Bazı kuantum mekaniksel sistemlerinin bilgisayar benzetişimi
Computer simulation of some quantum mechanical systems
- Tez No: 66534
- Danışmanlar: PROF. DR. METİN DEMİRALP
- Tez Türü: Yüksek Lisans
- Konular: Mühendislik Bilimleri, Engineering Sciences
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Matematik Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Sistem Analizi Bilim Dalı
- Sayfa Sayısı: 123
Özet
ÖZET Kuantum mekaniksel sistemlerin bilgisayar benzetişimi, bu sistemlerin öğrenilmesinde ve teorik çalışmaları desteklemek açısından büyük önem taşımak tadır. Bu tez çalışmasının amacını kuantum mekaniksel sistemlerin bilgisayarla benzetişimleri için bir alt yapı oluşturma, gerekli sayısal algoritmaların öğrenilmesi ve kodlanmalarının yapılması oluşturmaktır. Kuantum Mekaniğinde bir parçacığın dinamiği Schrödinger denklemi ile belirlenir. Bu denklem nonlineer bir kısmi türevli diferansiyel denklemdir ve potansiyel fonksiyona bağlı olarak çoğu zaman analitik çözümleri elde edilememektedir. Bu açıdan sayısal yöntemlerle çözülmeleri ve elde edilen sonuçların sunulması büyük önem taşımaktadır. Schrödinger denkleminin çözümüne yönelik algoritmalar birkaç grupta toplanmaktadır. Bu çalışmada bu yöntemler araştırılmış ve bazılarının bilgisayar programları yapılmıştır. Gerekli benzetişim ortamının UNIX işletim sistemi ve X Window grafiksel kullanıcı arayüzü olmasına karar verilmiş ve yazılımlar C dilinde gerçekleştirilmiştir. Geliştirilen yazılımla tek boyutlu quantum mekaniksel sistemlerin bilgisayar ortamında benzetişimi yapılabilmektedir. Yazılım eğitim amaçlı ve teorik sonuçların irdelenmesi amacı ile de kullanılabilecek niteliktedir. vıı
Özet (Çeviri)
SUMMARY Quantum Mechanics has changed our way of looking into physical systems. The classical mechanics, we can call it the physics of Newton, is determin istic, logical and certain. The new quantum physics is characterized by its indeterminism, illogicality and uncertainty. So the visualization of quantum mechanical systems is very important for the understanding of the ideas be hind quantum mechanics. The dynamics of a quantum mechanical particle is determined by the Schrö- dinger Equation. The Schrödinger Equation is a nonlinear partial differantial equation and in general it is not possible to solve it analiticaly. So implementing the numerical solutions are very important. There are many numerical algorithms to solve the Schrodinger equation. In this thesis, first of all a literature searching has been made. The available algo rithms are considered. And some of these algorithms are coded into computer programs. The environment for the computer simulation is UNIX operating system with the X Window System graphical user interface. And the programs are coded in C programming language. A standalone simulation software named QMSim that can be used to simulate one dimensional quantum mechanical sys tems has been developed. After a short overview of the thesis we begin with introducing basic pos tulates of quantum mechanics which serve to formalize the rules of quantum mechanics. First postulate states that to any well defined observable A in physics there corresponds an operator A such that measurement of A yields values which are eigenvalues of A. Second postulate of quantum mechanics states that for every dynamical system there exist a single valued integrablc and continuous function ij) from which all of the physical characteristics of the system can be obtained. This function is called Wave Function or State Function of the system. The third postulate states that the average of a physi cal observable is equal to the expectation value of the corresponding operator. Specificially, if a system is in the state ?/>, the average of any physical observable vmC relevant to that system at that time is = f^*C^dx The fourth postulate of quamtum mechanics specifies the time development of the wave function tp; the wave function for a system develops in time according to the equation 00(M) ft2 d2^(x,t), T" in-dT- = -^-d^r- + v(x,twx,t) This equation is called the One Dimensional Time Dependent Schrödinger Equation where V(x,t) is the potential function of the system and the ini tial wave function is given by ^(z,0) = ^0(a0 Available numerical solutions of this equation are one of the main subjects of this thesis. As stated before, in general it is not possible to solve the Schrödinger Equa tion analitically and the importance of solving the Schrödinger Equation which detemines the quantum mechanical behaviour of the systems, directed many researchers mostly working in applied chemistry and physics fields, to develop numerical solution methods in the past and today. The first step in a numerical solution of the Schrödinger Equation is to represent the wave function in computers memory. Since the wave function tp(x,t) is a continious function defined all over the phase space, one needs infinite memory to represent it in the computer completely. The approach to this problem is to represent the wave function on some finite number of points in the phase space. These points are called grid points and this process is called discretization. Both time and position are discretizated. The grid length for position is Aa; and for time is At. If X{ shows the discretizated position points we represent the value of wave function at the grid point Xj at time step n with V>J = ij>(xj,nAt) The methods for solving the Schrödinger Equation are grouped into two classes. Starting from the initial wave function given at time t = 0, if the time evoluation of wave function can be constructed after each time step by direct evaluation of an epression obtained from the discretizated equations, that kind of a method is called an explicit method. If some solution of a system of equations is necessary, it is an implicit method. IXIn the literature searching, many approaches for the numerical solution of Schrödinger Equation have been found beginning with the year 1967. Some of these methods are listed below. 