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
