Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Entanglement and Invariance of Qubit-Qubit, Qubit-Qutrit and Qutrit-Qutrit Quantum States(01. Izmir Institute of Technology, 2024) Kızılkaya, Betül; Pashaev, OktayMevcut tez, saf iki kübit, kübit-kütrit ve iki kütrit durumlarının dolanıklık özelliklerinin incelenmesine ayrılmıştır. Dolanıklık, esasen bileşik kuantum durumlarının klasik olmayan bir özelliğidir ve kuantum hesaplamasında ve kuantum bilgi teorisinde önemli bir rol oynar. Durumların dolanıklığını karakterize etmek için, saf bileşik durumun dolanıklığını karışık azaltılmış yoğunluklu matrisle ilişkilendiren azaltılmış yoğunluk matrisi yaklaşımını kullanırız. Azaltılmış yoğunluk matrisinin Von Neumann entropisi ve karesel eşzamanlılık olarak doğrusal entropi, dolanıklığı ölçmek için kullanılır. Üniter bir kübit ve bir kütrit kapılarını kullanarak, dönüşümler altında dolanıklığın değişmezliğini gösteririz. Bu, aynı seviyedeki dolanık durumların sürekli olarak parametrelendirilmiş kümesini oluşturmamızı sağlar. Sonuçları, verilen karışık durum için arıtmanın hesaplanması ve manyetik alanda iki kübit spin XYZ modeli için ortalama enerjinin maksimum dolaşık minimumunu bulmak için uyguluyoruz.Master Thesis Classical and Quantum Euler Equation(Izmir Institute of Technology, 2007) Eti, Neslihan; Pashaev, OktayIn the present thesis we give generalization of analytical mechanics to describe dynamical systems with dissipation. The Lagrangian function in this case is determined by nonstationary pseudo-Riemannian metric for the kinetic energy, and by general quadratic form, nondiagonal in the generalized coordinates and velocities. Skew symmetric nondiagonal terms in our approach play the role of dissipation coefficients. As an application we study in details the classical damped harmonic oscillator. We show that two known formulations of this oscillator, the Bateman dual and the Caldirola Kanai formulations are particular realizations of our general approach. The Hamiltonian formulation and quantization of the model in both representations are given. Moreover Ostrogradsky generalization of Lagrangian and Hamiltonian formalism for description of systems with higher order derivatives and its application to the constant coefficient equations of an arbitrary order are considered. We construct related with the last one the Euler differential equation of an arbitrary order and its Lagrangian and Hamiltonian structure. Quantum Euler systems are introduced and solved for the stationary Schrodinger picture. Nonstationary nonlinear quantum models corresponding to arbitrary Euler Hamiltonian are solved exactly in the Heisenberg picture.Master Thesis Exactly Solvab Q-Extended Nonlinear Classical and Quantum Models(Izmir Institute of Technology, 2011) Nalcı, Şengül; Pashaev, OktayIn the present thesis we study q-extended exactly solvable nonlinear classical and quantum models. In these models the derivative operator is replaced by q-derivative, in the form of finite difference dilatation operator. It requires introducing q-numbers instead of standard numbers, and q-calculus instead of standard calculus. We start with classical q-damped oscillator and q-difference heat equation. Exact solutions are constructed as q-Hermite and Kampe-de Feriet polynomials and Jackson q-exponential functions. By q-Cole-Hopf transformation we obtain q-nonlinear heat equation in the form of Burgers equation. IVP for this equation is solved in operator form and q-shock soliton solutions are found. Results are extended to linear q-Schrödinger equation and nonlinear q-Maddelung fluid. Motivated by physical applications, then we introduce the multiple q-calculus. In addition to non-symmetrical and symmetrical q-calculus it includes the new Fibonacci calculus, based on Binet-Fibonacci formula. We show that multiple q-calculus naturally appears in construction of Q-commutative q-binomial formula, generalizing all well-known formulas as Newton, Gauss, and noncommutative ones. As another application we study quantum two parametric deformations of harmonic oscillator and corresponding q-deformed quantum angular momentum. A new type of q-function of two variables is introduced as q-holomorphic function, satisfying q-Cauchy-Riemann equations. In spite of that q-holomorphic function is not analytic in the usual sense, it represents the so-called generalized analytic function. The q-traveling waves as solutions of q-wave equation are derived. To solve the q-BVP we introduce q-Bernoulli numbers, and their relation with zeros of q-Sine function.