Dış alanla etkileşimli kuvantum harmonik salınıcının ileriye doğru evrimi
Başlık çevirisi mevcut değil.
- Tez No: 75476
- Danışmanlar: PROF. DR. METİN DEMİRALP
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mühendislik Bilimleri, Mathematics, Engineering Sciences
- Anahtar Kelimeler: Kuantum mekaniği, Kuantum teorisi, Quantum mechanics, Quantum theory
- Yıl: 1998
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Mühendislik Bilimleri Ana Bilim Dalı
- Bilim Dalı: Sistem Analizi Bilim Dalı
- Sayfa Sayısı: Belirtilmemiş.
Özet
20. yy'ın ilk çeyreği fizik dünyasında yeni kapıların aralanmasına ve o zamana değin aydınlanamayan bir takım fiziksel olayların açıklanmasına yardımcı olacak teorilerin ortaya atılmasına şahit olmuştur. Kuvantum teorisinin temelleri bu yüzyılda atılmıştır. Kuvantum mekaniği atomik yapıların davranışlarına açıklık getirmektedir. Kuvantum mekaniğindeki gelişme dış bir alan altında bulunan molekülsel sistemlerin yapılarındaki değişimin nasıl belirleneceğini de açık olarak ortaya koymuştur. Bu çalışmada molekülsel sistem olarak kuvantum harmonik salınıcı alınmış ve bir dış alan altında değişiminin belirlenmesi amaçlanmıştır. Bu amaç doğrultusunda harmonik salınıcının toplam enerjisini karakterize eden Hamiltonyenine, sisteme uygulanan dış alanın potansiyeli eklenerek hareket denklemi (Schrödinger denklemi) oluşturulmuştur. Bu denklem çözülerek sistemin t=0 anından t=T anma kadar olan hareketini ya da diğer bir deyişle ileriye doğru evrimini karakterize eden dalga fonksiyonu w(x,t) bulunmuştur. Tez, aslında, optimal kontrol altında bulunan bir harmonik salınıcımın istenilen belli bir düzeye ulaştırılması için nasıl bir dış alan etkisine alınması amacına yöneliktir. Bu tür problemler ileriye ve geriye doğru evrim bileşenlerine sahiptir. Burada ileriye doğru evrim bileşeni incelenmektedir. Bu doğrultuda, verilen bir dış alan için hesaplama yöntemi geliştirilmektedir Harmonik salınıcıya uygulanan dış alanın genliğinin bilinmemesi halinde problem tam olarak çözülmüş sayılamayacağından genlik E=sabit ve E=coscot alınarak bu tezin üçüncü bölümünde hareket denklemleri oluşturulmuş ve çözülmüştür. E=sabit durumunda hareket denkleminin çözümü aynı zamanda Runge-Kutta yöntemiyle, mathematica dilinde programlanmış ve sayısal sonuçlara ulaşılmıştır. Bu sonuçlar Ek-A'da verilmiştir..
Özet (Çeviri)
The quantum theory was introduced to the world of physics in the first quarter of the 20th century. Until then, ordinary physical phenomena were explained by classical laws and physics was seen as an essentially complete science with little prospect of further developments. However, the classical laws failed to explain the behaviour of objects at atomic level. The study of quantum mechanics was initiated in 1900, the year Max Planck announced the results of his theoretical research into the radiation and absorption of a black body. He proposed that radiant energy was not emitted continuously but discretely (discrete units of light energy called quanta). He developed an equation which fully accounted for this hypothesis. The hv terms in this equation is called a quantum of energy. The constant h = 6,62x1 0"34 j.s is known as the Planck constant. According to Planck's hypothesis, electromagnetic radiation is made of clusters of quanta and each quantum is a particle of energy that can be expressed as: E = hv (2) la equation (2), v is the frequency of the accompanying electromagnetic radiation for that energy. In 1905, Einstein used Planck's hypothesis of quanta to explain the photoelectric phenomenon. He assumed that the energy of light was not distributed over the entire wave front. Instead, light was made from packets of energy quanta. The energy of each photon was E = hv. viaMeasurements made in those years showed that loss of energy in electrons occured in discontinuous quantities. Rutherford, the famous physicist, proved that the atom had a positively charged nucleus in the centre and the nucleus itself had electrons orbiting it. The classical laws of physics dictated that the electrons must emit their energy continuously until their kinetic energy was exhausted. Electrons without energy would then become motionless and fall towards the nucleus, causing atoms to collapse. Since that wasn't the case, other explanations had to be researched. Denmark's Bohr found an answer when he incorporated Planck's quantum theory into Rutherford's atomic model. He proposed that the electrons were restricted to orbits that were fixed in position. According to his theory, electrons do not radiate energy when moving in their fixed orbits, but only when they change orbits. Since the orbits are fixed in position, the radiated energy exists only in fixed quantities. The electrons in the lowest permitted orbit have no lower orbit to change to and since their energy is quantised, they do not gradually spiral into the nucleus. In other words the value hv can only take specific values. Louis de Broglie expanded the scope of all these theories when he proposed that matter, like light or any other type of radiated energy, has both particle and wave characteristics. According to de Broglie there is a relation between the momentum of the particle P, the Planck constant h and the wavelength of the particle that can be expressed as: <* = ) (3) and also he said that the wave property is also valid for electrons. According to de Broglie for any given particle's momentum