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Nuclear Pasta with a Touch of Quantum Towards Fully Antisymmetrised Dynamics for Bulk Fermion Systems

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Nuclear Pasta with a Touch of Quantum Towards Fully Antisymmetrised Dynamics for Bulk Fermion Systems Nuclear Pasta with a touch of Quantum Towards fully antisymmetrised dynamics for bulk fermion systems Klaas Vantournhout Proefschrift ingediend tot het behalen van de academische graad van Doctor in de Wetenschappen: Fysica Academiejaar 2009-2010 Promotor: Prof. Dr. Jan Ryckebusch Promotor: Prof. Dr. Natalie Jachowicz THIS PAGE INTENTIONALLY LEFT BLANK voor mijn ouders Philosophy is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. Galileo Galilei, Il Saggiatore (The Assayer, 1623) THIS PAGE INTENTIONALLY LEFT BLANK DANKWOORD Mijn proefschrift, 160 bladzijden! Een pad doorheen de werelden van fysica, wiskunde en sterrenkunde; geplaveid met continue, binaire en kwantummechanische tegels. Een zes jaar durende queeste naar vergelijkingen. Bloed, zweet en tranen heeft het gekost. Lezen, schrijven, herschrijven, beschrijven, verschrijven, verwijderen, vergis- sen, verfrommelen, ontfrommelen, ontrafelen, ontwarren, verwarren, vertalen, her- talen en ook een beetje euforie . de eerste print. Een eindwerk schrijf je niet alleen. Velen hebben bijgedragen aan dit proces. Veel dank ben ik verschuldigd aan mijn beide promotoren, professoren Jan Ryckebusch en Natalie Jachowicz. De wetenschappelijke vrijheid die ik kreeg, jullie steun, vertrouwen en inzicht waren voor deze scriptie heel belangrijk. I especially would like to express my gratitude to Professor Hans Feldmeier and Doctor Thomas Neff. Your continuous support and interest for this work is greatly appreci- ated. During the handful of conferences where we met, as well as ones in GSI, you always found the time to discuss my work with a remarkable enthusiasm and deeper knowledge behind the physics of FMD. This often shed a new light on this dissertation and brought its development one step further. Het is mij ook een eer om professor Kris Heyde te vermelden in dit woord van dank. Uw interesse, steun en onwaarschijnlijk geheugen hebben mij vaak geholpen. Het was een genoegen om ontelbare keren uw bureau plat te lopen, wetende dat daar het antwoord verscholen zat. Aan de vele collega’s en ex-collega’s, na zes jaar te veel om op te noemen, een welge- meende dank! Van de oud gedienden wens ik in het bijzonder Stijn en Veerle (meneer en madam), Peter (pière) en Kris (jezus) in de verf te zetten voor de vele wereldveran- derende en -verbeterende theoriën die we ontwikkelden tussen pot en pint. Hierbij v Dankwoord staat een dagje uit in West-Vlaanderen (Vleteren) centraal met paters, garres en ander goddelijk nat. Mijn (ex)bureaugenoten, Wim en Christophe mag ik zeker niet ver- geten. Het was een waar genoegen met jullie samen te zuchten, zwoegen en zweten maar ook te genieten, van de winterspelen. Wim, je muziekkeuze is . Christophe, jammer dat je onze tempel der wetenschappen hebt moeten verlaten, was het door Wim zijn muziekkeuze, of die van mij? In elk geval, met jullie altijd amuzeleute. Ten slotte wens ik ook nog Arne (wombat) te danken. Je bent iemand die ik enorm waardeer. Ernst en leute, altijd tegen de verzuring! The interest in astronomy and science has always been kept alive thanks to the many friends of the International Astronomical Youth Camp. You have given me so much and have supported me in this everlasting quest. I am grateful for that. To Sebastian Bürgel I would like to say the following. You see, I actually finished! Mijn studies zouden er nooit geweest zijn zonder de inzet en gedrevenheid van Dhr. Baesens. Gerard, het is dankzij u dat ik de grote stap van het technisch onderwijs naar het universitaire heb kunnen maken. Dankzij uw enthousiasme voor wiskunde, uw passie voor het analytische, ontdekte ik de schoonheid van een formule wat zich duidelijk vertaalt in dit werk. Een bijzonder woord van dank gaat uit naar mijn ouders. Niet enkel hebben jullie met mij moeten samenleven in de soms woelige tijden, maar desondanks alles bleven jullie me steunen. Jullie bijdrage tot deze verhandeling is van onschatbare waarde. Zonder jullie was dit doctoraat nooit af! Zonder jullie was ik nooit wat ik ben. Puzzles, complex problems and other brainteasers have always attracted me. Monika, you are my true Gordian knot. Not one to be cut, but to cherish and that not only because you helped correcting this work. Een tevreden promovendus dankt U, Klaas Vantournhout 27 november 2009 vi CONTENTS Dankwoordv Contents vii 1 Introduction1 2 Molecular dynamics for Fermions 15 2.1 Introduction . 15 2.2 Time-dependent variational methods . 17 2.3 The time-dependent variational principle . 