Quantum mechanical calculations for Bose-Einstein ......Quantum mechanical calculations for...

Quantum mechanical calculations forBose-Einstein condensates with

electromagnetically induced 1/r-interaction

Diploma Thesis ofIoannis Papadopoulos

March 12, 2007

First Supervisor : Prof. Dr. Gunter WunnerSecond Supervisor : Prof. Dr. Manfred Fahnle

Universitat Stuttgart1. Institut fur Theoretische Physik

Pfaffenwaldring 57 // IV70550 Stuttgart, Germany

Statutory Declaration

I explain the fact that I wrote this work independently and used no other sourcesor aids than those indicated.

Stuttgart March 12, 2007 Ioannis Papadopoulos

Contents

List of Figures v

List of Tables vii

1 Introduction 1

2 Theoretical basics on BEC 52.1 The cause of the gravity-like attraction between neutral atoms . . 52.2 Identity of particles . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 The indistinguishability of identical particles . . . . . . . . 82.3 Bose-Einstein condensation in atomic gases . . . . . . . . . . . . . 9

2.3.1 The ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . 92.3.2 The weakly-interacting Bose gas . . . . . . . . . . . . . . . 11

2.4 Extended Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . 112.4.1 Variational procedure . . . . . . . . . . . . . . . . . . . . . 132.4.2 Bogoliubov approximation . . . . . . . . . . . . . . . . . . 15

2.5 Quantum statistics - The ideal Bose gas in the grand canonicalensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 General features of the extended Gross-Pitaevskii equation 213.1 Several properties of the GP equation . . . . . . . . . . . . . . . . 21

3.1.1 Symmetry leads to spherical coordinates . . . . . . . . . . 213.1.2 Invariance of the GP system . . . . . . . . . . . . . . . . . 23

3.2 Scaling behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 “Atomic” units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Energy functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.1 The N dependency of the mean-field energy E . . . . . . . 293.5 Radius rrms and peak density ρmax . . . . . . . . . . . . . . . . . 34

3.5.1 The N dependency of the radius rrms . . . . . . . . . . . . 343.5.2 The N dependency of the peak density ρmax . . . . . . . . 35

3.6 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . 353.7 Asymptotic behavior of the wave function . . . . . . . . . . . . . 36

3.7.1 Limit r → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 37

i

ii Contents

3.7.2 Limit r →∞ . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 Derivation of an analytical wave function . . . . . . . . . . . . . . 39

4 Approximation method for stationary states 474.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 STO trial wave function . . . . . . . . . . . . . . . . . . . 484.1.2 GTO trial wave function . . . . . . . . . . . . . . . . . . . 60

4.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Variational and numerical calculations 735.1 The numerical procedure . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1 Normalization of the numerical solutions . . . . . . . . . . 745.2 The variational procedure . . . . . . . . . . . . . . . . . . . . . . 775.3 Calculations for different number of particles N . . . . . . . . . . 77

5.3.1 Calculations for N = 102 . . . . . . . . . . . . . . . . . . . 785.3.2 Calculations for N = 104 . . . . . . . . . . . . . . . . . . . 795.3.3 Calculations for N = 106 . . . . . . . . . . . . . . . . . . . 815.3.4 Calculations for N = 108 . . . . . . . . . . . . . . . . . . . 825.3.5 “TF-G” region: N2 a

au= 10+04 . . . . . . . . . . . . . . . . 84

5.3.6 “G” region: N2 aau

= 10−05 . . . . . . . . . . . . . . . . . . 865.4 Energies, sizes and peak densities for different scattering lengths . 89

5.4.1 Relative error between the numerical and variational cal-culations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Comparison between ψTF−G and ψnum . . . . . . . . . . . . . . . 995.6 Accuracy of the numerical energy values . . . . . . . . . . . . . . 1025.7 Tangent bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Beyond mean-field approximation 1076.1 Quantum corrections to the ground state of self-trapped BEC . . 107

7 Summary 1117.1 Purpose of the study . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A Integral representation of ψ(r) and U(r) 115

B Derivatives for the Taylor expansion 117

C Integrals 121

D Non-normalized solutions 123

E Wave functions and potentials 125

Contents iii

F Graphs 129

G Comparison between ψTF−G and ψnum 137

H Numerical integration 139

Bibliography 141

iv Contents

List of Figures

1.1 Triads of laser beams cross the condensate. . . . . . . . . . . . . . 1

2.1 The velocity distribution of rubidium atoms in the experiment by[Anderson et al. (1995)] is shown. The left frame represents a gas ata temperature just above condensation at about 400 nK; the centreframe describes a gas just after the appearance of the condensateat about 200 nK; in the right frame, after further evaporationsample of nearly pure condensate at about 50 nK is released. From[Cornell (1996)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 A 7-particle state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Orientation of the axes. . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1 Effective gravity-like potential Vu(r) for a STO (black curve). Theblue lines show the asymptotic behaviour of Vu(r) for r → 0 andr →∞, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1 Wave functions at different scattering lengths. . . . . . . . . . . . 76

5.2 Self-trapping potentials at different scattering lengths. The asymp-totic −1/r behaviour is shown for comparison. . . . . . . . . . . . 76

5.3 Phase diagram: The total number N of atoms as a function ofthe s-wave scattering length a

au. The two asymptotic regions are

separated by the line N2 aau

= 1. . . . . . . . . . . . . . . . . . . . 88

5.4 Mean-field energy as a function of the condensate radius [O’Dellet al. (2000)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5 Solid violet lines for variational computations with k− and dashedwith k+; green lines for numerical computations. . . . . . . . . . . 105

D.1 Non-normalized wave functions S and potentials U for b = 0; Sarbitrary but constant, as U(r = 0) is varied. . . . . . . . . . . . . 124

E.1 Wave function and corresponding potentials for N2 aau

= −1. . . . 125


= −6 · 10−01. 125

v

vi List of Figures


= −3 · 10−01. 126


= 0. . . . . 126


= 1 · 10+01. 126


= 1 · 10+02. 127


= 5 · 10+02. 127


= 1 · 10+03. 127


= 3 · 10+03. 128


= 5 · 10+03. 128


= 1 · 10+04. 128

F.1 Green lines: numerical computations; blue lines: variational com-putations; red lines: relative error between the numerical and vari-ational computations. . . . . . . . . . . . . . . . . . . . . . . . . . 129






F.7 Green lines: numerical computations; blue lines: variational com-putations for; red lines: relative error between the numerical andvariational computations. . . . . . . . . . . . . . . . . . . . . . . . 135

G.1 Wave functions as a function of the condensate radius; dashed linesfor analytical form (5.21); solid lines for numerical computations. 137

G.2 Relative error between the numerical and the analytical wave func-tion (5.21) as a function of r. . . . . . . . . . . . . . . . . . . . . 138

List of Tables

4.1 Comparison between STO and GTO . . . . . . . . . . . . . . . . 714.2 Comparison between STO and GTO . . . . . . . . . . . . . . . . 72

5.1 Numerical calculations for b = 0. U0 and S0 are the initial values,U(r = 0) is the correct final value, ν is the parameter for thescaling, and S0 is the scaled value at r = 0. . . . . . . . . . . . . . 75

5.2 Numerical calculations for N = 102 . . . . . . . . . . . . . . . . . 785.3 Numerical calculations for N = 102 . . . . . . . . . . . . . . . . . 785.4 Variational calculations with k+ for N = 102 . . . . . . . . . . . . 795.5 Variational calculations with k+ for N = 102 . . . . . . . . . . . . 795.6 Numerical calculations for N = 104 . . . . . . . . . . . . . . . . . 795.7 Numerical calculations for N = 104 . . . . . . . . . . . . . . . . . 805.8 Variational calculations with k+ for N = 104 . . . . . . . . . . . . 805.9 Variational calculations with k+ for N = 104 . . . . . . . . . . . . 815.10 Numerical calculations for N = 106 . . . . . . . . . . . . . . . . . 815.11 Numerical calculations for N = 106 . . . . . . . . . . . . . . . . . 815.12 Variational calculations with k+ for N = 106 . . . . . . . . . . . . 825.13 Variational calculations with k+ for N = 106 . . . . . . . . . . . . 825.14 Numerical calculations for N = 108 . . . . . . . . . . . . . . . . . 835.15 Numerical calculations for N = 108 . . . . . . . . . . . . . . . . . 835.16 Variational calculations with k+ for N = 108 . . . . . . . . . . . . 835.17 Variational calculations with k+ for N = 108 . . . . . . . . . . . . 845.18 Numerical calculations for N2 a

au= 10+04 . . . . . . . . . . . . . . 84

5.19 Numerical calculations for N2 aau

= 10+04 . . . . . . . . . . . . . . 85

5.20 Variational calculations with k+ for N2 aau

= 10+04 . . . . . . . . . 85


= 10+04 . . . . . . . . . 85


= 10+04 . . . . . . . . . 86


= 10−05 . . . . . . . . . . . . . . 86


= 10−05 . . . . . . . . . . . . . . 87


= 10−05 . . . . . . . . . 87


= 10−05 . . . . . . . . . 87


= 10−05 . . . . . . . . . 88

vii

viii List of Tables

5.28 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . 905.29 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . 915.30 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . 925.31 Variational calculations with k+ . . . . . . . . . . . . . . . . . . . 945.32 Variational calculations with k+ . . . . . . . . . . . . . . . . . . . 955.33 Variational calculations with k+ . . . . . . . . . . . . . . . . . . . 965.34 Error between the numerical and variational calculations . . . . . 975.35 Error between the numerical and variational calculations . . . . . 985.36 Wave function as a function of the condensate radius for a

au= 5·10+03100

5.37 Wave function as a function of the condensate radius for aau

= 8·10+03101

5.38 Wave function as a function of the condensate radius for aau

= 1·10+041015.39 Comparison of the energies for different accuracies by b = 6 . . . . 1025.40 Rounded energy eigenvalues for different accuracies by b = 6 . . . 1025.41 Numerical results for the energy and the chemical potential for

different negative scattering lengths . . . . . . . . . . . . . . . . . 1045.42 Variational results with k− for the energy and the chemical poten-

tial for different negative scattering lengths . . . . . . . . . . . . 104

6.1 Quantum corrections for the contact potential . . . . . . . . . . . 109

Zusammenfassung

Einer Publikation von O’Dell [O’Dell et al. (2000)] folgend, nach der sich eineArt

