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  • QUANTUM MECHANICAL COMPUTATION OF BILLIARD SYSTEMS WITH

    ARBITRARY SHAPES

    İNCİ ERHAN

    DECEMBER 2003

  • QUANTUM MECHANICAL COMPUTATION OF BILLIARD SYSTEMS WITH

    ARBITRARY SHAPES

    A THESIS SUBMITTED TO

    THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

    OF

    THE MIDDLE EAST TECHNICAL UNIVERSITY

    BY

    İNCİ ERHAN

    IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

    DOCTOR OF PHILOSOPHY

    IN

    THE DEPARTMENT OF MATHEMATICS

    DECEMBER 2003

  • Approval of the Graduate School of Natural and Applied Sciences

    Prof. Dr. Canan ÖZGEN

    Director

    I certify that this thesis satisfies all the requirements as a thesis for the degree of

    Doctor of Philosophy.

    Prof. Dr. Şafak ALPAY

    Head of Department

    This is to certify that we have read this thesis and that in our opinion it is fully

    adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy.

    Prof. Dr. Hasan TAŞELİ

    Supervisor

    Examining Committee Members

    Prof. Dr. Hasan TAŞELİ

    Prof. Dr. Münevver TEZER

    Prof. Dr. Marat AKHMET

    Assoc. Prof. Dr. Hakan TARMAN

    Assoc. Prof. Dr. N. Abdülbaki BAYKARA

  • Abstract

    QUANTUM MECHANICAL COMPUTATION OF

    BILLIARD SYSTEMS WITH ARBITRARY SHAPES

    İnci Erhan

    Ph.D., Department of Mathematics

    Supervisor: Prof. Dr. Hasan TAŞELİ

    December 2003, 88 pages

    An expansion method for the stationary Schrödinger equation of a particle moving

    freely in an arbitrary axisymmetric three dimensional region defined by an analytic

    function is introduced. The region is transformed into the unit ball by means of co-

    ordinate substitution. As a result the Schrödinger equation is considerably changed.

    The wavefunction is expanded into a series of spherical harmonics, thus, reducing

    the transformed partial differential equation to an infinite system of coupled ordi-

    nary differential equations. A Fourier-Bessel expansion of the solution vector in terms

    of Bessel functions with real orders is employed, resulting in a generalized matrix

    eigenvalue problem.

    The method is applied to two particular examples. The first example is a pro-

    late spheroidal billiard which is also treated by using an alternative method. The

    numerical results obtained by both methods are compared. The second example is a

    billiard family depending on a parameter. Numerical results concerning the second

    example include the statistical analysis of the eigenvalues.

    Keywords: Billiard Systems, Schrödinger Equation, Eigenfunction Expansion, Eigen-

    value Problem, Spherical Harmonics

    iii

  • Öz

    KEYFİ ŞEKİLLİ BİLARDO SİSTEMLERİNİN KUANTUM

    MEKANİKSEL HESAPLAMALARI

    İnci Erhan

    Doktora, Matematik Bölümü

    Tez Yöneticisi: Prof. Dr. Hasan TAŞELİ

    Aralık 2003, 88 sayfa

    Analitik bir fonksiyon ile tanımlanan ve dönel simetrisi olan keyfi bir üç boyutlu

    bölgede, serbest hareket eden bir parçacığın Schrödinger denklemi için açılım yöntemi

    verilmektedir. Bir koordinat dönüşümü vasıtası ile bölge birim küreye dönüştürül-

    mektedir. Bunun sonucunda, Schrödinger denklemi de önemli ölçüde değişikliğe

    uğramaktadır. Dalga fonksiyonu küresel harmonikler cinsinden seriye açılmaktadır

    ve böylece, değişmiş olan kısmi diferansiyel denklemi sonsuz boyutlu bir diferan-

    siyel denklem sistemine indirgemektedir. Çözüm vektörü için reel mertebeli Bessel

    fonksiyonları cinsinden Fourier-Bessel açılimları kullanilmaktadır ve denklem sistemi

    genelleştirilmiş bir matris özdeğer problemine dönüştürülmektedir.