19 2.4 The motion of a single particle . 26 2.5 The dynamics of a fermion system . 32 2.6 Final remark on FMD . 37 3 Fermion matter in bulk 41 3.1 Introduction . 41 3.2 Lattices and periodic quantum systems . 45 3.3 Truncated antisymmetrised periodic boundary conditions . 47 3.4 The Bloch representation of TAPBC . 60 3.5 Periodic boundary conditions in FMD . 63 3.6 The Bloch overlap in PBC . 68 3.7 Fermionic behaviour of a particle in one dimension . 72 3.8 Fermions and Brillouin zones . 78 3.9 Free fermions and periodic boundary conditions . 85 4 Conclusion and Outlook 91 vii Contents A Notations and conventions 97 A.1 Units and conversion factors . 97 A.2 Abbreviations . 97 A.3 Math notations, operations and functions . 98 A.4 Other notations . 99 B Reference Formulae 101 B.1 Parabolic cylinder functions . 101 B.2 Gaussian integrals . 102 B.3 Multivariate Gaussian integrals . 103 B.4 The discrete Fourier transform in lattice representation . 104 B.5 The Jacobi Theta functions . 106 B.6 The Riemann Theta function . 109 C Gaussian wave packets and related integrals 111 C.1 Representation of the Gaussian wave packet . 111 C.2 Overlap, mean positions and variance . 112 C.3 The equations of motion for a Gaussian wave packet . 113 C.4 Physical interpretation of the equations of motion . 115 C.5 The equation of motion in a damped harmonic oscillator . 117 C.6 The two-body-matrix element . 119 D The Wigner distribution of a Gaussian wave packet 121 D.1 Definition of the Wigner distribution . 121 D.2 The Wigner distribution of a one-dimensional Gaussian wave packet . 122 D.3 The Wigner distribution of a free Gaussian wave packet . 123 D.4 The Wigner distribution for a harmonic oscillator . 124 E Matrix elements of antisymmetric wave functions 127 E.1 The antisymmetric many-body wave function . 127 E.2 The overlap-matrix element Φ Ψ ....................... 128 E.3 The expectation value of a one-bodyh j i operator . 129 E.4 The expectation value for a two-body operator . 130 F Circulant matrices and their properties 133 F.1 Toeplitz and circulant matrices . 133 F.2 Eigendecomposition of a circulant matrix . 135 F.3 Inverting a circulant matrix . 136 F.4 The block-circulant matrix . 136 F.5 Eigendecomposition of a block-circulant matrix . 137 viii Contents F.6 Inverting a block-circulant matrix . 138 Nederlandstalige samenvatting 141 Bibliography 147 ix THIS PAGE INTENTIONALLY LEFT BLANK 1 INTRODUCTION Preface Mighty oaks from little acorns grow. This proverb reflects the evolution of the vast and ever-changing panoply presented to us while gazing at the heavens. It represents a cycle of life embracing our own. Massive gas clouds understood as a mere collection of dust and the tiniest of particles are the incubators of stars sprouting towards an active and varied life for millions of years. ‘Several billion trillion tons of superhot exploding hydrogen nuclei synthesise into ever-heavier elements, while often rising slowly above a horizon, managing to look small, cold and slightly damp’ [Ada82]. Then they collapse and die. In an act of despair, as a last struggle against gravity, they go supernova, one of the most glorious events in the universe. In an unrivalled explosion, they light up in the skies for days and nights on end while seeding ashes into the vast emptiness around, waiting to be used again in a new life cycle of a stellar system. Cocooned within those ashes lies a new type of star dormant. A neutron star, born the moment another star died. One star’s breath is another star’s death! (Fig. 1.1) Neutron stars Neutron stars demonstrate how matter behaves under extreme conditions. Contain- ing roughly 1.5 solar masses of matter within a radius of about 12 km, they reach densities as high as five, ten or even twenty times the nuclear saturation density ρ0 corresponding to 0.16 nucleons per fermi cubed. Without any doubt, they consti- tute one of the densest forms of matter in the observable universe. Their discovery started rather unusual. Before their actual theoretical prediction in 1934 by Baade and Zwicky [Baa34a, Baa34b], Landau anticipated the existence of neutron stars in 1931. That was before Chadwick announced in 1932 his discovery of the neu- tron [Lan32, Cha32, Hae07], the dominant nucleonic component of neutron stars. 1 Chapter 1: Introduction Figure 1.1: This wide-field composite image was made with X-ray (blue/ROSAT & Chandra), radio (green/Very Large Array), and optical (red/Digitized Sky Survey) ob- servations of the supernova remnant IC 443, i.e. the Jellyfish nebula. The pullout, also a composite with a Chandra X-ray close-up, shows a neutron star that is spewing out a comet-like wake of high-energy particles as it races through space.
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