”Gravitationswechselwirkung“ in einem Bose-Einstein-Kondensat (BEC) aus-

bilden kann, wenn man dieses mit Lasern unter bestimmten Winkeln bestrahlt,haben wir die Schrodinger-Gleichung aufgestellt, die in der Fachliteratur alserweiterte Gross-Pitaevskii Gleichung bezeichnet wird und die Losungen be-rechnet. Wellenfunktionen und die dazugehorigen Potentiale, Energien, Dichtenund andere Großen werden fur verschiedene Teilchenzahlen sowohl numerischals auch analytisch berechnet. Die numerischen Berechnungen werden mitHilfe eines Runge-Kutta Verfahrens vierter Ordnung mit adaptiver Schrittweitedurchgefuhrt. Fur die analytischen Berechnungen machten wir uns das Vari-ationsprinzip zunutze. Fur den Ansatz einer Gauss-Wellenfunktion wurde dasEnergiefunktional des Systems minimiert. Variationsresultate wurden mit denender exakten numerischen Rechnung verglichen. Ab einer bestimmten negativenStreulange kollabiert das Kondensat. Dabei tritt eine Tangenten-Bifurkation auf.Diese Bifurkation wurde sowohl numerisch als auch analytisch analysiert.

Die Physik solcher Bose-Kondensate mit einer 1/r Wechselwirkung - wobeidas Fallenpotential abgeschaltet werden kann, da sich die Kondensate durch ihreeigene

”Gravitationswechselwirkung“ selber einfangen - kann am besten durch

die Einfuhrung von”atomaren“ Einheiten verstanden werden. Diese sind die

”Rydberg“-Energie Eu und der Bohr-Radius au. Dadurch wird eine grundle-

gende Skalierungseingenschaft aufgezeigt. Es wird dargelegt, dass die Physiksolcher Kondensate nur von dem einzigen Parameter N2a/au abhangt, wobei Ndie Teilchenzahl ist und a die Streulange. Dieser Parameter definiert zwei neuephysikalische Grenzgebiete. Fur N2a/au À 1 befinden wir uns im sogenannten

”TF-G“-Gebiet, bei dem die kinetische Energie vernachlassigt werden kann. Die

”Gravitationswechselwirkung“ steht dabei im Gleichgewicht mit dem Streupoten-

tial. Fur den Fall, dass N2a/au ¿ 1, liegt das”G“-Gebiet vor, bei dem das

Streupotential weggelassen werden kann. Die kinetische Energie befindet sichmit der

”Gravitationswechselwirkung“ im Gleichgewicht.

Chapter 1

Introduction

In recent years the research on Bose-Einstein condensation (BEC) has becomewide-spread and famous both from the experimental and the theoretical point ofview. There exist many and different experiments with ultracold atoms since thefirst realization of BEC in the year 1995. The study of ultracold quantum gases,such as the superfluidity of 4He and 3He-4He mixtures, has been advanced fordecades, whereas the current research deals with condensed matter systems, in-cluding spin-polarized hydrogen, laser-cooled atoms, excitons, subatomic matterand high-temperature superconductors [Griffin et al. (1995)]. The present thesisanalyzes a new phenomenon of ultracold quantum gases that was proposed in arecent publication.

FIG. 1.1: Triads of laser beams cross the condensate.

O’Dell suggest [O’Dell et al. (2000)] that an inverse square attractive force

1

2 Chapter 1. Introduction

can be induced between ultracold quantum gases using extremely off-resonant1

electromagnetic fields. One could use this effect to perform various experiments,that simulate a gravity-like interaction between large numbers of particles.

Condensates are samples of ultra-cold atoms that occupy the same quantummechanical state2 and therefore show remarkable properties. If these condensatesare irradiated by intense, off-resonant laser beams, a long-range attractive forcebetween each and every other atom can be induced. This is shown in FIG.1.1. The potential, that is proportional to 1/r, can simulate a form of gravitybetween quantum particles, the same as that of a Bose star. The strength ofthis artificial induced “gravity” is much greater than the genuine gravitationalattraction between the particles. As we elaborate in chapter 2.1, a static electricfield induces a force between atoms that decays with the inverse fourth powerof their separation. But in an oscillating field of a laser beam there exists anadditional force that decays as the inverse square. The sum of these forces iseither attractive or repulsive for a given pair of atoms, depending on the anglesbetween the atoms and the electric field. By shining a whole set of lasers onthe condensate from many directions, one can average over all angles, to obtaina net attraction that is purely inverse square. We will see that Bose-Einsteincondensates subjected to this artificial gravity-like interaction are self-bound,i.e. remain bound together under their own gravity, with the external trappingfield turned off [Anglin (2000)]. The cases of self-trapping of the condensateare physically best understood by introducing appropriate “atomic” units. Twonew physical regions, the “TF-G” and “G” region, with unique scaling propertiesemerge where the BEC is self-bound even if the trap potential is switched off.The physics of self-trapping depends only on the parameter N2a/au, where N isthe particle number, a the scattering length and au the “Bohr” radius.

The mathematical treatment of the many-body problem leads to the extendedGross-Pitaevskii (GP) equation that is a simple nonrelativistic asymptotic exam-ple to analyze such condensates within the mean-field approximation (MFA). TheGP equation is a nonlinear Schrodinger3 equation so that the principle of super-position cannot be used. A superposition of stationary eigenstates is in general nomore stationary. Moreover there exists no exact quantum mechanical research onthis equation with the additional gravity-like potential in the literature. The aimof this work is to solve numerically the radial GP equation for different number ofatoms, to calculate energies, sizes and peak densities and to compare them withvariational calculations in the asymptotic cases of the two new physical regionsfor Bose condensates with attractive gravity-like interaction.

Chapter 2 describes the theoretical basics of the present thesis, and the GP

1The resonant frequency of the electromagnetic fields should not agree with the resonantfrequency of the condensates, otherwise the condensates would become excited, which one wantsto avoid.

2as long as the condition is fulfilled that they are bosons, see chapter 2.2.3Schrodinger (1887-1961) was an Austrian theoretical physicist.

3

equation with electromagnetically induced 1/r-interaction is derived. Chapter 3discusses the properties of the GP equation, and the new scaling property arisingfrom the introduction of “atomic”units is presented. An analytical solution of theGP equation in the “TF-G” region has also been obtained. Chapter 4 describesa variational approach to solve the Schrodinger equation, where the energies,sizes and peak densities depend on the parameter k that in term again dependsonly on the parameters N and a/au if the trap is switched off. The resultsare compared with different approaches. Chapter 5 presents the variational andnumerical solutions and calculations. Chapter 6 gives a look beyond the mean-field theory and chapter 7 finally summarizes all the results that this thesis hasexplored.

The diploma thesis will make use of the following conventions:

• All formulas are in S.I. units.

• Vectors −→r will be presented bold r.

• Angular brackets denote the expectation value of an observable, e.g. theexpectation value of the number density operator

∫d3r ψ(r)†ρ(r)ψ(r) is

denoted by 〈ρ(r)〉.• The nth partial derivative with respect to the variable x will be represented

by ∂nx .

• ψN denotes that the wave function is normalized to the particle number N ,whereas ψ without subscript denotes that the wave function is normalizedto unity.

• ∫R3 d

3r =∫ ∫

dr dΩ r2 = 4π∫dr r2 because of the use of spherical coor-

dinates [Bronstein et al. (2001)], where∫dΩ = 4π is the integral over the

solid angle dΩ.

4 Chapter 1. Introduction

Chapter 2

Theoretical basics on BEC

The following sections contain all the theories and methods, that are used inthis thesis. At the beginning we will see how the gravity-like attraction betweenneutral atoms arises. After the cause of the 1/r potential has been explained, webegin with the postulate of the indistinguishability of identical particles that leadsus finally to quantum statistics. We give a review of Bose-Einstein condensationin atomic gases, where we discuss the ideal Bose gas and the weakly-interactingBose gas. It follows a quantum mechanical treatment of many-body systems andwe derive the extended Gross-Pitaevskii (GP) equation, which we need for themathematical analysis.

2.1 The cause of the gravity-like attraction be-

tween neutral atoms

There always exists a binding attraction between atoms, which can be only ob-served, however, when other interactions are not operating, viz. when the atomsare neutral. The physical cause of this attraction are charge fluctuations in theatoms by the zero-point motion. The atoms induce dipole moments in each other,and the induced moments cause an attractive interaction between the atoms.This is the so-called Van der Waals-London interaction or dipole-dipole interac-tion, that is attractive in the near-zone (kr ¿ 1) and depends by the minus sixthpower on the separation of the atoms1. Such dipole moments can also be in-duced by external electromagnetic radiation with intensity I, wave vector k, andpolarization e. By means of quantum electrodynamics, which is a satisfactoryframework for studying effects including retardation, and within fourth order per-turbation theory [Thirunamachandran and Craig (1984)], the interaction energybetween two neutral atoms, or equivalently, between two non-polar molecules is

1Only if retarded effects are neglected, otherwise it is modified to 1/r7 [Casimir and Polder(1948)]. The potential of the Van der Waals-London interaction is called Lennard-Jones poten-tial.

