Kuantum mekaniğinde yarı-klasik yaklaşım yöntemleri ve uygulamaları
Küçük Resim Yok
Tarih
2010
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Ege Üniversitesi
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Bu tezde, kuantum mekaniğinin Zamandan Bağımsız Schrödinger Denklemi'nin (ZBSD) yarı-klasik yaklaşımla çözülmesine ilişkin, JWKB ve MAF yöntemleri çalışılmıştır. Geleneksel JWKB yöntemi ile yapılan ZBSD çözümlerinin, klasik erişilemez bölgede gösterdiği uyumsuzluklar ve bu uyumsuzlukları gidermek üzere önerilen düzeltmeler, Başlangıç Değer Problemlerine (BaşDP) dönme noktası parametresinin de dahil edildiği iki değişkenli ve üç boyutlu grafiksel incelemeyle ortaya konulmuştur. Asimptotik uyumsuzlukların yarı-klasik sebepleri ve önerilen düzeltmelerin ispatı, yine burada önerilen JWKB açılım terimleri olan S_ij terimlerinden oluşturulan M_ij=S ?_(i-1,j) (i=1,2,3 ve j=1,2 olmak üzere) matris elemanları ile gösterilmiştir. Aynı asimptotik düzeltmelerin, Bağlı Durum Problemleri (BağDP) için de geçerli olduğu, Harmonik osilatör ve V şeklinde potansiyel kuyuları üzerine uygulanarak gösterilmiştir. BağDP'ne ilişkin asimptotik düzeltmelerin yanı sıra, JWKB parite düzeltmesi ve klasik dönme noktalarına ait faz değişim düzeltmeleri de yapılmıştır.ZBSD'nin çözümünde, MAF yöntemi ile tam çözümü mümkün kılan bir kriter tanımı yapılmış, bu kritere uyan potansiyeller incelenerek sınıflandırılması yapılmış ve bunlardan bazılarının BaşDP için MAF çözümleri yapılarak tamlığı gösterilmiştir. Sonuç olarak, genel yöntemlerle (Örn.; Frobenius yöntemi) analitik tam çözümü mümkün olmayan yeni potansiyeller ailesi tespit edilmiş olup, bazıları için MAF çözümü yapılarak tamlığı gösterilmiştir. Bu potansiyellerden, lineer potansiyel ile elde edilen V şeklinde potansiyel kuyusu BağDP için MAF yöntemiyle tam çözümler yapılmış olup, benzer şekilde diğerleriyle de oluşturulacak BağDP'leri için de tam çözümün mümkün olduğu gösterilmiştir
In this thesis, JWKB and MAF methods for solutions of the quantum mechanical Time Independent Schrödinger?s Equation (TISE) by the semiclassical approximation methods have been studied. The anomalies of the traditional JWKB solutions of the TISE in the classically inaccessible region, and suggested modifications for the asymptotic matching have been studied by three-dimensional, two-variable graphical analyses by using the turning point parameter as the additional, second variable in the Initial Value Problems (IVPs). Semiclassical reasons of these anomalies and proofs of the suggested modifications have been shown by the suggested M_ij=S ?_(i-1,j) matrix elements made up of the JWKB expansion terms, S_ij (where i=1,2,3 and j=1,2). The same modifications have also been shown to be valid for the Bound State Problems (BSPs) by applying to the harmonic oscillator and V shaped potential wells. Besides the asymptotic matching for the BSPs, parity matching and phase matching regarding to the turning point phase changes have also been studied.A criteria definition to determine the potentials enabling exact solution for the MAF method in the solution of the TISE has been made, such potentials have been studied for the IVPs, and some of which have been solved exactly by the MAF method. As a result, a new family of potentials whose exact analytical solutions by the general methods (i.e., Frobenius method) can not be obtained have been detected, some of which have been solved by the MAF method and their exactness have been shown. The V shaped potential well made up of the linear potentials among these exactly solvable potentials has been solved exactly by the MAF method and it has been shown that, other BSPs made up of the other such potentials can be solved exactly by the MAF method.
In this thesis, JWKB and MAF methods for solutions of the quantum mechanical Time Independent Schrödinger?s Equation (TISE) by the semiclassical approximation methods have been studied. The anomalies of the traditional JWKB solutions of the TISE in the classically inaccessible region, and suggested modifications for the asymptotic matching have been studied by three-dimensional, two-variable graphical analyses by using the turning point parameter as the additional, second variable in the Initial Value Problems (IVPs). Semiclassical reasons of these anomalies and proofs of the suggested modifications have been shown by the suggested M_ij=S ?_(i-1,j) matrix elements made up of the JWKB expansion terms, S_ij (where i=1,2,3 and j=1,2). The same modifications have also been shown to be valid for the Bound State Problems (BSPs) by applying to the harmonic oscillator and V shaped potential wells. Besides the asymptotic matching for the BSPs, parity matching and phase matching regarding to the turning point phase changes have also been studied.A criteria definition to determine the potentials enabling exact solution for the MAF method in the solution of the TISE has been made, such potentials have been studied for the IVPs, and some of which have been solved exactly by the MAF method. As a result, a new family of potentials whose exact analytical solutions by the general methods (i.e., Frobenius method) can not be obtained have been detected, some of which have been solved by the MAF method and their exactness have been shown. The V shaped potential well made up of the linear potentials among these exactly solvable potentials has been solved exactly by the MAF method and it has been shown that, other BSPs made up of the other such potentials can be solved exactly by the MAF method.
Açıklama
Anahtar Kelimeler
Fizik ve Fizik Mühendisliği, Physics and Physics Engineering, Schrödinger denklemi, Schrödinger equation