1. Simple Discretization Method 2. Crank-Nicholson Method 3. Second Order Differencing Method 4. Fourier Methods 5. Split Operator Method 6. Lanczos Method The first three of these methods are explained in detail in the thesis. The third one, Second Order Differencing method is programmed and used in the simulation software QMSim developed for the thesis. The simple discretization method is obtained simply by replacing the time and position derivatives in the Schrödinger Equation by their finite difference representations. It is an explicit and unfortunately highly unstable method. So it can not be used in the realistic simulations of quantum mechanical systems. The Crank-Nicholson method is obtained by replacing the Cayley form of e-itH 0perator in the formal operator solution of Schrödinger Equation. This is an implicit method and solving a tridiagonal system of linear equations is necessary at each time step. The Second Order Differencing method is obtained by using the linear terms in the series expansion of the e_ operator in the formal operator solution of the Schrödinger Equation and doing some calculations. It is an exlicit method. It is easier to code the algorithm into a computer program and has been coded in the QMSim as a numerical method to be used in the simulation. The computing environment in which the simulation software has been de veloped is a Personal Computer with Linux operating system and with X Win dow System as the graphical user interface. The programs are written in C programming language which is necessary to use to write programs that use X Window System functions. Below, some properties of the Linux Operating System and X Window System are given. Linux is a kind of UNIX operating system developed for 386 based personal computers (PC). First version of Linux is designed and developed by Linus Trovalds in 1992. It has been brought to its professional form by many volun teers around the world mostly on Internet. Linux is free, it can be downloaded from some sites on Internet. It is a complete operating system and It has al most all the utilities those used in the UNIX. These properties make the Linuxvery popular especially at the universities. That is the reason to choose Linux as the operating system for the simulation software developed in this thesis. The most comonly used graphical interface used on UNIX as well as Linux is the X Window System (also called X Windows or simply X). X Windows is a software development system used to produce graphical user interfaces in a networked environment. It has client-server architecture. Applications that benefit from X Windows functions can be written in C programming language. X Windows applications developed in one system can be ported to another system that support X windows without much difficulties. The QMSim program has been developed on a PC with Linux and recompiled successfully on an IRIX system. In the X Window system all communications between the client and sever are arranged with a protocol named X Protocol. A C language interface to this protocol is Xlib. Xlib has all the primitive functions to perform all the func tions from communications to graphical activities. But Xlib is a very low level library. Xt is a higher level library built on Xlib, which has many programming functions and which introduces some programming elements named widgets to be used in application programs. Xt itself is not enough for all user needs but it gives a base to write more complicated and more professionally looking graphical elements. Using Xt's and Xlib's functions one can write customized graphical programming elements and tools. Athena widget set is one of them built on Xt and has a lot of objects like menus, text boxes, panes etc. This widget set is being used in QMSim. Motif and Open Look are another examples for the widget sets built on Xt. Xlib and Xt have lots of functions and these functions can only be explained in many volumes of books. In chapter three of the thesis most of the functions used in the QMSim is explained shortly. As already stated before a simulation software named QMSim has been de veloped. Using this software one can simulate one dimensional quantum me chanical systems. It is possible to choose potential function and initial wave function from different predefined ones or define new ones. Parameter values can be changed. An effect function which can be regarded as a time dependent potential function can also be incuded in the simulation to simulate forced sys tems. The result of the simulation is the graph of either square of the module of wave function or real and imaginary parts of the wave function at each time step. Wave function, potential function and effect function are displayed in different windows that can be resized, moved or closed. Program is written in C programming language and requires X Windows and Athena Widget set to compile and run. The program consist of many files. The complete list are given in the appendix. To define new functions one can edit the file QMSfunc.c and redefine the functions, then recompile the program. The program is designed in a modular manner so that new subroutines can be XIadded or old ones can be replaced by the new ones easily. Even different numerical solution methods can be applied. Some simulation results where the current numerical method is Second Order Differencing method are given at chapter five. The program's main components are listed below: 1. Graphic Windows: Seperate windows that show the wave function, potential function and effect function. These windows are that kind of graph windows denned in the file GraphUtl.c and can show graphs of real or complex arrays. They can be resized and moved. 2. Numerical Solution Subroutine: Obtaining the new wave function is done by calling this subroutine. If different methods are defined the active method is used. 3. Menu: Chosing the different functions and program options are performed by using the menu options. 4. Simulation Control Functions: Necessary functions to start, stop and reset the simulation are placed here. 5. Information windows: The objects which show the current values of the parameters and which allow to edit their values are placed here. 6. Event Manager: This is the main control routine of the simulation. All events are processed properly and simulation is run according to the user re sponses. The main routines take place in the file QMSim.c. There are initialization routines, menu and window definitions, event management functions in this file. The header files contain the variables, constants and function prototypes used in the program. xu
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