5

6 Chapter 2. Theoretical basics on BEC

found to be in terms of Cartesian components i, j

U(r) =

(I

4πcε20

)α2(k)

∑i,j

e∗i ejVij(k, r) cos(kr). (2.1)

Here is r the interatomic axis, α(k) the isotropic, dynamic, polarizability of theatoms at frequency ck and Vij the retarded dipole-dipole interaction tensor

Vij(k, r) =1

r3

[(δij − 3rirj) (cos(kr) + kr sin(kr))− (δij − rirj) k

2r2 cos(kr)]

(2.2)where ri = ri

r. The influence of retardation2 on the energy of interaction U(r) be-

tween two neutral atoms was analyzed by Casimir and Polder [Casimir and Polder(1948)]. As Thirunamachandran pointed out [Thirunamachandran (1980)] , thereexist several cases where the interaction energy can be analyzed. The incidentbeam can be linearly or circularly polarized, the direction of propagation of thebeam can be parallel or orthogonal to r, viz. the polarization direction can beorthogonal or parallel to r, respectively, and the orientation of the molecular paircan be kept fixed or randomly oriented with respect to the direction of propa-gation of the incident radiation. If the molecular pair is fixed with respect tothe incident radiation direction then we obtain the well known 1/r3 variationof the interaction energy at near-zone separation in the limit kr ¿ 1. But forgaseous and liquid systems the dipoles have to be randomly orientated; that is,all directions of r are equally probable. From this it follows that when an averageover all orientations of the interatomic axis with respect to the incident radiationdirection is taken, we obtain a 1/r law for the interaction energy. The instanta-neous, nonretarded part 1/r3 (δij − 3rirj) in (2.2) vanishes and we obtain, in thenear zone, the weaker attractive 1/r potential instead of the 1/r3 part. One canrealize the suppression of the 1/r3 interaction in an experiment by considering aspatial configuration of external fields. This leads us to the near-zone potential

U(r) = −3Ik2α2

16πcε20

1

r

[7

3+ (sin θ cosφ)4 + (sin θ sinφ)4 + (cos θ)4

], (2.3)

where we take advantage of Taylor expansions of Eqs. (2.1) and (2.2) in powersof kr and using the identity

e∗(±)i (k)e

(±)j (k) =

1

2

[(δij − kikj

)± iεijkkk

], (2.4)

where + and − correspond to left and right circular polarizations, respectively[O’Dell et al. (2000)]. The interaction is attractive for any orientation (θ, φ) of rrelative to the beams. If we want the purely 1/r potential, the angular anisotropy

2Retardation effects allow for the finite speed of propagation of electromagnetic influences.

2.1. The cause of the gravity-like attraction between neutral atoms 7

in (2.3) can be canceled out by introducing certain Euler angles (α, β, γ). Theseangles define the different orientations of the laser beams. One configurationthat cancels the anisotropy is using 18 laser beams, as mentioned by O’Dell[O’Dell et al. (2000)], rotated through the following Euler angles: (0, π/4, π/8),(0, π/4,−π/8), (0, π/4, 3π/8), (0, π/4,−3π/8), (0, 0, π/8), (0, 0,−π/8). Undersuch angles the gravity-like interatomic potential becomes

U(r) = − 11

4π

Ik2α2

cε20

1

r= −u

r, (2.5)

where r is the distance between the atoms3 and with the gravitational-like cou-pling u

u = − 11

4π

Ik2α2

cε20

. (2.6)

The energy of the gravity-like interaction between the pair of particles i, j withinteratomic distance r = |ri − rj| is then given by

U(ri, rj) = − u

|ri − rj| (2.7)

or equivalent by

U(r, r′) = − u

|r − r′| ≡ Wu(r, r′). (2.8)

The suppression of the 1/r3 interaction is achievable with lasers in the far infrared,or with a microwave source that satisfies the near-zone condition, like e.g. aCO2 laser with k = 2π

10.6 µmand intensity of, e.g. I = 108 W

cm2 . With a static

polarizability of, e.g. α = 24.08 · 10−24 cm3 for sodium atoms, we obtain theenergy of

U(100 nm) = −2 · 10−15 eV, (2.9)

at r = 100 nm, the mean separation in a typical BEC. This energy is comparablewith the Van der Waals-London energy at this distance. The distinction lies inthe interaction between atoms. The 1/r potential acts over the entire sample,whereas the dipole-dipole interaction is acting only for nearest neighbors. Fromthis follows that one will observe the transition from external trapping to self-binding in the future, when experiments will be achieved.

3If u = Gm2 and a = 0 (that is the s-wave scattering length, see below), where G =6.673 · 10−11Nm2kg−2 [Misner et al. (1973)] is the gravitational constant and m the mass ofthe atoms, we obtain the Newton-Schrodinger equation, which has been extensively discussedin the literature [Diosi (1987); Jones (1995); Moroz et al. (1998); Tod and Moroz (1999); Soni(2002); Harrison et al. (2003); Greiner (2005); Knapp (2005); Greiner and Wunner (2006)].


2.2 Identity of particles

As is well known, the classical treatment of a many-body system uses the Boltz-mann statistics. This is based on the assumption, that the particles are distin-guishable. However, quantum mechanical particles are indistinguishable. The’correct’ statistics has to include this fact. In quantum mechanics we need eventwo statistics – one for fermions and one for bosons.

2.2.1 The indistinguishability of identical particles

The principle of the indistinguishability of identical particles or the symmetrypostulate for identical particles, as it is called, plays a fundamental role in thequantum mechanical investigation of systems composed of identical particles. Ifa system consists of identical particles4, there can be only certain vectors of itsstate space that describe its physical states. The physical vectors will thus beeither symmetrical or antisymmetrical when two particles are exchanged depend-ing on the nature of the particles. The particles are called bosons if the physicalvectors are symmetric and fermions, if they are antisymmetric. The principleof the indistinguishability of identical particles thus restricts the state space ofone system of identical particles. It is not, as for distinguishable particles, thedirect product space H of the state spaces of the individual particles. In fact,it is composed of a subspace of H, HS or HA, depending on whether the par-ticles are bosons or fermions, where the subscripts S and A denote symmetricand antisymmetric spaces [Cohen-Tannoudji et al. (1999)]. This postulate dividesthe existing particles in nature in two categories. All known particles satisfy thefollowing empirical rule5: Particles with half-integer spin are fermions (electrons,protons, neutrons, etc.) and those with integer spin are bosons (photons, mesons,etc.).

The quantum statistics must take the principle of the indistinguishability intoaccount. According to Pauli’s principle, systems with identical bosons and sys-tems with identical fermions must be treated differently. In a system consisting ofidentical fermions, two or more particles cannot be in the same state at the sametime. This leads to different statistical properties: Bosons, described by symmet-rical wave functions, obey the Bose-Einstein statistics, while fermions, describedby antisymmetrical wave functions, obey the Fermi-Dirac statistics. The physicalproperties of systems with identical fermions and identical bosons, respectively,are very different; this can be observed e.g. at low temperatures. For identi-cal bosons, it is feasible to concentrate them in the single state of the smallestpossible energy i.e. to share the same quantum state (this phenomenon is called

4Identical particles are said to be particles that agree in all their properties, i.e. mass, spin,charge, volume, moment of inertia, magnetic moment.

5By means of the spin-statistics theorem, that is verified in Quantum Field Theory [Pauli(1940)], one can understand this rule as a result of very abstract assumptions.

2.3. Bose-Einstein condensation in atomic gases 9

Bose-Einstein condensation), whereas fermions are subjected to the restrictionsof Pauli’s principle. In this thesis we deal solely with BEC, i.e. we only considerbosons. Fermions have been mentioned just for the sake of completeness.

2.3 Bose-Einstein condensation in atomic gases

In contrast to fermions all identical bosons of one system fall, at absolute zerotemperature, in the same quantum state, the ground state. In this state allparticles are absolutely indistinguishable and build a form of “super-particle“.Not only elementary particles, but also composite particles as atomic nuclei oratoms behave as fermions or bosons, depending on whether their wave functionis antisymmetrical or symmetrical. For this reason Einstein and Bose predicted1924 that bosonic atoms in gas phase at very low temperatures fall in a collectiveground state and thus build a new form of matter.

The first experimental evidence of Bose-Einstein condensation was accom-plished by Eric A. Cornell and Carl E. Wieman at the University of Colorado inBoulder [Anderson et al. (1995)] in 1995. FIG. 2.1 shows one of the first picturesof the atomic clouds of rubidium. They cooled down vapors of rubidium to ex-tremely low temperatures and observed the velocity distribution of the atoms. Itfollowed a series of experiments [Davis et al. (1995)]. The atoms were allowed toexpand by switching off the confining trap and then imaged with optical meth-ods. A sharp peak in the velocity distribution was then observed below a certaincritical temperature, depending on the particle density.

2.3.1 The ideal Bose gas

First of all we neglect the atom-atom interaction, as it is performed in [Dalfovoet al. (1999)], so that we can introduce analytically the Bose-Einstein condensatewave function. The condensate is trapped in a harmonic potential of the form

Vext(r) =1

2mω2r2 =

1

2m

(ω2xx

2 + ω2yy

2 + ω2zz

2). (2.10)

The solution of the Schrodinger equation for the many-body Hamiltonian, whichis the sum of single-particle Hamiltonians, obtained by putting all the N nonin-teracting bosons in the lowest single-particle state (nx = ny = nz = 0), viz.

