A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation


The Gross-Pitaevskii equation (GPE), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GPE consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a spectral method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained. This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.

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G. Rawitscher, "A Spectral Integral Equation Solution of the Gross-Pitaevskii Equation," Applied Mathematics, Vol. 4 No. 10C, 2013, pp. 70-77. doi: 10.4236/am.2013.410A3009.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. N. Bose, “Plancks Gesetz und Lichtquantenhypothese,” Zeitschrift für Physik, Vol. 26, No. 1, 1924, pp. 178-181. http://dx.doi.org/10.1007/BF01327326
[2] A. Einstein, “Quantentheorie des einatomigen idealen Gases,” Sitzungsberichte der Preuβischen Akademie der Wissenschaften, 1924, p. 261.
[3] A. Einstein, “Quantentheorie des einatomigen idealen Gases. 2. Abhandlung,” Sitzungsberichte der Preuβischen Akademie der Wissenschaften, 1925, p. 3.
[4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman and E. A. Cornell, “Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor,” Science, Vol. 269, No. 5221, 1995, pp. 198-201.
[5] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, “BoseEinstein Condensation in a Gas of Sodium Atoms,” Physical Review Letters, Vol. 75, No. 22, 1995, pp. 3969-3973. http://dx.doi.org/10.1103/PhysRevLett.75.3969
[6] E. P. Gross, “Structure of a Quantized Vertex in Boson Systems,” Nuovo Cimento, Vol. 20, No. 3, 1961, pp. 454-477.
[7] E. P. Gross, “Hydrodynamics of a Superfluid Condensate,” Jurnal of Mathematical Physics, Vol. 4, No. 2, 1963, pp. 195-207. http://dx.doi.org/10.1063/1.1703944
[8] L. P. Pitaevskii, “Dyamics of Collapse of a Confined Bose Gas,” Physics Letters A, Vol. 221, No. 1-2, 1996, pp. 14-18. http://dx.doi.org/10.1016/0375-9601(96)00538-5
[9] M. Edwards and K. Burnett, “Numerical Solution of the Nonlinear Schrodinger Equation for Small Samples of Trapped Neutral Atoms,” Physical Review A, Vol. 51, No. 2, 1995, pp. 1382-1386.
[10] P. A. Ruprecht, M. J. Holland, K. Burnett and M. Edwards, “Time-Dependent Solution of the Nonlinear Schrodinger Equation for Bose-Condensed Trapped Neutral Atoms,” Physical Review A, Vol. 51, No. 6, 1995, pp. 4704-4711.
[11] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stingari, “Theory of Bose-Einstein Condensation in Trapped Gases,” Reviews of Modern Physics, Vol. 71, No. 3, 1999, pp. 463-512.
[12] Y.-S. Choi, I. Koltracht, P. J. McKenna and N. Savytska, “Global Monotone Convergence of Newton Iteration for a Nonlinear Eigen-Problem,” Linear Algebra and Its Applications, Vol. 357, No. 1-3, 2002, pp. 217-228.
[13] Y.-S. Choi, J. Javanainen, I. Koltracht, M. Kostrun, P. J. McKenna and N. Savytska, “A Fast Algorithm for the Solution of the Time-Dependent Gross-Pitaevskii Equation,” Journal of Computational Physics, Vol. 190, No. 1, 2003, pp. 1-21.
[14] C. M. Dion and E. Cancès, “Spectral Method for the Time-Dependent Gross-Pitaevskii Equation with a Harmonic Trap,” Physical Review E, Vol. 67, No. 4, 2003, Article ID: 046706.
[15] B. I. Schneider and D. L. Feder, “Numerical Approach to the Ground and Excited States of a Bose-Einstein Condensed Gas Confined in a Completely Anisotropic Trap,” Physical Review A, Vol. 59, No. 3, 1999, pp. 2232-2242.
