Penerapan metode relaksasi Gauss-Seidel untuk menyelesaikan persamaan Schrödinger
DOI:
https://doi.org/10.12928/jrkpf.v6i2.14613Abstract
Abstrak. Metode numerik, seperti metode finite difference, finite element dan Fourier, untuk menyelesaikan persamaan Schrodinger telah banyak digunakan sebelumnya. Metode finite difference time domain (FDTD) telah dikembangkan oleh Sudiarta dan Geldart (2007). Metode FDTD telah berhasil diaplikasikan untuk berbagai sistem kuantum, satu partikel ataupun lebih. Salah satu kelemahan metode FDTD adalah pada kasus tertentu seperti potensial kotak dan potensial osilator harmonik ditemukan iterasi FDTD lebih lambat menuju konvergen sehingga memerlukan waktu komputasi yang lebih lama. Untuk mengatasi hal tersebut, metode relaksasi Gauss-Seidel digunakan. Pada paper ini, metode relaksasi diaplikasikan untuk menyelesaikan persamaan Schrodinger satu partikel pada berbagai potensial.
Kata kunci: persamaan Schrodinger, metode relaksasi Gauss-Seidel, metode FDTD
Abstract. Numerical methods, such as finite difference, finite element and Fourier methods to solve the Schrodinger equation have been used previously. The finite difference time domain (FDTD) method has been developed by Sudiarta and Geldart (2007). The FDTD method has been successfully applied to various quantum systems, for one particle or more. One of the weaknesses of the FDTD method is that in certain cases such as the box potential and harmonic oscillator it has been found that FDTD iterations are slower to converge and thus require longer computation time. To overcome this, the relaxation Gauss-Seidel method can be used. In this paper, the relaxation method was applied to solve the Schrodinger equation for one particle in various potential wells.
Keywords: Schrodinger equation, relaxation Gauss-Seidel method, FDTD method
References
R. A. Serway, J. W. Jewett , “Mekanika kuantum,†in Fisika untuk Sains dan Teknik, 6th Ed. Jakarta: Salemba Teknika, 2010, ch. 41, sec. 4, pp. 342–344.
R. Becerril, F.S. Guzman, A. Rendon-Romero, S. Valdez-Alvarado, “Solving the time-dependent Schrodinger equation using finite difference methods,†in Revista Mexicana De Fisica E 54, 2008, pp. 120-132.
F. S. Levin, J. Shertzer, “Finite-element solution of the Schrodinger equation for the helium ground state,†The American Physical Society, Vol. 32, No. 6, 1985, pp. 3285-3290.
J. Ackermann, R. Roitzsch, “A two-dimensional multilevel adaptive finite element method for the time-independent Schriidinger equation,†Chemical Physics Letters, Berlin, Vol. 214, No. 1, 1993, pp. 109-117.
C. C. Marston, G. G. BalintKurti, “The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions,†in Journal of Chemical Physics, Vol. 91, No. 6, 1989, pp. 3571-3576.
D. Kosloff, R. Kosloff, “A fourier method solution for the time dependent Schrodinger equation as a tool in molecular dynamics,†in Journal of Computational Physics, vol. 52, 1983, pp. 35-53.
A. Soriano, E. A. Navarro, J. A. PortÃ, V. Such, “Analysis of the finite difference time domain technique to solve the Schrödinger equation for quantum devices,†in Journal of Applied Physics, Vol. 95, No. 12, 2004, pp. 8018.
W. Dai, G. Li, R. Nassar, S. Su, “On the stability of the FDTD method for solving a time-dependent Schrodinger equation,†in Wiley InterScience, 2005, pp. 1140-1154.
I. W. Sudiarta, D. J. W. Geldart, “Solving the schrӧdinger equation using the finite difference time domain method,†in Journal of Physics A: Math. Theor. 40, 2007, pp. 1885-1896.
R. Kosloff, H. Tal-Ezer, “A direct relaxation method for calculating eigenfunctions and eigenvalues of the Schrodinger equation on a grid,†Chemical Physics Letters, Vol. 127, No. 3, 1986, pp. 223-230.
C. Bottcher, M. R. Strayer, A. S. Urnar, P. G. Reinhard, “Damped relaxation techniques to calculate relativistic bound states,†The American Physical Society, Vol. 40, No. 8, 1989, pp. 4182-4189.
D. V. Schroeder, “the Variational-relaxation algorithm for finding quantum bound states,†in the American Journal of Physics, 2017.
A. Trellakis, A. T. Galick, A. Pacelli, U. Ravaioli, “Iteration scheme for the solution of the two-dimensional Schrödinger-Poisson equations in quantum structures,†Journal of Applied Physics, Vol. 81, No. 12, 1997, pp. 7880-7884.
F. Marsiglio, “The harmonic oscillator in quantum mechanics: A third way,†in American Journal of Physics, Vol. 77, No. 3, 2009, pp. 253-258.
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