Ψ(r1, . . . , rN) =N∏i=1

ψ0(ri), (2.11)

results in the well known harmonic oscillator energy eigenvalues

εnxnynz =

(nx +

1

2

)~ωx +

(ny +

1

2

)~ωy +

(nz +

1

2

)~ωz, (2.12)


FIG. 2.1: The velocity distribution of rubidium atoms in the experiment by[Anderson et al. (1995)] is shown. The left frame represents a gas at a temperaturejust above condensation at about 400 nK; the centre frame describes a gas justafter the appearance of the condensate at about 200 nK; in the right frame, afterfurther evaporation sample of nearly pure condensate at about 50 nK is released.From [Cornell (1996)].

where nx, ny, nz are non-negative integers, and the ground state wave function

ψ0(r) =(mωhoπ~

) 34exp

[−m

2~(ωxx

2 + ωyy2 + ωzz

2)], (2.13)

with the geometric mean of the trapping frequencies

ωho = (ωxωyωz)13 . (2.14)

For the ideal gas case this is exactly the Bose-Einstein condensate wave function6,which is occupied with a large number of particles. The density distribution thenbecomes

ρ(r) = N |ψ0(r)|2 (2.15)

and its value grows with N .

6In chapter 4 we use ”Slater Type Orbitals“ (STO) and ”Gaussian Type Orbitals“ (GTO)for an analytical approach.

2.4. Extended Gross-Pitaevskii equation 11

2.3.2 The weakly-interacting Bose gas

Now we add interactions between the particles to introduce the main conceptsfor the treatment of the condensate wave function of a real gas. We discussdilute Bose-Einstein condensates in an approximation where the temperaturegoes to zero, viz. T → 0. The formalism introduced is strictly valid in thislimit of zero temperature, when all particles are in the condensate. The dynamicbehavior and the generalization to finite temperatures are discussed at greatlength in [Pitaevskii and Stringari (2003); Dalfovo et al. (1999)]. Furthermoremean-field theory will be our framework, in which the Hartree-Fock equations forthis analysis will be derived. This equations are solved numerically by means ofa fourth-order Runge-Kutta routine with adaptive step-size control in chapter 5.

2.4 Extended Gross-Pitaevskii equation

In condensates the atoms interact with each other at very short distance. Allproperties of these dilute Bose gases can be made understandable, if we take onlytwo-body collisions into account, that are characterized by the s-wave scatteringlength a. First however we summarize the general basics of a system consisting ofN bosons. In quantum mechanics bosons are well-defined by the symmetry of theN -particle wave function Ψ(r1, r2, . . . , ri, . . . , rj, . . . , rN) under permutations oftwo particles (i ↔ j). The wave function Ψ describes a system of N bosonsinteracting via an effective potential W (ri, rj), an external harmonic potential,in which the particles are being confined,

Vext(r) =1

2mω2r2, (2.16)

where m is the atomic mass and ω = ωx = ωy = ωz the oscillator frequency andthe kinetic energy

T (r) =p2

2m= − ~

2

2m∆, (2.17)

with Planck’s constant ~ = 1.0545 · 10−34Js [Schwabl (2002)] and the Laplacian7

∆, that solves the time-dependent Schrodinger equation

i~∂Ψ

∂t= HΨ (2.18)

with the many-body Hamiltonian

H =N∑i=1

[T (ri) + Vext(ri)

]+

∑i<j

W (ri, rj). (2.19)

7The divergence of the gradient of a scalar function is called the Laplacian. The Laplacianfor a scalar function is a scalar differential operator. In subsection 3.1.1 we introduce theLaplacian in spherical coordinates.


The summation rule i < j prevents the interaction energy between pairs of bosonsbeing counted twice. If instead of using this rule, we sum over all indices i andj, with the limitation that i 6= j, we must set a factor of 1/2 in front of theinteraction sum and obtain

H =N∑i=1

[T (ri) + Vext(ri)

]+

1

2

N∑i,j=1i6=j

W (ri, rj). (2.20)

The Hamiltonian H measures the energy of the atoms, where the second sum-mand in (2.20) leads us to nonlinearity.

Quantum field theory formulation

In the following we follow [Dalfovo et al. (1999)] by changing into the states of theoccupation number representation (into the Fock space) of the Heisenberg picture,instead of the treatment of the many-body wave function Ψ(r1, r2, . . . , rN). Thisis usually connected with the method of second quantization [Schwabl (2003);Nolting (2004c)] and the whole treatment is often more handy. In second quan-tization (2.20) is transformed into

H =

∫dr Ψ†(r)

[T (r) + Vext(r)

]Ψ(r)

+1

2

∫dr dr′ Ψ†(r)Ψ†(r′)W (r − r′)Ψ(r′)Ψ(r),

(2.21)

where Ψ(r) and Ψ†(r) are the boson field operators that annihilate and createa particle at the position r, respectively. We introduce the bosonic creation andannihilation operators a†k and ak, where the field operator can be rewritten as

Ψ(r) =∑

k Ψk(r)ak, where Ψk(r) are single-particle wave functions and thatobey the commutation rules

[ak, a†l ] = δkl, [ak, al] = [a†k, a

†l ] = 0. (2.22)

In Fock space they are defined through the relations

a†k |n0n1, . . . , nk, . . .〉 =√nk + 1 |n0n1, . . . , nk + 1, . . .〉 , (2.23)

ak |n0n1, . . . , nk, . . .〉 =√nk |n0n1, . . . , nk − 1, . . .〉 (2.24)

where nk are the eigenvalues of the operator nk = a†kak giving the number ofatoms in the single-particle k state. Now it is very difficult or even impossibleto solve the Schrodinger equation for large numbers of N . So we have to makeuse of mean-field theory. Mean-field approaches belong to the standard methodsof quantum mechanics to describe interacting Bose-condensed gases. One simple


approach is the Hartree-Fock approximation, where the wave function is assumedto take the form of a product of one-particle wave functions. The many-bodyproblem is reduced to the one-body problem by considering each particle on itsown. It starts with a many-body wave function Ψ consisting of a product ofsingle-particle wave functions that are not equal,

Ψ(r1, . . . , rN) =N∏i=1

ψi(ri), (2.25)

where we assume that the single-particle wave functions are normalized8 to unity∫d3r |ψi(ri)|2 = 1. (2.26)

Because of the symmetry postulate for identical particles we have to symmetrizethe right-hand side of (2.25) with respect to the particle labels. Now we simplifythis product ansatz for the ground state of the many-boson system assuming allparticles occupy the same single-particle state, ψ1 = ψ2 = ... = ψN = ψ, sothe symmetry requirement is fulfilled automatically [Friedrich (2005)]. We thenobtain

Ψ(r1, . . . , rN) =N∏i=1

ψ(ri). (2.27)

The case of identical one-particle functions ψ1 = ψ2 = ... = ψN = ψ leads us toone Hartree-Fock equation for the wave function ψ, that is derived below.

2.4.1 Variational procedure

First the Hartree-Fock equation for the BEC is derived by minimizing the expec-tation value of the Hamiltonian (2.20) with respect to variations of the single-particle wave functions9. This leads us to an equation for ψ as is pointed out by[Reidl (2002)]. Thereafter we just use the boson field operators Ψ†(r) and Ψ(r)with the commutation rules

[Ψ(r), Ψ†(r)] = δ(r − r′), [Ψ(r), Ψ(r′)] = [Ψ†(r), Ψ†(r′)] = 0 (2.28)

when we have continuous states, e.g. the position eigenstates |r1, r2, . . .〉. Thefield operators satisfy the Heisenberg equation with the many-body Hamiltonian(2.21):

i~∂

∂tΨ(r, t) = [Ψ, H]

=

[T (r) + Vext(r) +

∫dr′ Ψ†(r′, t)W (r′, r, t)Ψ(r′, t)

]Ψ(r, t).

(2.29)

8Functions that satisfy the normalization condition are called square-integrable. Any phys-ical wave function must be square-integrable.

9For more details on the variational principle see section 4.1.


We know that in a BEC all particles are located in the same quantum state. Itis a kind of a macroscopically occupied state, that is typically the one-particleground state |ψ〉 = a†0 |0〉 of the interacting system:

|Ψ〉 =1√N !

(a†0

)N|0〉 (N À 1), (2.30)

where |0〉 is the vacuum, representing the state containing zero particles.In the case of weak interacting Bose-gases the many-body state functionΨ(r1, r2, . . . , rN) factoring approximately (2.27) and for the ground state func-tion ψ(r) of the interacting system the energy

⟨Ψ | H | Ψ

⟩(2.31)

of the Hamiltonian (2.20) takes the minimal value. With the condition 〈ψ | ψ〉 = 1and the Lagrange parameter defining the chemical potential µ as it is shown in[Nolting (2004c)], the variation of ψ(r) at the minimum of (2.31) leads to theequation

[T (r) + Vext(r) + (N − 1)

∫d3r′ |ψ(r′)|2W (r′, r)

]ψ(r) = µψ(r), (2.32)

with|ψ(r′)|2 = ψ(r′)ψ(r′)∗, (2.33)

where ψ∗(r′) is the complex conjugate10 of the wave function ψ(r′). Thus thewave function is complex, with real and imaginary parts. As from now we omitthe imaginary parts so that we just only keep the real parts of the wave function.The chemical potential is a measure for the occupation of the ground state anda basic parameter for Bose-Einstein condensates. It is interpreted as the groundstate orbital energy in the Hartree-Fock approach. In the thermodynamical limitN → ∞, the prefactor (N − 1) of (2.32) can be replaced with N , so that weeventually obtain the Hartree-Fock equation

[T (r) + Vext(r) +N

∫d3r′ |ψ(r′)|2W (r′, r)

]ψ(r) = µψ(r) (2.34)

with the Hartree-Fock potential

Veff (r) =

∫d3r′ |ψ(r′)|2W (r′, r) , (2.35)

that is the mean-field contribution of the condensate wave function ψ(r).

