidelberg (1971).
[20] E. Anderson et al. LAPACK Users’ Guide, 3rd ed., SIAM, Philadelphia (2000).
[21] B. T. Smith et al, Matrix eigensystems routines — EISPACK guide, Lecture Notes in Computer Science, vol. 6, Berlin, Springer, Sec. Ed. (1976); B. W. Garbow, J. M. Boyle, J. J. Dongara, C. B. Moler, Matrix eigensystems routines – EISPACK guide extension, Lecture Notes in Computer Science, vol. 61, Berlin, Springer (1977).
[22] C. F. Gerald, P. O. Wheatley, Applied Numerical Analysis, Addison-Wesley, Reading (1989).
[23] M. Just, M. Sc. Thesis, Numerical Methods of Solving Schrödinger Equation. Program Package: MARKS, Report SPR-333/1998, Institute of Physics, Wroclaw University of Technology,
Wroclaw 1998 (in Polish).
[24] A. Baker and P. Graves-Morris, Padé Approxitnants. Encyclopaedia of Mathematics and its
Applications, vol. 13 and 14, Addison-Wesley (1981).
[25] P. Dean, Rev. Mod. Phys., 44, 127 (1972); see also W. Salejda, Int. J. Mod. Phys., B9, 1429
(1995); B9, 1453 (1995); B9, 1475 (1995).
[26] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia (1996).
[27] K. S. Dy, S.Y. Wu, T. Spratlin, Phys. Rev. B20, 4237 (1979).
[28] Z. Zheng, J. Phys. C: Cond. Matt., 19, L689 (1986).
[29] H. Konwent, Phys. Lett. A118, 467 (1986).
[30] H. Konwent, Acta Phys. Pol. A71, 637 (1987).
[31] H. Konwent, Phys. Stat. Sol. B138, K7 (1986).
[32] H. Konwent, P. Machnikowski, P. Magnuszewski, A. Radosz, J. Phys. A: Math.Gen.. A31, 7541 (1998).
An implementation of numerical algebraic methods of solving a stationary one-dimensional Schrödinger equation (SODSE) is presented. In the framework of the proposed approach, SODSE is converted into an algebraic eigenvalue problem, which represents a discrete version of studied problem on an equally spaced grid. The AMSSE program written in Delphi calculates eigenvalues and corresponding eigenvectors by means of various methods and algorithms described here. It is an efficient and valuable computational environment, which can be used in science and nanotechnology. Arbitrary potentials can be introduced into AMSSE program in the form of analytic formulae or data tables, or with the mouse. The user-friendly graphical interface takes advantage of full capabilities of the Windows operating system. Main program features are described. Efficiency and accuracy of different numerical algorithms are comprehensively tested and compared. Factors influencing accuracy are discussed. Examples are widely presented. Matrix approach extension to the case of an effective-mass equation is mentioned.
[1] L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics. Vol. Ill Quantum Mechanics, 4th
ed,, Nauka, Moscow (1989) (in Russian).
[2] E. Merzbacher, Quantum Mechanics, 3rd ed., John Wiley & Sons, New York (1998).
[3] R. L. Liboff, Introductoiy Quantum Mechanics, 3rd ed., Addison-Wesley Longman, New York, (1998).
[4] W. Greiner, Quantum Mechanics. An Introduction, 2nd corrected ed., Springer-Verlag, Berlin
(1993).
[5] S. Flügge, Practical Quantum Mechanics, Springer-Verlag, Berlin (1999).
[6] R. Shankar, Principles of Quantum Mechanics, 2nd ed., Plenum Press, N.Y. and London (1994).
[7] F. Mandl, Quantum Mechanics, Wiley, Chichester (1992).
[8] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, Reading (1994).
[9] F. Schwabl, Quantum Mechanics, 2nd revised ed., Springer-Verlag, Berlin (1995).
[10] J. F. Van der Maelen Uria, S. Garcia-Granda, A. Menendez-Veläzquez, Am. J. Phys. 64, 327
(1996).
[11] W. Salejda, M. H. Tyc, J. Andrzejewski, M. Kubisa, J. Misiewicz, M. Just, K. Ryczko, Acta
Phys. Pol., 95, 881 (1999).
[12] J. Killingbeck, Microcomputer Algorithms, Hilger, Bristol, 1991; J. Killingbeck and G. Jolicard, Phys. Lett. A172, 313 (1993).
[13] P. Harrison, Quantum Wells, Wires and Dots. Theoretical and Computational Physics, Wiley,
Chichester, Ch. 2 and 3 (2000).
[14] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. Art of
Scientific Computing. Cambridge University Press, Cambridge, Ch. 16 (1992).
[15] R. Guardiola, J. Ros, J. Comput. Phys. 45, 374 (1982).
[16] B. Lindberg, J. Chem. Phys. 88, 3805 (1988).
[17] G. C. Groenenboom and H. M. Buck, J. Chem. Phys. 92, 4374 (1990).
[18] Ch. 11 in [14],
[19] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford 1965; see also
Handbook for Automatic Computations, vol. 2, Linear Algebra, J. H. Wilkinson and C. Reinsch
(Eds.), Springer-Verlag, Heidelberg (1971).
[20] E. Anderson et al. LAPACK Users’ Guide, 3rd ed., SIAM, Philadelphia (2000).
[21] B. T. Smith et al, Matrix eigensystems routines — EISPACK guide, Lecture Notes in Computer Science, vol. 6, Berlin, Springer, Sec. Ed. (1976); B. W. Garbow, J. M. Boyle, J. J. Dongara, C. B. Moler, Matrix eigensystems routines – EISPACK guide extension, Lecture Notes in Computer Science, vol. 61, Berlin, Springer (1977).
[22] C. F. Gerald, P. O. Wheatley, Applied Numerical Analysis, Addison-Wesley, Reading (1989).
[23] M. Just, M. Sc. Thesis, Numerical Methods of Solving Schrödinger Equation. Program Package: MARKS, Report SPR-333/1998, Institute of Physics, Wroclaw University of Technology,
Wroclaw 1998 (in Polish).
[24] A. Baker and P. Graves-Morris, Padé Approxitnants. Encyclopaedia of Mathematics and its
Applications, vol. 13 and 14, Addison-Wesley (1981).
[25] P. Dean, Rev. Mod. Phys., 44, 127 (1972); see also W. Salejda, Int. J. Mod. Phys., B9, 1429
(1995); B9, 1453 (1995); B9, 1475 (1995).
[26] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia (1996).
[27] K. S. Dy, S.Y. Wu, T. Spratlin, Phys. Rev. B20, 4237 (1979).
[28] Z. Zheng, J. Phys. C: Cond. Matt., 19, L689 (1986).
[29] H. Konwent, Phys. Lett. A118, 467 (1986).
[30] H. Konwent, Acta Phys. Pol. A71, 637 (1987).
[31] H. Konwent, Phys. Stat. Sol. B138, K7 (1986).
[32] H. Konwent, P. Machnikowski, P. Magnuszewski, A. Radosz, J. Phys. A: Math.Gen.. A31, 7541 (1998).