    Yöntem iki özel örneğe uygulanmaktadır. Birinci örnek “prolate spheroid” şek-

    lindeki bilardo sistemidir ve ayni anda alternatif bir yöntemle daha incelenmektedir.

    Her iki yöntemle elde edilen sayısal sonuçlar karşılaştırılmaktadır. İkinci örnek bir

    parametreye bağlı bir bilardo ailesidir. Bu örneğe ait sayısal sonuçlar özdeğerlerin

    istatistiksel analizini de içermektedir.

    Anahtar Kelimeler: Bilardo Sistemleri, Schrödinger Denklemi, Özfonksiyon Açı

    lımı, Özdeğer Problemi, Küresel Harmonikler

    iv

  • To my lovely Selin,

    v

  • Acknowledgments

    I would like to express my sincere gratitude to my supervisor, Prof. Dr. Hasan

    TAŞELİ, for his precious guidance and encouragement throughout the research and

    also for his friendly attitude. I would like to thank Prof. Dr. Münevver Tezer,

    Assoc. Prof. Dr. N. A. Baki Baykara, Assoc. Prof. Dr. Hakan Tarman and Prof.

    Dr. Marat Akhmet for their valuable suggestions. I also thank Christine Böckmann,

    Arkadi Pikovsky and all the members of the Institute of Numerical Mathematics and

    the Department of Nonlinear Dynamics in Potsdam University. I would also like to

    express my thanks to the members of TÜBİTAK-BAYG for the grant which gave

    me the possibility to join the study group in Potsdam University and especially to

    Ayşe Ataş for her helps.

    I offer my special thanks to my mother Gülten Müftüoğlu, my father Fehmi Müf-

    tüoğlu for their love throughout my whole life. I especially thank my husband Adem

    Tarık Erhan for his precious love, and encouragement during my long period of

    study. I also thank my mother-in-law Necla Erhan who helped me by taking care of

    my daughter during the period of preparing and writing the thesis.

    I thank all the members of the Department of Mathematics and my friends for their

    suggestions.

    Last, but not least, I deeply thank my friend Ömür Uğur for his helps during the

    period of writing this thesis.

    vi

  • Table of Contents

    Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

    Öz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

    Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

    Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

    CHAPTER

    1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

    1.1 Billiard systems in classical and quantum mechanics . . . . . . . . . . 2

    1.2 Statistical Properties of the Energy Spectra of Quantum Systems and

    Random Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5

    1.3 Mathematical Model of a Quantum Billiard System Defined by a

    Shape Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    2 Development of a Method for Solving Three-Dimensional

    Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2.1 The Shape Function and the New Coordinates . . . . . . . . . . . . . 10

    2.2 An Eigenfunction Expansion for the Transformed Equation . . . . . . 14

    2.3 Reduction to a System of Ordinary Differential Equations . . . . . . . 18

    vii

  • 2.4 Transformation of the Boundary and Square Integrability Conditions 21

    2.5 The Special Case of Spherical Billiard and the Truncated System of

    ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.6 Investigation of the Coefficient Matrices . . . . . . . . . . . . . . . . 26

    2.7 Truncated Solution in One Dimensional Subspace . . . . . . . . . . . 29

    2.8 Reduction of the System to a Matrix Eigenvalue Problem . . . . . . . 31

    3 Two Examples: Prolate Spheroid and a Parameter-

    Depending Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

    3.1 Prolate Spheroidal Billiard: Solution by the Method of Chapter 2 . . 38

    3.2 Prolate Spheroidal Billiard: Solution by the Method of Moszkowski . 42

    3.3 A Billiard Family Depending on a Parameter: Classical and Quantum

    Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

    4 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . 56

    4.1 Numerical Results for the Prolate Spheroidal Billiard . . . . . . . . . 56

    4.2 Numerical Results for the δ-billiard . . . . . . . . . . . . . . . . . . . 66

    4.3 Statistical analysis of the spectra . . . . . . . . . . . . . . . . . . . . 74

    5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

    viii

  • List of Tables

    4.1 First 60 even eigenvalues for a prolate spheroidal billiard with a = 1

    and b = 1.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

    4.2 First 60 odd eigenvalues for a prolate sp