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]

  • Mohammad Javad Mahmoodabadi 1
  • Fatemeh Sadeghi Googhari 2
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
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