10The complex conjugate is given by changing the sign of the imaginary part of a complexfunction.


2.4.2 Bogoliubov approximation

The Hartree-Fock equation (2.34) can also be obtained by using a Bogoliubov11

approximation [Dalfovo et al. (1999)]. He formulated his basic idea for a mean-field description of a dilute Bose gas in 1947. The basic approximation of theSchrodinger equation is to replace the bosonic field operators in lowest orderwith their expectation values

Ψ(r, t) ≡ 〈Ψ(r, t)〉, (2.36)

with〈Ψ(r, t)〉 ≡ ψ(r, t). (2.37)

The function ψ(r, t) is a classical field and represents the order parameter ofBEC theory. It is often characterized as the wave function of the condensate.The density contribution of the condensate is fixed by

ρ(r) = N |ψ(r, t)|2. (2.38)

Now we can derive the equation for the condensate wave function ψ(r, t) via thetime evolution of the field operator Ψ(r, t). We consider the Heisenberg equation(2.29) and replace the field operator Ψ(r, t) with the classical field ψ(r, t). In adilute and cold gas, only binary collisions between particles are relevant and thesecollisions are characterized by a single parameter, the s-wave scattering length12

a, independently of the details of the two-body potential Wc(r′, r)13, arising from

the binary collisions between the particles. This allows us [Fermi (1936)] forlow-energy scattering to replace the precise interatomic potential Wc (r

′, r) witha shape-independent approximation, also called the pseudopotential, which usesa zero-range potential, as opposed to an extended potential with a well-definedshape [McKinney et al. (2004)]. Hence the two-body pseudopotential has theform

Wc(r′, r) = gδ(3)(r − r′), (2.39)

where the coupling constant g is related to the scattering length a through

g =4π~2a

m. (2.40)

If we now take into account that the condensate is irradiated by electromagneticfields, viz. lasers, as described in chapter 2.1, we can just add the gravity-likepotential to the contact potential (2.39),

Wu(r′, r) = − u

|r − r′| . (2.41)

11Bogoliubov (1909-1992) was a Russian-Ukrainian mathematician and theoretical physicist.12Typical values for, a, lie in the range of a few nm.13The subscript c denotes that this two-body potential is named contact potential.


The effective potential in (2.29) then reads

W (r′, r) = Wc(r′, r) +Wu(r

′, r) = gδ(3)(r − r′)− u

|r − r′| . (2.42)

The use of the effective potential (2.42) in (2.29) is compatible with the replace-ment of Ψ with ψ and leads us to the central equation that describes Bose-Einsteincondensates of interacting particles with electromagnetically induced ”gravity“:

[T (r) + Vext (r) +N

∫d3r′ |ψ(r′)|2

(gδ(3)(r − r′)− u

|r − r′|)]

ψ(r)

= i~∂

∂tψ(r, t). (2.43)

The time independent version of (2.43) is obtained using the ansatz ψ(r, t) =ψ(r) exp(−iµt/~):


∫d3r′ |ψ(r′)|2

(gδ(3)(r − r′)− u

|r − r′|)]

ψ(r)

=


(g|ψ(r)|2 − u

∫d3r′

|ψ(r′)|2|r − r′|

)]ψ(r) = µψ(r), (2.44)

with the effective self-binding potential (2.35)

Veff (r) = Vc(r) + Vu(r) = g|ψ(r)|2 − u

∫d3r′

|ψ(r′)|2|r − r′| , (2.45)

the mean-field Hamiltonian

Hmf = T (r) + Vext (r) +N

(g|ψ(r)|2 − u

∫d3r′

|ψ(r′)|2|r − r′|

), (2.46)

and the normalization condition∫d3r |ψ(r)|2 !

= 1. (2.47)

Equation (2.44) is the extended Gross-Pitaevskii (GP) equation. The originalGP equation was derived only for the contact interaction independently by Gross(1961, 1963) and Pitaevskii (1961) [Pitaevskii and Stringari (2003)]. The GPequation (2.44), which is a nonlinear Schrodinger equation, can also be formulatedfor the renormalized wave function

ψN(r) =√Nψ(r), (2.48)

where ψN is the wave function of the condensate in the mean-field approach andψ the one-particle ground state wave function. This version reads

[T (r) + Vext (r) + g|ψN(r)|2 − u

∫d3r′

|ψN(r′)|2|r − r′|

]ψN(r) = µψN(r), (2.49)

2.5. Quantum statistics - The ideal Bose gas in the grand canonical ensemble 17

with the normalization condition∫d3r |ψN(r)|2 = N. (2.50)

What are now the conditions of applicability of (2.49)? First, the total number ofatoms N should be large enough, so that we can use the concept of Bose-Einsteincondensation. Second, the diluteness of the condensate must be satisfied in orderto replace the field operator with the classical field and third, the temperature ofthe sample must be sufficiently low.

Limits of the GP equation

The GP equation presents an excellent qualitative and quantitative description ofBose-Einstein condensates. But there still arise some deviations in the followingexamples [Weidemuller and Zimmermann (2003)]:

1. Through the dynamics of the system there occur deviations due to quantumcorrections of the order of a few percent.

2. Finite temperature effects are not treated by the GP equation but must beadded by some extensions of this theory.

3. The GP equation is only valid in the mean-field approach. In strongly cor-related systems, e.g. BEC in strong periodic potentials [Orzel et al. (2001)]and [Greiner et al. (2002)] one must go beyond mean-field theory to describephysics correctly.

2.5 Quantum statistics - The ideal Bose gas in

the grand canonical ensemble

Quantum statistics of Bose gases are described in [Heer (1972); Trebin (1991);Friedrich (2005)]. Here we give a short review. At first we consider the Bosegas at T = 0. For that we do not need quantum statistics. We would obtain the

same result, if we used the Maxwell-Boltzmann statistics NMBk (εk) = exp

(µ−εk

kBT

),

where kB is Boltzmann’s constant. But the Bose statistics leads already below acritical temperature Tc to a macroscopically occupied ground state. Most easilywe can demonstrate that fact for the ideal, viz. noninteracting, gas in the grandcanonical ensemble. There, the partition function Z expresses the probabilities,that characterize all possible realizations of the many body state, and are deter-mined by the temperature T , the chemical potential µ and the volume14 V . A

14The volume dependency of the partition function is hidden in the one-particle energies εi.


single-particle state can be occupied arbitrarily by bosons. The number of parti-cles is not subjected to restrictions, as for fermions. For this reason, we can sumover all particle numbers in the grand canonical partition function. We obtain ageometric series, and the product of the several sums gives the grand canonicalpartition function

ZBE(T, V, µ) =∞∏i=1

( ∞∑n=0

(exp

(µ− εikBT

))n)

=∞∏i=1

1

1− exp(µ−εi

kBT

) , (2.51)

for bosons, where i indicates the quantum mechanical one-particle state. The ν-particle state is characterized by the occupation number n

(ν)i . For instance FIG.

2.2 shows a 7-particle state:

i = 2

i = 1

i = 4

i = 5

i = 6

n

n

n

n

4

5

6

2(6)

(6)

(6)

(6)

i = 3 n3

= 2

= 0

= 2

= 1

= 1

n1(6) = 1

(6)

ε

ε

ε

ε

ε

ε6

5

4

3

2

1

E

= 7ν

FIG. 2.2: A 7-particle state.

If we now want to know the occupation of a particular state k with the energyεk, we obtain it by direct application of (2.51)15:

NBEk = −kBT ∂

∂εkln ZBE =

exp(µ−εk

kBT

)

1− exp(µ−εk

kBT

) (2.52)

or assigned

NBEk =

1

exp(εk−µkBT

)− 1

. (2.53)

(2.53) is called the Bose-Einstein statistics. In this case the chemical potentialhas to be smaller than the lowest single particle energy, otherwise the Z-factors

15For more details see [Trebin (1991)], where you can find an exact derivation of the partitionfunction of bosons, fermions and even the Maxwell-Boltzmann limit. You will find there thederivation of formula (2.54), too.

2.5. Quantum statistics - The ideal Bose gas in the grand canonical ensemble 19

in (2.51) would diverge and NBEk would be negative. Hence we choose the energy

scale so that the ground state has the energy ε1 = 0. Thus it applies alwaysµ < 0. The occupation probability tends to infinity when εk → µ.

