Computation of the Schrödinger Equation via the Discrete Derivatives Representation Method: Improvement of Solutions Using Particle Swarm Optimization

ABSTRACT

We develop the discrete derivatives representation method (DDR) to find the physical structures of the Schrödinger equation in which the interpolation polynomial of Bernstein has been used. In this paper the particle swarm optimization (PSO for short) has been suggested as a means to improve qualitatively the solu-tions. This approach is carefully handled and tested with a numerical example.

We develop the discrete derivatives representation method (DDR) to find the physical structures of the Schrödinger equation in which the interpolation polynomial of Bernstein has been used. In this paper the particle swarm optimization (PSO for short) has been suggested as a means to improve qualitatively the solu-tions. This approach is carefully handled and tested with a numerical example.

Cite this paper

nullA. Zerarka, H. Saidi, A. Attaf and N. Khelil, "Computation of the Schrödinger Equation via the Discrete Derivatives Representation Method: Improvement of Solutions Using Particle Swarm Optimization,"*Journal of Modern Physics*, Vol. 1 No. 1, 2010, pp. 44-47. doi: 10.4236/jmp.2010.11005.

nullA. Zerarka, H. Saidi, A. Attaf and N. Khelil, "Computation of the Schrödinger Equation via the Discrete Derivatives Representation Method: Improvement of Solutions Using Particle Swarm Optimization,"

References

[1] A. Zerarka, S. Hassouni, H. Saidi and Y. Boumedjane, “Energy Spectra of the Schrödinger Equation and the Differential Quadrature Method,” Communication in Nonlinear Science and Numerical Simulation, Vol. 10, 2005, pp. 737-745.

[2] A. Zerarka and A. Soukeur, “A Generalized Integral Quadratic Method: I. An Efficient Solution for One-Di- mensional Volterra Integral Equation,” Communication in Nonlinear Science and Numerical Simulation, Vol. 10, 2005, pp. 653-663.

[3] F. Iachello, “Algebraic Methods in Quantum Mechanics with Applications to Nuclear and Molecular Structure,” Nuclear Physics A, Vol. 560, 1993, pp. 23-34.

[4] M. Selg, “Numerically Complemented Analytic Method for Solving the Time-Independent One-Dimensional Sch rödinger Equation,” Physical Review E, Vol. 64, No. 5, 2001, 056701.

[5] S. H. Dong, “Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials,” International Journal of Theoretical Physics, Vol. 39, 2000, p. 1119.

[6] S. K. Bose and N. Gupta, “Exact Solution of Nonrelativistic Schrödinger Equation for Certain Central Physical Potentials,” Il Nuovo Cimento B, Vol. 113, 1998, p. 299.

[7] J. Kennedy and R. C. Eberhart, “Particle Swarm Optimization,” Proceedings IEEE International Conferences Neural Networks, Piscataway, 1995, pp. 1942-1948.

[8] Maitland, et al., “Intermolecular Forces,” University Press, Oxford, 1987.

[9] Y. P. Varshni, “The First Three Bound States for the Potential ,” Physics Letters A, Vol. 183, 1993, pp. 9-13.

[10] S. H. Dong, “Quantum Monodromy of Schrödinger Equation with the Decatic Potential,” International Journal of Theoretical Physics, Vol. 41, No. 1, January 2002, pp. 89-99.

[11] R. C. Eberhart and J. Kennedy, “A New Optimizer Using Particles Swarm Theory,” Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995, pp. 39-43.

[12] R. C. Eberhart and Y. Shi, “Parameter Selection in Particle Swarm Optimization,” Lecture Notes in Computer Science-Evolutionary Programming VII, Porto, V. W., Saravanan, N. Waagen, D., Eiben, A. E., Springer, Vol. 1447, 1998, pp. 591-600.

[13] T. I. Cristian “The Particle Swarm Optimization Algorithm: Convergence Analysis and Parameter Selection,” Information Processing Letters, Vol. 85, No. 6, 2003, pp. 317-325.

[1] A. Zerarka, S. Hassouni, H. Saidi and Y. Boumedjane, “Energy Spectra of the Schrödinger Equation and the Differential Quadrature Method,” Communication in Nonlinear Science and Numerical Simulation, Vol. 10, 2005, pp. 737-745.

[2] A. Zerarka and A. Soukeur, “A Generalized Integral Quadratic Method: I. An Efficient Solution for One-Di- mensional Volterra Integral Equation,” Communication in Nonlinear Science and Numerical Simulation, Vol. 10, 2005, pp. 653-663.

[3] F. Iachello, “Algebraic Methods in Quantum Mechanics with Applications to Nuclear and Molecular Structure,” Nuclear Physics A, Vol. 560, 1993, pp. 23-34.

[4] M. Selg, “Numerically Complemented Analytic Method for Solving the Time-Independent One-Dimensional Sch rödinger Equation,” Physical Review E, Vol. 64, No. 5, 2001, 056701.

[5] S. H. Dong, “Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials,” International Journal of Theoretical Physics, Vol. 39, 2000, p. 1119.

[6] S. K. Bose and N. Gupta, “Exact Solution of Nonrelativistic Schrödinger Equation for Certain Central Physical Potentials,” Il Nuovo Cimento B, Vol. 113, 1998, p. 299.

[7] J. Kennedy and R. C. Eberhart, “Particle Swarm Optimization,” Proceedings IEEE International Conferences Neural Networks, Piscataway, 1995, pp. 1942-1948.

[8] Maitland, et al., “Intermolecular Forces,” University Press, Oxford, 1987.

[9] Y. P. Varshni, “The First Three Bound States for the Potential ,” Physics Letters A, Vol. 183, 1993, pp. 9-13.

[10] S. H. Dong, “Quantum Monodromy of Schrödinger Equation with the Decatic Potential,” International Journal of Theoretical Physics, Vol. 41, No. 1, January 2002, pp. 89-99.

[11] R. C. Eberhart and J. Kennedy, “A New Optimizer Using Particles Swarm Theory,” Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995, pp. 39-43.

[12] R. C. Eberhart and Y. Shi, “Parameter Selection in Particle Swarm Optimization,” Lecture Notes in Computer Science-Evolutionary Programming VII, Porto, V. W., Saravanan, N. Waagen, D., Eiben, A. E., Springer, Vol. 1447, 1998, pp. 591-600.

[13] T. I. Cristian “The Particle Swarm Optimization Algorithm: Convergence Analysis and Parameter Selection,” Information Processing Letters, Vol. 85, No. 6, 2003, pp. 317-325.