[16] T. Bergeman, D. L. Feder, N. L. Balazs and B. I. Schneider, “Bose Condensates in a Harmonic Trap near the Critical Temperature,” Physical Review A, Vol. 61, No. 6, 2000, Article ID: 063605.
[17] A. Gammal, T. Frederico, L. Thomio and P. Chomaz, “Atomic Bose-Einstein Condensation with Three-Body Interactions and Collective Excitations,” Journal of Physics B, Vol. 33, No. 19, 2000, p. 4053.
[18] Z. Marojevic, E. Goklü and C. Lammerzahl, “Energy Eigenfunctions of the 1D Gross-Pitaevskii Equation,” Computer Physics Communications, Vol. 184, No. 8, 2013, pp. 1920-1930.
[19] M. Caliari and S. Rainer, “GSGPEs: A MATLAB Code for Computing the Ground State of Systems of Gross-Pitaevskii Equations,” Computer Physics Communications, Vol. 184, No. 3, 2013, pp. 812-823.
[20] W. Bao, D. Jaksch and P. A. Markowich, “Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation,” Journal of Computational Physics, Vol. 187, No. 1, 2003, pp. 318-342.
[21] J. C. Gunn and J. M. F. Gunn, “An Exactly Soluble Hartree Problem in an External Potential,” European Journal of Physics, Vol. 9, No. 1, 1988, p. 51.
[22] J. W. Negele, “Structure of Finite Nuclei in the LocalDensity Approximation,” Physical Review, Vol. 1, No. 4, 1970, pp. 1260-1321.
[23] J. W. Negele and D. Vautherin, “Density-Matrix Expansion for an Effective Nuclear Hamiltonian,” Physical Review C, Vol. 5, No. 5, 1971, pp. 1472-1493.
[24] D. Vautherin and D. M. Brink, “Hartree-Fock Calculations with Skyrme’s Interaction. I. Spherical Nuclei,” Physical Review C, Vol. 5, No. 3, 1972, pp. 626-647.
[25] B. D. Esry, “Hartree-Fock Theory for Bose-Einstein Condensates and the Inclusion of Correlation Effects,” Physical Review A, Vol. 55, No. 2, 1997, pp. 1147-1159.
[26] A. Deloff, “Semi-Spectral Chebyshev Method in Quantum Mechanics,” Annals of Physics, Vol. 322, No. 6, 2007, pp. 1373-1419.
[27] L. N. Trefethen, “Spectral Methods in MATLAB,” Society for Industrial and Applied Mathematics, Philadelphia, 2000. http://dx.doi.org/10.1137/1.9780898719598
[28] J. P. Boyd, “Chebyshev and Fourier Spectral Methods,” 2nd Revised Edition, Dover Publications, Mineola, 2001.
[29] R. A. Gonzales, J. Eisert, I. Koltracht, M. Neumann and G. Rawitscher, “Integral Equation Method for the Continuous Spectrum Radial Schroedinger Equation,” Journal of Computational Physics, Vol. 134, No. 1, 1997, pp. 134-149. http://dx.doi.org/10.1006/jcph.1997.5679
[30] R. A. Gonzales, S.-Y. Kang, I. Koltracht and G. Rawitscher, “Integral Equation Method for Coupled Schrodinger Equations,” Journal of Computational Physics, Vol. 153, No. 1, 1999, pp. 160-202.
[31] G. H. Rawitscher, B. D. Esry, E. Tiesinga, J. P. Burke Jr. and I. Koltracht, “Comparison of Numerical Methods for the Calculation of Cold Atom Collisions,” Journal of Chemical Physics, Vol. 111, No. 23, 1999, pp. 10418-104226. http://dx.doi.org/10.1063/1.480431
[32] G. H. Rawitscher, C. Merow, M. Nguyen and I. Simbotin, “Resonances and Quantum Scattering for the Morse Potential as a Barrier,” American Journal of Physics, Vol. 70, No. 9, 2002, pp. 935-944.
