# حل عددی معادله شرودینگر وابسته به زمان با استفاده از ترکیب روش تفاضل محدود و الگوریتم بهینه‌سازی تجمعی ذرات

نوع مقاله : مقاله پژوهشی کامل

نویسندگان

1 دانشکده مکانیک، دانشگاه صنعتی سیرجان، سیرجان، ایران

2 دانشکده فیزیک، دانشگاه شهیدباهنر، کرمان، ایران

چکیده

در این مقاله، با استفاده از یک روش عددی جدید به حل معادله شرودینگر وابسته به زمان پرداخته شده است. روش ارائه شده، حاصل ترکیب یک الگوریتم فراابتکاری قوی با سرعت و دقت بالا و روش تفاضل محدود است. به این منظور، ابتدا فضای حل متغیرهای مسأله‌ی مورد نظر با استفاده از روش تفاضل محدود شبکه‌بندی و سپس، معادله شرودینگر با شرایط مرزی مشخص به یک مسئله بدون قید تبدیل شده است. در ادامه، به کمک روش ضریب پنالتی، شرایط مرزی ارضاء و یک تابع هدف مناسب تعریف شده است. در پایان، با استفاده از یک مدل بهبود یافته از الگوریتم تجمعی ذرات به بهینه‌سازی تابع هدف مورد نظر پرداخته شده است. در چندین مثال مختلف مقدار خطای حاصل از مقایسه مقدار دقیق تابع و مقدار عددی محاسبه شده بیانگر موفقیت روش عددی پیشنهادی در حل مسئله شرودینگر وابسته به زمان است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Numerical solution of time-dependent Schrodinger equation by combination of the finite difference method and particle swarm optimization

نویسندگان [English]

1 Department of Mechanical Engineering, Sirjan Univercity of Technology, Sirjan, Iran
2 Department of Physics, Shahid Bahonar University of Kerman, Kerman, Iran
چکیده [English]

In this paper, a new numerical method is introduced to solve the time-dependent nonlinear Schrödinger equation. The proposed method is a combination of a novel metaheuristic optimization algorithm with the finite difference method. First, the regarded Schrödinger equation with the ralated boundary and initial conditions are converted into an unconstrained problem. For this purpose, the boundary and initial conditions are satisfied using the penalty method and a proper objective function is defined through the discretized governing equation. Then, a successful version of the particle swarm optimization is implemented to minimize the identified error function and find the best nodal values. The simulation results for several cases are illustrated to depict the effectiveness and capability of the introduced sterategy for solving the time-dependent nonlinear Schrödinger equation.

کلیدواژه‌ها [English]

• Time-dependent Schrödinger equation
• Particle swarm optimization algorithm
• Finite difference method
• Penalty method