Now we consider the ideal Bose gas as a system of free particles of mass mmoving in a large cube. Changing from the sum to an integral, we calculate thetotal particle number by integrating over the continuous energy values ε. Withoutderivation we obtain, by using a Taylor expansion in z,

N =

∞∫

0

NBEk D(ε)dε =

g0V

4π2

(2m

~2

) 32

∞∫

0

√ε

z−1exp(

εk

kBT

)− 1

dε

=g0V

λ3dB

F 32(z), (2.54)

with the density of states

D(ε) =g0V

4π2

(2m

~2

) 32 √

ε, (2.55)

where g0 is the degree of spin degeneracy, the polylogarithmic function

F 32(z) = z +

z2

2√

2+

z3

3√

3+ · · · ≡

∞∑n=1

zn

n32

, (2.56)

the thermal wave length or de Broglie wave length

λdB ≡√

2π~2

mkBT, (2.57)

and the fugacity

z ≡ exp

(µ

kBT

). (2.58)

By inserting µ = 0 in (2.54) we define the critical temperature Tcr, by

ρ =N

V=

g0

λ3dB

∞∑n=1

1

n32

= g0

(mkBTcr2π~2

) 32∞∑n=1

1

n32

(2.59)

or rewritten

Tcr =2π~2

mkB

(ρ

g0

∑∞n=1

1

n32

) 23

. (2.60)

As the fugacity approaches unity, F 32(z) becomes the Riemann zeta function

ζ(x) =∞∑n=1

1

n(2.61)


and for the value x = 32

we obtain

ζ

(3

2

)= F 3

2(1) ≈ 2.612, (2.62)

which is the maximum value of this function. The critical temperature Tcr isreached when the number density ρ is equal to the inverse cube of the thermalwave length, except for the factor of ζ

(32

),

ρ = g0

ζ(

32

)

λ(Tcr)3≈ 2.612

λ(Tcr)3. (2.63)

At that point the occupation of the lowest state begins which means the onset ofthe Bose-Einstein condensation.

If we now reduce the temperature below Tcr, the chemical potential remainsstill zero. The number Nexc of particles in excited states is given by

Nexc

V= g0

(mkBT

2π~2

) 32∞∑n=1

1

n32

=

(T

Tcr

) 32 N

V, T ≤ Tcr. (2.64)

The number N0 of particles which occupy the non-degenerate ground state is

N0

N= 1−

(T

Tcr

) 32

, T ≤ Tcr. (2.65)

For T → 0 all particles are located into the ground state. This is the extremecase of a degenerate Bose gas.

Loosely speaking Bose-Einstein condensates represent a macroscopic quantumstate, in which all atoms swing in perfect consonance as matter wave.

Chapter 3

General features of the extendedGross-Pitaevskii equation

In this chapter we analyze the radial GP equation, benefit from the special sym-metry of the system, and rewrite it into a dimensionless form using“atomic units”.We introduce a new scaling of the system that fulfills the normalization condi-tion of the condensate wave function and consider two limiting cases of the GPequation. Finally we derive an analytical wave function for the condensate in acertain limit-region.

3.1 Several properties of the GP equation

As mentioned in the previous chapter the GP equation describes particles movingin their own gravity-like potential. That means that they are trapped even if theexternal potential Vext (r) is switched off. Hence we can omit Vext (r), so that thegravity-like potential becomes the trapping potential.

3.1.1 Symmetry leads to spherical coordinates

The extended Gross-Pitaevskii equation (2.49) in the absence of an external po-tential Vext reads

[T (r) + g|ψN(r)|2 − u

∫d3r′

|ψN(r′)|2|r − r′|

]ψN(r) = µψN(r). (3.1)

By inserting (2.17) and (2.40) we obtain[− ~

2

2m∆ +

4π~2a

m|ψN (r)|2 − u

∫d3r′

|ψN (r′) |2|r − r′|

]ψN (r) = µψN (r) , (3.2)

which can be written as

∆ψN (r) = −2m

~2

[u

∫d3r′

|ψN (r′) |2|r − r′| + µ− 4π~2a

m|ψN (r)|2

]ψN (r) . (3.3)

21

22 Chapter 3. General features of the extended Gross-Pitaevskii equation

The integro-differential equation (3.3) is equivalent to a system of two equations

∆ψN (r) = −2m

~2

[−Vu(r) + µ− 4π~2a

m|ψN (r)|2

]ψN (r) , (3.4a)

∆Vu (r) = 4πu|ψN (r) |2, (3.4b)

with the gravity-like potential

Vu (r) = −u∫d3r′

|ψN (r′) |2|r − r′| . (3.5)

(3.4b) is called the gravitational Poisson’s equation. The proof that (3.5) solves(3.4b)

Vu (r) = −u∫d3r′

|ψN (r′) |2|r − r′| ,

∆Vu (r) = −u∫d3r′ |ψN (r′) |2∆ 1

|r − r′|= 4πu

∫d3r′ |ψN (r′) |2δ(3)(r − r′)

= 4πu|ψN (r) |2

¤

makes use of the Green’s function relation [Jackson (2002)]

δ(3)(r − r′) = − 1

4π∆

1

|r − r′| . (3.6)

Now we take into account the symmetry of the effective self-binding potential(2.45) that depends only on the distance from the origin. Hence the whole prob-lem can be analyzed much easier, if we change from rectangular coordinates tospherical coordinates by regarding the special symmetry of the central potential1.Our goal is to solve the radial GP equation, so we introduce firstly the Laplacianin spherical coordinates [Nolting (2004c)]:

∆ =1

r

∂2

∂r2r +

1

r2

(∂2

∂θ2+

1

tan θ

∂

∂θ+

1

sin2 θ

∂2

∂φ2

). (3.7)

The Laplacian2 can be split into both radial and angular components

∆ =

(∂2

∂r2+

2

r

∂

∂r

)+

1

r2 sin2 θ

[sin θ

∂

∂θ

(sin θ

∂

∂θ

)+

θ2

θφ2

]= ∆r + ∆θφ. (3.8)

1A spherical symmetric potential is called central potential.2(3.7) provides the Laplacian for nonvanishing r and is not defined at the origin r = 0, which

plays a special role in spherical coordinates.

3.2. Scaling behavior 23

If we now omit the angular components, we obtain the radial Laplacian

∆r =

(∂2

∂r2+

2

r

∂

∂r

)=

1

r2∂r

(r2∂r

). (3.9)

We finally arrive at the radial GP equation in integro-differential form[− ~

2

2m∆r +

4π~2a

m|ψN (r)|2 − u

∫d3r′

|ψN (r′) |2|r − r′|

]ψN (r) = µψN (r) , (3.10)

or equivalently the radial GP system

∆rψN (r) = −2m

~2

[−Vu + µ− 4π~2a

m|ψN (r)|2

]ψN (r) , (3.11a)

∆rVu (r) = 4πu|ψN (r) |2. (3.11b)

We use the radial integro-differential form for our analytical and the radial GPsystem for our numerical calculations. The GP equation that is normalized tounity reads

[− ~

2

2m∆r +N

(4π~2a

m|ψ(r)|2 − u

∫d3r′

|ψ(r′)|2|r − r′|

)]ψ(r) = µψ(r). (3.12)

3.1.2 Invariance of the GP system

Under the transformation

(ψN (r) , Vu (r)) 7→ (−ψN (r) , Vu (r)) (3.13)

the system (3.11) remains invariant. This means that if we have a solution(ψN (r) , Vu (r)) we get automatically another solution by (−ψN (r) , Vu (r)). Thiscan be traced back to the fact that simply the square of the wave function|ψN (r) |2 has a physical meaning as a probability distribution to find a parti-cle at position r. Therefore we just assume that

ψN(0) ≥ 0. (3.14)

3.2 Scaling behavior

The GP equation has a remarkable scaling behaviour that allows for the transi-tion from non-normalized to normalized solutions. To analyze this behavior weintroduce new variables for ψ and r. The equation remains invariant in the newvariables. We define

ψN = λψN (3.15a)

r = νr (3.15b)


and insert (3.15b) in (3.9) to obtain the Laplacian in the scaled variable r

∆r =

(∂2

∂ (νr)2 +2

νr

∂

∂νr

)=

1

ν2

(∂2

∂r2+

2

r

∂

∂r

)=

1

ν2∆r. (3.16)

Inserting (3.15) and (3.16) in (3.10) the equation reads:

[− 1

ν2

~2

2m∆r +

4π~2a

mλ2|ψN (r) |2 − u

∫ν3d3r′

λ2|ψN (r′) |2ν|r − r′|

]λψN (r)

= µλψN (r) .

(3.17)

Finally we multiply (3.17) with ν2, divide by λ, and obtain the GP equation

[− ~

2

2m∆r +

4π~2a

mλ2ν2|ψN (r) |2 − u

∫d3r′

λ2ν4|ψN (r′) |2|r − r′|

]ψN (r) = µψN (r) ,

(3.18)with the abbreviation

µ = µν2. (3.19)

Now we require that (3.18) must be equivalent to (3.10). From this follows that

λ2ν4 = 1 (3.20)

viz.

λ =1

ν2(3.21)

and

λ2ν2 =1

ν2. (3.22)

Altogether we can see by inserting (3.21) in (3.15a) and (3.20) and (3.22) in (3.18)that the quantities

ψN = ν2ψN , (3.23a)

r =r

ν, (3.23b)

µ = ν2µ, (3.23c)

a =a

ν2, (3.23d)

fulfill the following rescaled GP equation

[− ~

2

2m∆r +

4π~2a

m|ψN (r) |2 − u

∫d3r′

|ψN (r′) |2|r − r′|

]ψN (r) = µψN (r) , (3.24)

3.2. Scaling behavior 25

with the normalization condition (2.50)

∫d3r |ψN (r) |2 = ν

∫d3r |ψN (r) |2 !

