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LIE ALGEBRAS in CLASSICAL and QUANTUM MECHANICS By LIE ALGEBRAS IN CLASSICAL AND QUANTUM MECHANICS by Matthew Cody Nitschke Bachelor of Science, University of North Dakota, 2003 A Thesis Submitted to the Graduate Faculty of the University of North Dakota in partial ful¯llment of the requirements for the degree of Master of Science Grand Forks, North Dakota May 2005 This thesis, submitted by Matthew Cody Nitschke in partial ful¯llment of the require- ments for the Degree of Master of Science from the University of North Dakota, has been read by the Faculty Advisory Committee under whom the work has been done and is hereby approved. (Chairperson) This thesis meets the standards for appearance, conforms to the style and format require- ments of the Graduate School of the University of North Dakota, and is hereby approved. Dean of the Graduate School Date ii PERMISSION Title Lie Algebras in Classical and Quantum Mechanics Department Physics Degree Master of Science In presenting this thesis in partial ful¯llment of the requirements for a graduate degree from the University of North Dakota, I agree that the library of this University shall make it freely available for inspection. I further agree that permission for extensive copying for scholarly purposes may be granted by the professor who supervised my thesis work or, in his absence, by the chairperson of the department or the dean of the Graduate School. It is understood that any copying or publication or other use of this thesis or part thereof for ¯nancial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of North Dakota in any scholarly use which may be made of any material in my thesis. Signature Date iii TABLE OF CONTENTS LIST OF FIGURES :::::::::::::::::::::::::::::::::::: v LIST OF TABLES ::::::::::::::::::::::::::::::::::::: vi ACKNOWLEDGEMENTS :::::::::::::::::::::::::::::::: vii ABSTRACT :::::::::::::::::::::::::::::::::::::::: viii CHAPTER 1: INTRODUCTION :::::::::::::::::::::::::::: 1 CHAPTER 2: LIE ALGEBRA ::::::::::::::::::::::::::::: 4 2.1 Basic Concepts . 4 2.2 Structure of Lie Algebras . 6 2.2.1 The Killing Form . 9 2.3 Familiar Physical Examples . 11 CHAPTER 3: CONFINED PARTICLE IN A MAGNETIC FIELD ::::::::: 17 3.1 A Generating Function . 27 3.2 Selection Rules . 34 3.3 A Charged Particle with Spin . 37 CHAPTER 4: COMPLETELY SOLVABLE THREE-BODY PROBLEM :::::: 42 4.1 The Three Body Problem . 44 CHAPTER 5: CONCLUSION :::::::::::::::::::::::::::::: 55 REFERENCES :::::::::::::::::::::::::::::::::::::: 56 iv LIST OF FIGURES Figure Page 1. Plot of Energy as a Function of the Larmor Frequency. 35 2. Plot of Energy Including Spin as a Function of the Larmor Frequency. 41 3. 2D parametric plots of possible xy motion generated by combinations of os- cillatory elementary solutions, which determine three separate modes of the system. Paths (a) - (c) correspond to orbits generated by solutions v11 - v18 (double arrows indicate direction of motion), while path (d) represents the mode generated by v5 and v6. .......................... 54 v LIST OF TABLES Table Page 1. Complete Set of Ladder Operators . 22 2. Selection Rules . 38 3. Eigenvalues and Eigenvectors of the 3-D three body problem . 48 4. Eigenvalues and Eigenvectors of the entire Cartan subalgebra . 50 5. Phase Space Coordinates in terms of Elementary Solutions . 53 vi ACKNOWLEDGEMENTS I expresses sincere gratitude particularly to my graduate advisor Dr. Bill Schwalm. He demonstrated a great deal of patience and understanding during the course of this project. Also, his bottomless reserve of creativity and intuition contributed greatly to the content of this work. If not for his continuous guidance, this project would not have been possible. I would also like to thank Dr. Ju Kim for making it possible for me to continue my studies at the University of North Dakota. vii ABSTRACT A nonrelativistic quadratic Hamiltonian constitutes the simplest class of completely solv- able problems in classical and quantum mechanics. Such a Hamiltonian is a sum of terms, each of which is a quadratic combination of positions and momenta. Physical systems gov- erned by quadratic Hamiltonians include the n-dimensional harmonic oscillator and a particle in a constant magnetic ¯eld. These simple systems can be used e®ectively to model more complicated physical systems. They also form a starting point for approximate treatments of many phenomena in condensed matter and statistical physics. Thus, it is important to de- velop as complete an understanding as possible of simple quadratic models. The purpose of this work is to present two examples of the use of Lie algebra to analyze physical phenomena completely. Two systems are considered. One is a charged harmonic oscillator in a constant magnetic ¯eld, treated quantum mechanically, and the other is a solvable three-body problem, treated in Hamiltonian dynamics. In each case, the procedure is to address the problem as one of constructing representations of a Lie algebra. A maximal set of mutually commuting quantities (or