#### مراجع

 D.J. Griffiths, Introduction to quantum mechanics, Pearson Prentice Hall, (2010).
 S.H. Dong, Factorization method in quantum mechanics, Springer, (2007).
 M. Aktas, R. Sever, Exact supersymmetric solution of Schrodinger equation for central confining potentials by using the Nikiforov-Uvarov method, Journal of Molecular Structure 710 (2004) 223-228. https://doi.org/10.1016/j.theochem.2004.09.011
 B.J. Falaye, K.J. Oyewumi, M. Abbas, Exact solution of Schrödinger equation with q-deformed quantum potentials using Nikiforov-Uvarov method, Chinese Physics B 22 (2013) 110301.
 H. Karayer, D. Demirhan, F. Büyükkılıç, Solution of Schrödinger equation for two different potentials using extended Nikiforov-Uvarov method and polynomial solutions of biconfluent Heun equation, Journal of Mathematical Physics 59 (2018) 053501.     https://doi.org/10.1063/1.5022008
 C.O. Edet, P.O. Okoi, Any l-State Solutions of the Schrodinger Equation for q-Deformed Hulthen Plus Generalized Inverse Quadratic Yukawa Potential in Arbitrary Dimension, Revista Mexicana de Física 65 (2019) 333-344. https://doi.org/10.31349/RevMexFis.65.333
 E.E. Perepelkin, B.I. Sadovnikov, N.G. Inozemtseva, A.A. Tarelkin, A new class of exact solutions of the Schrödinger equation, Continuum Mechanics and Thermodynamics 31 (2019) 639-667. https://doi.org/10.1007/s00161-018-0716-9
 C.A. Onate, K.J. Oyewumi, B.J. Falaye, Approximate solutions of the Schrödinger equation with the hyperbolical potential: supersymmetric approach, Few-Body Systems 55 (2014) 61-67. http://dx.doi.org/10.1007/s00601-013-0731-0
 H. Hassanabadi, B.H. Yazarloo, S. Zarrinkamar, M. Solaimani, Approximate analytical versus numerical solutions of Schrödinger equation under molecular Hua potential, International Journal of Quantum Chemistry 112 (2012) 3706-3710. https://doi.org/10.1002/qua.24064
 R.H. Hammed, Approximate Solution of Schrödinger Equation With Manning–Rosen Potential in Two Dimensions by using the shifted 1/N expansion method, Journal of Basrah Researches (Sciences) 38 (2012) 51-59.
 D. Xu, J. Stare, A.L. Cooksy, Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method, Computer Physics Communications 180 (2009) 2079-2094. https://doi.org/10.1016/j.cpc.2009.06.010
 S. Dong, J. Garcia-Ravelo, S.H. Dong, Analytical approximations to the l-wave solutions of the Schrödinger equation with an exponential-type potential, Physica Scripta 76 (2007) 393. https://doi.org/10.1088/0031-8949/76/4/019
 H. Egrifes, D. Demirhan, F. Büyükkiliç, Exact Solutions of the Schrödinger Equation for Two Deformed Hyperbolic Molecular Potentials, Physica Scripta 60 (1999) 195-198. https://doi.org/10.1238/PHYSICA.REGULAR.060A00195
 F. Yaşuk, C. Berkdemir, A. Berkdemir, Exact solutions of the Schrödinger equation with non-central potential by the Nikiforov–Uvarov method, Journal of Physics A: Mathematical and General 38 (2005) 6579. https://doi.org/10.1088/0305-470/38/29/012
 L. Chai, S. Jin, P.A. Markowich, A hybrid method for computing the Schrödinger equations with periodic potential with band-crossings in the momentum space, Communications in Computational Physics 24 (2018) 989-1020. https://doi.org/10.4208/cicp.2018.hh80.01
 U.S. Okorie, A.N. Ikot, M.C. Onyeaju, E.O. Chukwuocha, Bound state solutions of Schrödinger equation with modified Mobius square potential (MMSP) and its thermodynamic properties, Journal of molecular modeling 24 (2018) 289. https://doi.org/10.1007/s00894-018-3811-8
 C.A. Onate, J.O. Ojonubah, Eigensolutions of the Schrödinger equation with a class of Yukawa potentials via supersymmetric approach, Journal of Theoretical and Applied Physics 10 (2016) 21-26. https://doi.org/10.1007/s40094-015-0196-2
 S.M. Ikhdair, R. Sever, Bound states of a more general exponential screened Coulomb potential, Journal of mathematical chemistry 41 (2007) 343-353. https://doi.org/10.1007/s10773-008-9806-y