= N, (3.25)

where ν is the normalization factor. The proof that (3.23) let (3.10) invariant isthe following:

[− ~

2

2m∆r +

4π~2a

m|ψN (r)|2 − u

∫ |ψN (r′) |2|r − r′| d3r′

]ψN (r) = µψN (r) ,

[− ~

2

2m

1

ν2∆r +

4π~2

mν2a

1

ν4|ψN (r) |2 − u

∫ 1ν4 |ψN (r′) |2ν|r − r′| ν3d3r′

]1

ν2ψN (r)

=µ

ν2

1

ν2ψN (r) ,

[− ~

2

2m

1

ν2∆r +

4π~2

mν2a

1

ν4|ψN (r) |2 − u

∫ 1ν4 |ψN (r′) |2ν|r − r′| ν3d3r′

]ψN (r)

=µ

ν2ψN (r) ,

[− ~

2

2m∆r +

4π~2

ma|ψN (r) |2 − u

∫ |ψN (r′) |2|r − r′| d3r′

]ψN (r) = µψN (r) .

¤

From the normalization condition follows

ν =N∫

d3r |ψN (r) |2 ≡N

‖ ψN ‖2, (3.26)

and by inserting (3.26) in (3.23) we obtain the following scaling behavior

ψN =N2

‖ ψN ‖4ψN , (3.27a)

r =‖ ψN ‖2

Nr, (3.27b)

µ =N2

‖ ψN ‖4, µ (3.27c)

a =‖ ψN ‖4

N2a. (3.27d)

The scaling required for the scale invariance is

(ψN , r, µ, a) → (ν2ψN ,r

ν, ν2µ,

a

ν2) =: (ψN , r, µ, a). (3.28)


3.3 “Atomic” units

It is common in the literature [O’Dell et al. (2000)] to use as natural units forBEC systems the quantum energy ~ω and the oscillator width aosc belongingto the harmonic trap potential. In the case of self-trapping we see that thesebecome “bad” units because the trapping potential is switched off, viz. ~ω → 0and aosc = ~

mω0→∞. Therefore it is more adequate to choose suitable “atomic”

units, so that the extended Gross-Pitaevskii equation (3.12) that is normalizedto unity can be made dimensionless. We introduce the following quantities

αf =u

~c, (3.29)

au =~

mcαf=λCαf

=~2

um, (3.30)

Eu =1

2α2fmc

2 =1

2

u2m

~2=

1

2

~2

ma2u

, (3.31)

where αf is the “gravitational fine-structure” constant3, au the “gravitationalBohr” radius and Eu the “gravitational Rydberg” energy, respectively [Jackson(2002); Haken and Wolf (2000)]. The “gravitational Rydberg” energy can also bedescribed as a function of the Compton wavelength λC

Eu =1

2α2fmc

2 (3.29)=

1

2u2 mc

~2c2(3.30)=

1

2

~4

m2a2u

mc2

~2c2

=1

2a2u

~2mc2

m2c2=

1

2

(~

aumc

)2

mc2 =1

2

(λCau

)2

mc2, (3.35)

where λC = ~mc

is a typical wavelength for a quantum object of mass m. Itdescribes the idea of de Broglie assuming that all matter has a wave-like nature.

The transition to dimensionless “atomic” units can now be made with the

3We make use of the analogy between electrostatics and gravitation,

e2

4πε0⇔ Gm2, (3.32)

thus the usual fine-structure constant

αe =e2

4πε0

1~c

(3.33)

is converted into

αf =Gm2

~c. (3.34)

Note that in our case u = − 114π

Ik2α2

cε206= Gm2 (cf. (2.6)).

3.3. “Atomic”units 27

transformation equations

E =E

Eu, (3.36a)

r =r

au, (3.36b)

the substitution rules

∆r =

(∂2

∂r2+

2

r

∂

∂r

)=

1

au∆r, (3.37)

d3r = a3u d

3r, (3.38)

ψ(r) =1

a32u

ψ(r), (3.39)

and the relation [Nolting (2004b)]

δ(3)(r − r′) = δ(3)(aur − aur′) =

1

|a3u|δ(3)(r − r′). (3.40)

First we transform the effective potential (2.42)

W (r′, r) =W

Eu=

4πa~2

m

1

a3u

δ(3)(r − r′)1

Eu− u

au|r − r′|1

Eu

=4πa~2

m

u2m2

~4

1

auδ(3)(r − r′)

2~2

u2m− u

au|r − r′|2~2

u2m

= 8πa

auδ(3)(r − r′)− 2

|r − r′| , (3.41)

and then the kinetic energy (2.17)

T (r) =T

Eu=

p2

2m

1

Eu= − ~

2

2m∆r

2~2

u2m= −~

2

m

1

uau

1

a2u

∆r = −auau

∆r

= −∆r. (3.42)

Next we transform the external potential (2.16) introducing therefore an oscillatorwidth

aosc =

√~mω

→ ω =~

ma2osc

, (3.43)

so that we obtain

Vext(r) =VextEu

=12mω2r2

12α2fmc

2=

1

α2f

ω2r2

c2(3.43)=

1

α2f

~2

m2a4osc

r2

c2

=1

α2f

λ2C

a2osc

r2

a2osc

=~2c2

G2m4

~2

m2c21

a2osc

r2

a2osc

=~4

u2m2

1

a2osc

r2

a2osc

=a2u

a2osc

r2

a2osc

=a2u

a4osc

r2 =a4u

a4osc

r2

a2u

=

(r

au

)2

γ2

= γ2r2 (3.44)


with

γ =a2u

a2osc

=

(auaosc

)2

. (3.45)

If au ¿ aosc and ~ω ¿ Eu respectively, γ = a2u

a2osc= ~ω

2Eu→ 0 and we can neglect

the external potential.Finally we insert (3.38), (3.39), (3.41), (3.42) and (3.44) into (2.34) so that

we obtain the GP equation in “atomic” units by omitting the ring above thesizes

µψ(r) =

[−∆r + γ2r2 +N

∫d3r′ |ψ(r′)|2

(8π

a

auδ(3)(r − r′)− 2

|r − r′|)]

=

[−∆r + γ2r2 +N

(8π

a

au|ψ(r)|2 − 2

∫d3r′

|ψ(r′)|2|r − r′|

)]ψ(r),

(3.46)

with the contact potential

Vc(r) = 8πa

au|ψ(r)|2, (3.47)

the gravity-like potential

Vu(r) = −2

∫d3r′

|ψ(r′)|2|r − r′| , (3.48)

and the effective self-binding potential

Veff (r) = Vc(r) + Vu(r) = 8πa

au|ψ(r)|2 − 2

∫d3r′

|ψ(r′)|2|r − r′| . (3.49)

Of course in the same way (3.11) can be made dimensionless leading us to

∆rψN(r) = −[−Vu(r) + µ+N8π

a

au|ψN (r) |2

]ψN(r), (3.50a)

∆rVu(r) = 8π|ψN(r)|2, (3.50b)

where we have omitted the external potential. Using the abbreviation

U(r) ≡ µ− Vu(r) (3.51)

we obtain

∆rψN(r) = −U(r)ψN(r) + 8πa

auψN(r)3, (3.52a)

∆rU(r) = −8π|ψN(r)|2. (3.52b)

3.4. Energy functional 29

We can now prove with (3.6) that (3.48) solves (3.50b)

∆rVu(r) = −2

∫d3r′ |ψN(r′)|2∆ 1

|r − r′| , (3.53)

∆rVu(r) = 8π|ψN(r)|2, (3.54)

and (3.52b)

∆rU(r) = ∆µ−∆Vu(r) = −∆Vu(r)

= −(−2

∫d3r′ |ψN(r′)|2∆ 1

|r − r′|), (3.55)

∆rU(r) = −8π|ψN(r)|2. (3.56)

3.4 Energy functional

For our numerical and variational calculations of the mean-field energy of a self-trapped BEC we have to use the following energy functional:

E[ψ] = N 〈µ〉 − N2

2〈Veff〉 = N

(〈µ〉 − N

2〈Veff〉

)= N

⟨µ− N

2Veff

⟩. (3.57)

Inserting all quantities leads to

E[ψ] = N

⟨−∆r + γ2r2 +

N

28π

a

au|ψ(r)|2 − N

22

∫d3r′

|ψ(r′)|2|r − r′|

⟩

= N

⟨−∆r + γ2r2 + 4Nπ

a

au|ψ(r)|2 −N

∫d3r′

|ψ(r′)|2|r − r′|

⟩, (3.58)

where ψ is the ground-state field, which is given by the solution of the GP equation(3.46). The stationary solution of (3.46) minimizes the energy functional (3.58).The chemical potential is given by

µ =∂E

∂N= −∆r + γ2r2 +N

(8π

a

au|ψ(r)|2 − 2

∫d3r′

|ψ(r′)|2|r − r′|

), (3.59)

which is in accordance with (3.46).

3.4.1 The N dependency of the mean-field energy E

In this subsection we want to derive the N dependency of the energy. Thereforewe first scale the gravity-like interaction Vu and the contact interaction Vc.


General

First we introduce the scaling (3.15)

ψ = λψ (3.60a)

r = νr, (3.60b)

as in section 3.2. The wave function ψ should be normalized in the new coordi-nates to unity4

1 =

∫d3r ψ2(r) =

1

ν3

1

λ2

∫d3r ψ2(r)

︸︷︷︸=1

=1

ν3λ2,

thusν3λ2 = 1. (3.61)

Now we assume a general two-particle interaction homogeneous of the degree ofn

W (νr, νr′) = νnW (r, r′). (3.62)

The wave function ψ(r) is a solution of the Schrodinger equation containing thisinteraction [

−∆ +

∫d3r′W (r, r′)ψ2(r′)

]ψ(r) = µψ(r), (3.63)

where ψ(r) is normalized to unity∫d3r ψ2(r) = 1. (3.64)

The Schrodinger equation (3.63) is scaled with (3.60) and we obtain− 1

ν2∆ + ν3

∫d3r′ W (νr, νr′)︸︷︷︸

νnW (r,r′)

λ2ψ2

λψ = µλψ. (3.65)

Dividing by λ, multiplying with ν2 and the equation (3.65) reads[−∆ + ν5+nλ2

∫d3r′W (r, r′)ψ2

]ψ = µψ, (3.66)

withµ = µν2. (3.67)

Next we wish (3.66) to satisfy (2.34). This is fulfilled, if

ν5+nλ2 = ν2+n = N

⇒ ν = N1

2+n , (3.68)

using (3.61).