quantities that have zero Lie bracket with one another) is determined. Then using undetermined coe±cients, the remaining variables are combined into ladder operators, in the quantum mechanical case, or fundamental solutions in the classical case. The result is a pair of detailed examples of Lie algebra techniques applied to physical systems. viii CHAPTER 1 INTRODUCTION Symmetry plays a central role in the analysis of physical systems and the foundation of physical laws. Lie groups and their corresponding Lie algebras constitute the underlying mathematical framework. Lie algebraic techniques have contributed greatly to a wide range of problems in high energy and condensed matter physics. It is useful to have a clear understanding of Lie algebras and Lie algebraic techniques in order to apply them to solve physical systems e®ectively. Much progress has been made in the development of Lie algebraic techniques applied to physics. For example, Lie transformation groups such as SU(2) and SO(3) are well known and used extensively. Consequently, a large amount of literature devoted to the subject is aimed particularly at physicists [1, 2, 3, 4, 5]. However, the connections between Lie algebra and physics are not accessible to typical physics graduate students without a thorough math- ematical background. On the other hand, when the mathematics is clearly explained for the average student of physics, worked examples may be in short supply. This paper attempts to mitigate this situation by providing two explicitly-worked examples of simple physical systems treated by Lie algebraic techniques. These simple examples are solved exactly and can be extended to model more complicated systems. Two systems are considered. One is a charged anisotropic, three-dimensional harmonic oscillator in a constant magnetic ¯eld, treated quantum mechanically, and the other is a 1 solvable three-body problem, treated in classical Hamiltonian dynamics. The problem of non-interacting, non-relativistic charged particles in an external magnetic ¯eld has been treated for example by Kennard in 1927 [6] and Johnson and Lippmann in 1949 [7]. These results have been extended more recently within a group theoretical framework by Beckers and Hussin [8]. Reference [8] is the primary motivation for the ¯rst part of the current work. The second portion of the present paper was intended to extend these results. After devoting nearly six months to the work reported here, I discovered many (not all) of the results had been published previously by N.F. Johnson and Payne [9] along with the 1993 follow-up paper by Hasse and N.F. Johnson [10]. The latter paper presents a solution for N harmonically interacting particles in a quantum dot, which is an interesting problem particularly to condensed matter physicists. The results in the third chapter of the present paper can be compared to the model in Ref.[10] for the case of three interacting particles. This thesis is organized as follows. In chapter I, the basic concepts of vector spaces and a Lie algebra are reviewed. This chapter serves as a framework underlying the entire paper, introducing explicitly the mathematical concepts employed throughout this work. Chapter II addresses the well-known problem of calculating the energy eigenvalues of a particle under the influence of a constant magnetic ¯eld subject to a quantum dot-like con¯nement potential. A generating function for all of the possible excited states is obtained, which may not have been published previously. Also, selection rules for allowed transitions are computed. The problem is treated quantum-mechanically using Lie algebraic techniques, with a ¯rst-order correction for spin-orbit coupling. The application of Lie algebra is continued in chapter III, this time to classical Hamiltonian dynamics. Here, a completely solvable three-body problem is addressed, using the techniques developed in the ¯rst chapter. This problem is 2 solvable due to the nature of the Hamiltonian, which is a quadratic form. Finally, a set of elementary solutions corresponding to quantum mechanical raising and lowering operators is obtained. It is shown that the results correlate directly to familiar results in appropriate limiting cases. 3 CHAPTER 2 LIE ALGEBRA In general, an algebra is a vector space with a multiplication de¯ned between vectors. The multiplication takes two vectors into another vector. The reader is assumed to be familiar with vector spaces, however for completeness I include some basic concepts. 2.1 Basic Concepts A vector space over the complex numbers C is a set V of vectors and two operations, vector addition (+), under which V forms an abelian group, and multiplication of a vector v by a scalar f. If v1 and v2 are in V , then so is v1 + v2, also the following properties hold: 1. v1 + (v2 + v3) = (v1 + v2) + v3 and v1 + v2 = v2 + v1 2. There exists a zero vector 0, such that v + 0 = v for any v 2 V . 3. Each v 2 V has an additive inverse, ¡v such that v + (¡v) = 0 which de¯nes subtraction v1 ¡ v2 = v1 + (¡v2).
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