 C. Tezcan, R. Sever, A general approach for the exact solution of the Schrödinger equation, International Journal of Theoretical Physics 48 (2009) 337-350.
 S.M. Ikhdair, R. Sever, Exact solutions of the modified Kratzer potential plus ring-shaped potential in the D-dimensional Schrödinger equation by the Nikiforov–Uvarov method, International Journal of Modern Physics C 19 (2008) 221-235. https://doi.org/10.1142/S0129183108012030
 S.M. Ikhdair, R. Sever, Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential, Central European Journal of Physics 6 (2008) 685-696. https://doi.org/10.2478/s11534-008-0024-2
 S.M. Ikhdair, R. Sever, Polynomial solutions of the Mie-type potential in the D-dimensional Schrödinger equation, Journal of Molecular Structure 855 (2008) 13-17. https://doi.org/10.1016/j.theochem.2007.12.044
 A. Sadighi, D.D. Ganji, Analytic treatment of linear and nonlinear Schrödinger equations: a study with homotopy-perturbation and Adomian decomposition methods, Physics Letters A 372 (2008) 465-469. https://doi.org/10.1016/j.physleta.2007.07.065
 N. Taghizadeh, M. Mirzazadeh, F. Farahrooz, Exact solutions of the nonlinear Schrödinger equation by the first integral method, Journal of Mathematical Analysis and Applications 374 (2011) 549-553. https://doi.org/10.1016/j.jmaa.2010.08.050
 S.O. Edeki, G.O. Akinlabi, S.A. Adeosun, On a modified transformation method for exact and approximate solutions of linear Schrödinger equations, In AIP Conference proceedings 1705 (2016). https://doi.org/10.1063/1.4940296
 A.R. Seadawy, Exact solutions of a two-dimensional nonlinear Schrödinger equation, Applied Mathematics Letters 25 (2012) 687-691. https://doi.org/10.1016/j.aml.2011.09.030
 N.K. Vitanov, Z.I. Dimitrova, Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation, Journal of Theoretical and Applied Mechanics 48 (2018) 59-68. https://doi.org/10.2478/jtam-2018-0005
 M.J. Mahmoodabadi, F. Shojaei, Z. Arasteh, Analysis of the Three-dimensional time-dependent Schrödinger equation by the meshless local Petrov- Galerkin method, Journal of Research on Many-Body Systems 8 17 (2018) 51-58.
https://doi.org/10.22055/JRMBS.2018.13884
 M. Dehghan, D. Mirzaei, The meshless local Petrov–Galerkin (MLPG) method for the generalized two-dimensional non-linear Schrödinger equation, Engineering Analysis with Boundary Elements 32 (2008) 747-756. https://doi.org/10.1016/j.enganabound.2007.11.005
 M. Dehghan, A. Shokri, A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Computers & Mathematics with Applications 54 (2007) 136-146. https://doi.org/10.1016/j.camwa.2007.01.038
 A. Bashan, N.M. Yagmurlu, Y. Ucar, A. Esen, An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method, Chaos, Solitons & Fractals 100 (2017) 45-56. https://doi.org/10.1016/j.chaos.2017.04.038
 A. Sóbester, P.B. Nair, A.J. Keane, Genetic programming approaches for solving elliptic partial differential equations, IEEE transactions on evolutionary computation 12 (2008) 469-478.  https://doi.org/10.1109/TEVC.2007.908467
 J. Kennedy, R. Eberhart, Particle Swarm Optimization. In Proceedings of IEEE International Conference on Neural Networks 4 (1995) 1942-1948. https://doi.org/10.1109/ICNN.1995.488968
 M.J. Mahmoodabadi, Z.S. Mottaghi, A. Bagheri, HEPSO: high exploration particle swarm optimization, Information Sciences 273 (2014) 101-111.
 D.M. Causon, C.G. Mingham Introductory finite difference methods for PDEs. Ventus Publishing ApS (2010).
 R. Becerril, F.S. Guzmán, A. Rendón-Romero, & S. Valdez-Alvarado, Solving the time-dependent Schrödinger equation using finite difference methods, Revista mexicana de física E 54 2 (2008) 120-132.

### سابقه مقاله

• تاریخ دریافت: 22 آذر 1398
• تاریخ بازنگری: 14 آذر 1399
• تاریخ پذیرش: 06 اردیبهشت 1400
• تاریخ اولین انتشار: 29 اردیبهشت 1400