[33] G. H. Rawitscher, S. Y. Kang and I. Koltracht, “A Novel Method for the Solution of the Schrodinger Equation in the Presence of Exchange Terms,” Journal of Chemical Physics, Vol. 118, No. 20, 2003, pp. 9149-9157.
[34] G. Rawitscher and W. Gloeckle, “Integrals of the TwoBody T Matrix in Configuration Space,” Physical Review, Vol. A 77, No. 1, 2008, Article ID: 012707.
[35] G. Rawitscher, “Calculation of the Two-Body Scattering K-Matrix in Configuration Space by an Adaptive Spectral Method,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 1, 2009, Article ID: pp. 015201.
[36] G. Rawitscher and J. Liss, “The Vibrating Inhomogeneous String,” American Journal of Physics, Vol. 79, No. 4, 2011, pp. 417-427. http://dx.doi.org/10.1119/1.3534837
[37] G. H. Rawitscher, “Solution of the Schroedinger Equation Containing a Perey-Buck Nonlocality,” Nuclear Physics A, Vol. 886, 2012, pp. 1-16.
[38] G. Rawitscher, “Iterative Evaluation of the Effect of Long-Range Potentials on the Solution of the Schrodinger Equation,” Physical Review A, Vol. 87, No. 3, 2013, Article ID: 032708.
[39] G. Rawitscher and I. Koltracht, “An Economical Method to Calculate Eigenvalues of the Schrodinger Equation,” European Journal of Physics, Vol. 27, No. 5, 2006, p. 1179. http://dx.doi.org/10.1088/0143-0807/27/5/017
[40] G. Rawitscher, “Iterative Solution of Integral Equations on a Basis of Positive Energy Sturmian Functions,” Physical Review E, Vol. 85, No. 2, 2012, Article ID: 026701.
[41] G. Rawitscher and I. Koltracht, “Description of an Efficient Numerical Spectral Method for Solving the Schroedinger Equation,” Computing in Science and Engineering, Vol. 7, 2005, p. 58.
[42] G. Rawitscher, “Applications of a Numerical Spectral Expansion Method to Problems in Physics: A Retrospective,” In: T. Hempfling, Ed., Operator Theory, Advances and Applications, Vol. 203, Birkauser Verlag, Basel, 2009, pp. 409-426.
[43] S. G. Cooper and R. S. Mackintosh, “Energy Dependent Potentials Determined by Inversion: The p+α Potential Up to 65 MeV,” Physical Review C, Vol. 54, No. 6, 1996, pp. 3133-3152.
[44] K. Amos, L. Canton, G. Pisent, J. P. Svenne and D. van der Knijff, “An Algebraic Solution of the Multichannel Problem Applied to Low Energy Nucleon—Nucleus Scattering,” Nuclear Physics A, Vol. 728, No. 1-2, 2003, pp. 65-95.
[45] M. I. Jaghoub and G. H. Rawitscher, “Evidence of Nonlocality Due to a Gradient Term in the Optical Model,” Nuclear Physics A, Vol. 877, 2012, pp. 59-69.
[46] S. K. Adhikari, “Numerical Study of the Spherically Symmetric Gross-Pitaevskii Equation in Two Space Dimensions,” Physical Review E, Vol. 62, No. 2, 2000, pp. 2937-2944.
[47] S. K. Adhikari, “Stability and Collapse of a Coupled Bose-Einstein Condensate,” Physics Letters A, Vol. 281, No. 4, 2001, pp. 265-271.
[48] F. Dalfovo and S. Stringari, “Bosons in Anisotropic Traps: Ground State and Vortices,” Physical Review A, Vol. 53, No. 4, 1996, pp. 2477-2485.
[49] G. Baym and C. J. Pethick, “Ground-State Properties of Magnetically Trapped Bose-Condensed Rubidium Gas,” Physical Review Letters, Vol. 76, No. 1, 1996, pp. 6-9.

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