P, there is a wavelength associated with it. Schrödinger and his followers founded the science of quantum mechanics on these principles. In 1926, Schrödinger used the principle of de Broglie waves when he developed his wave equation. Thanks to the development of quantum mechanics, it is now possible to explain the structural changes of a molecular system when subjected to an external field. The first section of this thesis examines the optimal control and behaviour of molecular systems under optical fields. Then, it discusses the quantum harmonic oscillator as a model and solves the equations connected with external fields. Given a molecular system, in order to solve the optimal control problem, the forward and backward evolution equations of that system must be solved first. The second section of this thesis explains the procedure for finding the optimal control equations of molecular systems. The effect of an external electromagnetic field on a given system is determined by two factors, the field potential and the system Hamiltonian. The molecular system's IXtotal energy is characterized by its Hamiltonian. The Hamiltonian is combined with the external field's scalar and vectorial potentials, giving a Schrödinger equation which is then solved. The Hamiltonian is given by H = H0+Ju(x)E(t) (4) Here, u(x) represents the time independent dipole function and E(t) is the amplitude function of the external field. HQ characterises the system's state dependent behaviour without the external field. The optimal control theory requires that after the Hamiltonian has been constructed, a suitable cost functional is chosen. The optimization is performed with respect to the cost functional. The cost functional is used to determine the molecular system's optimal control and it includes several terms the physical objective, two penalty terms and finally, the dynamical constraint. An operator (Q) characterises observable agent over the time interval 0 < t < T during the field-molecule interaction. Our aim is to make expectation value of a given operator close to its prescriped target value ( Q ) as much as possible. The objective term; j0=U(HxJ)\Q\v(x,t))-Q)2 (5) The field is wanted to be minimized for this reason the first penalty term is used. This term includes a weight function is represented by WE (t) and the first penalty term is given as follows; JE=\\WE{t)E\t)dt (6) 2 0 We choose another operator Q' which represents undesired observable agent. During the field-molecule interaction this operator is wanted to be rninimized. A weight function \Wp(t)) can be used through this minimization procedure. This can be expressed as the second penalty term; xJc> = T \Wp{t){v{x,t)\Q\y,{x,t)?dt (7) 2 The wave function must satisfy the fundamental equation of quantum mechanics. Consequently Schrödinger equation may be included the cost functional. Therefore we can write the following dynamical constraint; Jdk=k\\ \x\xj) ^0 x ih~H0-ttx)E(t) at y(x,t)dxdt (8) The symbol X*(x,t) represents the complex conjugate of X (x,t). The cost functional itself is given by; J = J0+JE+JDK+Jci (9) The system's optimally controlled motion is determined by the evolution equations. To find the evolution equations, the first variation of the cost functional must be set to zero. This is the procedure used to get the forward evolution equations in the second part of this thesis m^ = [H0+M(x)E(t)]y, (10) -m^ = [H0+^(x)E(t)y (11) The symbol y/* represents the complex conjugate of y/. The E(t) amplitude function in the above equations is the basic agent that needs to be determined. To determine it, the forward and backward evolution equations must be solved and combined. Only then the optimal control problem will become solved. In the third section of this thesis, a quantum harmonic oscillator is examined as a molecular system with a view to determining it's change under external fields. XIThe quantum harmonic oscillator is a system that is commonly used as a model for vibration. However, it is only an approximation to describe the characteristics of vibrating systems. To improve this model, terms not present in the harmonic oscillator must be included as correction terms. The first step in finding the forward evolution equation of a harmonic oscillator is the construction of its Hamiltonian _ -ft2 â2 kx1 2m âc2 2 So the Schrödinger equation of a harmonic oscillator interacting with an external field within dipole polarization approximation is given as..âu/ -h2 d2w kx2 &-%- = - - -Z-+-y + gE(t)xy (14) ât 2m âc' 2 The wave function which determines the system properties through expectation values is assumed to be in the following form y/{x,t) = e-a«)x-a*)x-a*) (15) Hence the Schrödinger equation is solved for the external field's constant and cosot values under this structure. At the end of the thesis the program which is presented in appendix-A is prepared in mathematica. The way which is called Runge-Kutta is used in this mathematica program. This program calculated ax, a,, a3 coefficients of the wave function which characterizes forward evolution of harmonic oscillator. This thesis do not include the backward evolution. Hence, it does not suffice for an optimal control approach where we need to evaluate the external field according to our goals. Here, we assumed that the field is given and seeked the solutions for forward evolution. xii
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