Bir boyutlu kuantum sistemlerde elektron iletimi
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Date
2017
Authors
Oymak, Hüseyin
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Abstract
İki ideal iletkene bağlanmış, bir boyutlu kuantum sistemlerde (veya bir boyutlu sistemlere indirgenebilen iki boyutlu sistemlerde) bir elektronun kuantum iletim problemini çalıştık. Klasik mekaniğe göre, akım kavraına yönelik en temel elektrik ve bazı diğer manyetik olguları inceledikten sonra, kuantum mekaniğin temel formalizmini verdik. Bunun için kuantum mekaniğin çalışma prensiplerini Dirac notasyonu kullanarak formüle ettik. Sonra bir boyutlu saçılma problemleri ile ilgilendik ve delta fonksiyonu içeren bazı örnekleri detaylı olarak çalıştık. Daha sonra, amonyak molekülü, hidrojen molekül iyonu ve hidrojen molekülü gibi iki durumlu kuantum sistemler üzerine yoğunlaştık. Devamında, bir elektronun bir boyutlu kristal örgü içindeki yayılımını tüm ince detayları ile araştırdık. Bir boyutlu doğrusal atom zinciri ve atom halkası yapılarının temellerini oturttuktan sonra, açıklayıcı problemler yoluyla incelediğimiz rezonant iletim konusuna geldik. Ardından ise Aharonov-Bohm (AB) halkalarının matematik ve fiziğini verdik; bunun için, elektromanyetik alan içinde hareket eden yüklü bir parçacıkla ilgili klasik Lagrange ve Hamilton dinamikleri ve bununla alakalı ayar dönüştürme konularına dokunduk. Son olarak, bir boyutlu kuantum sistemlerin iletim özelliklerini elde edebilmek için, zamandan bağımsız Schrödinger denklemini, iki noktadan yinelemeli başka bir denkleme çeviren, 'Ricatti oranlar yöntemi' olarak adlandırılan bir budama/kesme tekniğinin nasıl kullanılacağını gösterdik. Bu yöntemi, basit bir seyreltme/kaldırma tekniği yardımı ile, sonuçları iyi bilinen bazı bir boyutlu sistemlerde ve ayrıca tekli ve çiftli AB halkalarında test ettik.
We study the quantum transport of an electron through some one-dimensional (1D) quantum systems (or seemingly 2D systems which are practically reducible to 1D systems) coupled to two ideal leads. After reviewing the rudiments of electricity phenomena, leading to the concept of current, and some magnetism phenomena in terms of classical mechanics, we provide the basic formalism of quantum mechanics by elucidating its working machinery in terms of Dirac notation. We next deal with 1D scattering problems and study in detail some examples containing delta-function potentials. We then focus on the so-called two-state quantum systems, like the ammonia molecule, the hydrogen molecular ion, and the hydrogen molecule. Later on we investigate the propagation of an electron in a 1D crystal lattice with all of its fine details. After settling down the rudiments of 1D finite linear atomic-chain and atomic-ring structures, we come to the subject of resonant transport that we investigate through some illustrative problems. We then present the mathematics and physics of the Aharonov-Bohm (AB) rings, for which we touch the classical Lagrangian and Hamiltonian dynamics for a charged particle moving in an electromagnetic field and the relevant gauge transformations. Finally, for obtaining transport properties of 1D quantum systems, we demonstrate how to use a pruning technique, called Ricatti-ratios method, which changes the time-independent discrete Schrödinger equation into a two-point recursive equation. We test the method in some 1D well-known systems as well as in single and double AB rings by simply reducing them to one dimension with the help of a decimation method.
We study the quantum transport of an electron through some one-dimensional (1D) quantum systems (or seemingly 2D systems which are practically reducible to 1D systems) coupled to two ideal leads. After reviewing the rudiments of electricity phenomena, leading to the concept of current, and some magnetism phenomena in terms of classical mechanics, we provide the basic formalism of quantum mechanics by elucidating its working machinery in terms of Dirac notation. We next deal with 1D scattering problems and study in detail some examples containing delta-function potentials. We then focus on the so-called two-state quantum systems, like the ammonia molecule, the hydrogen molecular ion, and the hydrogen molecule. Later on we investigate the propagation of an electron in a 1D crystal lattice with all of its fine details. After settling down the rudiments of 1D finite linear atomic-chain and atomic-ring structures, we come to the subject of resonant transport that we investigate through some illustrative problems. We then present the mathematics and physics of the Aharonov-Bohm (AB) rings, for which we touch the classical Lagrangian and Hamiltonian dynamics for a charged particle moving in an electromagnetic field and the relevant gauge transformations. Finally, for obtaining transport properties of 1D quantum systems, we demonstrate how to use a pruning technique, called Ricatti-ratios method, which changes the time-independent discrete Schrödinger equation into a two-point recursive equation. We test the method in some 1D well-known systems as well as in single and double AB rings by simply reducing them to one dimension with the help of a decimation method.
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Fizik ve Fizik Mühendisliği, Rezonans, Physics and Physics Engineering, Resonance, Saçılma problemleri, Scattering problems
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