Some efficient schemes of variational iteration method for handling Schrödinger equations in quantum mechanics

Abstract

At the atomic level, one of the fundamental differential equations that describe the universe and everyday life is the Schrödinger equation in quantum mechanics. Solutions of Schrödinger equations have attracted the attention of researchers. In this study, the variational iteration method is used to handle three dimensional linear and nonlinear time dependent Schrödinger equations. Different iteration formulas have been constructed with the Lagrange multipliers. The accuracy of the approximate solutions obtained by using different iteration formulas of the variational iteration method is given with numerical examples. Comparisons of approximate solutions and exact solutions are shown with graphs. In addition, absolute error tables are included for these comparison results. As a result, it is seen that the variational iteration method gives approximations that converge to the exact solution more rapidly thanks to different iteration formulas.

Keywords

• Analytical solution
• variational iteration method
• Schrödinger equation

Kuantum mekaniğinde Schrödinger denklemlerini ele almak için varyasyonel iterasyon yönteminin bazı etkili şemaları

Abstract

Atomik düzeyde evreni ve günlük yaşamı tanımlayan temel diferansiyel denklemlerden biri kuantum mekaniğindeki Schrödinger denklemidir. Schrödinger denklemlerinin çözümleri araştırmacıların ilgisini çekmiştir. Bu çalışmada, üç boyutlu lineer ve lineer olmayan zamana bağlı Schrödinger denklemlerini ele almak için varyasyonel iterasyon yöntemi kullanılmıştır. Lagrange çarpanları ile farklı iterasyon formülleri oluşturulmuştur. Varyasyonel iterasyon yönteminin farklı iterasyon formülleri kullanılarak elde edilen yaklaşık çözümlerin doğruluğu sayısal örneklerle verilmiştir. Yaklaşık çözümlerin ve tam çözümlerin karşılaştırmaları grafiklerle gösterilmiştir. Ayrıca bu karşılaştırma sonuçları için mutlak hata tablolarına da yer verilmiştir. Sonuç olarak, varyasyonel iterasyon yönteminin farklı iterasyon formülleri sayesinde tam çözüme daha hızlı yakınsayan yaklaşımlar verdiği görülmektedir.

Keywords

• Analitik çözüm
• varyasyonel iterasyon yöntemi
• Schrödinger denklemi

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