4Note that in section 3.2 we assume that the wave function is normalized to N . Hence (3.61)is not in conflict with (3.20).


Gravity-like interaction

Now we assume the gravity-like interaction (2.41) that arises from the irradiationof the electromagnetic fields

Wu(r, r′) ∝ 1

|r − r′| (3.69)

and via (3.60b) follows

Wu(νr, νr′) ∝ 1

|νr − νr′| = ν−1 1

|r − r′| ∼ Wu(r, r′).

This expresses a homogeneity of degree of n = −1. Hence (3.68) becomes

ν = N1

2+n = N1

2−1 = N (3.70)

and (3.61), using (3.70),

λ2 =1

ν3=

1

N3

⇒ λ =1

N32

. (3.71)

We can now insert (3.70) and (3.71) in (3.60) to obtain

ψ = N32ψ, (3.72a)

r =r

N, (3.72b)

so that the wave function ψ(r) satisfies the Schrodinger equation (3.66)

[−∆ +N

∫d3r′Wu(r, r

′)ψ2(r′)]ψ(r) = µψ(r). (3.73)

From (3.72a) follows with (2.48) the N -dependency of the scaled wave functionthat is normalized to the particle number N

ψ

N32

=ψN

N32N

12

=ψNN2

, (3.74)

which is in accordance with (3.27a). The mean-field energy of the many-bodywave function

Ψ = ψ(r1)ψ(r2) · · · ψ(rN) (3.75)

with the purely gravity-like interaction

Vu(r) =

∫d3r′Wu(r, r

′)ψ2(r′) (3.76)


then becomes by means of (3.57)

Eu = N

(〈µ〉 − N

2〈Vu(r)〉

)

= N 〈µ〉 − N2

2

∫ ∫d3r′ d3r ψ2(r)ψ2(r′)Wu(r, r

′)

= N3 〈µ〉 − N2

2

1

ν6

1

λ4︸︷︷︸=1

1

νn


′)

= N3 〈µ〉 − N2

2

1

N−1


′)

= N3 〈µ〉 − N2

2N


′)

= N3

(〈µ〉 − 1

2


′)). (3.77)

Contact interaction

Now still another further interaction with different homogeneity degree is present

W (r, r′) = W (νr, νr′) = νmW (r, r′). (3.78)

We assume the contact interaction (2.39), so that we obtain

Wc(r, r′) = 8π

a

auδ(3)(r − r′), (3.79)

Wc(νr, νr′) = 8π

a

auδ(3)(νr − νr′) = 8π

a

au

1

ν3δ(3)(r − r′), (3.80)

implying a homogeneity of degree m = −3. The wave function ψ(r) is a solutionof the Schrodinger equation

[−∆ +

∫d3r′Wc(r, r

′)ψ2(r′) +

∫d3r′Wu(r, r

′)ψ2(r′)]ψ(r) = µψ(r), (3.81)

where ψ(r) is normalized to unity∫d3r ψ2(r) = 1. (3.82)

Scaling as before leads to 1

ν2∆ + ν3

∫d3r′ Wc(νr, νr

′)︸︷︷︸νmWc(r,r

′)

λ2ψ2(r′) + ν3

∫d3r′ Wu(νr, νr

′)︸︷︷︸νnWu(r,r′)

λ2ψ2

ψ(r)

= µν2ψ(r). (3.83)


Dividing by λ, multiplying with ν2 and the equation (3.83) changes into

(−∆ + ν5+mλ2

∫d3r′Wc(r, r

′)ψ2(r′) +N

∫d3r′Wu(r, r

′)ψ2(r′))ψ(r)

= µψ(r), (3.84)

using (3.67) and (3.73). The prefactor of the second summand of (3.84) becomes

ν5+mλ2 (3.61)= ν2+m. (3.85)

In our case we had m = −3 that lead us to

ν2+m = ν−1 (3.70)=

1

N. (3.86)

Hence the wave function ψ(r) satisfies the Schrodinger equation (3.84)

(−∆ +

1

N

∫d3r′Wc(r, r

′)ψ2(r′) +N

∫d3r′Wu(r, r

′)ψ2(r′))ψ(r)

= µψ(r),(−∆ +N

1

N2

∫d3r′Wc(r, r

′)ψ2(r′) +N

∫d3r′Wu(r, r

′)ψ2(r′))ψ(r)

= µψ(r),(−∆ +N

∫d3r′

1

N2Wc(r, r

′)ψ2(r′) +N

∫d3r′Wu(r, r

′)ψ2(r′))ψ(r)

= µψ(r)

with the scaled contact potential

1

N2Wc(r, r

′) =1

N28π

a

auδ(3)(r − r′) = 8π

(1

N2

a

au

)δ(3)(r − r′). (3.87)

From this follows the N dependency of the s-wave scattering

a

au→ 1

N2

a

au, (3.88)

as derived in (3.27d). The mean-field energy of the many-body wave function

Ψ = ψ(r1)ψ(r2) · · · ψ(rN) (3.89)

with the pure contact interaction

Vc(r) =

∫d3r′

1

N2Wc(r, r

′)ψ2(r′) (3.90)


then becomes because of (3.57)

Ec = N

(〈µ〉 − N

2〈Vc(r)〉

)

= N 〈µ〉 − N2

2

1

N2

∫ ∫d3r′ d3r ψ2(r)ψ2(r′)Wc(r, r

′)

= N3 〈µ〉 − 1

ν6

1

λ4︸︷︷︸=1

1

νm1

2


′)

= N3 〈µ〉 −N3 1

2


′)

= N3

(〈µ〉 − 1

2


′). (3.91)

By comparing with (3.77) we obtain the complete mean-field energy includingboth interactions:

E = N3

(µ− 1

2


′)

− 1

2


′)

). (3.92)

This implies the important scaling rule

E|(N,N2a/au) = N3E|(N=1,a/au). (3.93)

3.5 Radius rrms and peak density ρmax

In this section we derive the N dependency of the root-mean-square radius rrmsand the peak density ρ(0) = ρmax at the center of the condensate. For that wecarry out the same procedure as in subsection 3.4.1 for the mean-field energy E.

3.5.1 The N dependency of the radius rrms

The root-mean-square radius rrms is defined as

rrms =

√√√√⟨

Ψ

∣∣∣∣∣N∑i=1

r2i

∣∣∣∣∣ Ψ

⟩=

√√√√N∑i=1

⟨ψ

∣∣r2i

∣∣ψ⟩=

√N

∫d3r r2ψ2(r), (3.94)

3.6. Integral representation 35

where Ψ is the many-body wave function (3.89). We now insert (3.60) in (3.94)and obtain

rrms =

√√√√N1

ν3

1

λ2︸︷︷︸=1

1

ν2

∫d3r r2ψ2(r)

=

√N · 1

N2

∫d3r r2ψ2(r)

=

√1

N

∫d3r r2ψ2(r)

=1√N

√〈ψ(r) |r2|ψ(r)〉. (3.95)

Hence the radius rrms is proportional to the inverse root of N and the scalingrule reads: √

〈r2〉|(N,N2a/au) =1√N

√〈r2〉|(N=1,a/au). (3.96)

3.5.2 The N dependency of the peak density ρmax

We start with the definition of the peak density at the position r′

ρ(r′) =

⟨Ψ

∣∣∣∣∣N∑i=1

δ(r′ − ri)

∣∣∣∣∣ Ψ

⟩

= N

∫d3r δ(r′ − r)ψ2(r′). (3.97)

For the maximum peak density at the center of the BEC r′ = 0 we obtain

ρmax = ρ(0) = N

∫d3r δ(0− r)ψ2(0)

= ψ2(0)N =1

λ2ψ2(0)N = N3ψ2(0)N

= ψ2(0)N4. (3.98)

Hence the peak density ρmax is proportional to N4 and the scaling reads:

ρmax|(N,N2a/au) = N4ρmax|(N=1,a/au) = N4ψ2(0). (3.99)

3.6 Integral representation

Our goal is now to find an integral formulation of equations (3.52) [Tod andMoroz (1999)] which will be used below. For that we introduce the abbreviation

C = 8πa

au(3.100)


and we obtain

∆rψ(r) =1

r2∂r

(r2∂rψ(r)

)= −U(r)ψ(r) + Cψ(r)3, (3.101a)

∆rU(r) =1

r2∂r

(r2∂rU(r)

)= −8π|ψ(r)|2. (3.101b)

Now the solutions ψ and U have to be continuously differentiable at the origin,so that we require that the derivatives ∂rψ and ∂rU vanish at r = 0, viz.

∂rψ(r = 0) = 0, (3.102a)

∂rU(r = 0) = 0. (3.102b)

Thereafter we multiply (3.101) with r2 and by integration with respect to r wearrive at

∂rψ(r) = − 1

r2

r∫

0

x2U(x)ψ(x) dx+ C1

r2

r∫

0

x2ψ(x)3 dx, (3.103a)

∂rU(r) = −8π1

r2

r∫

0

x2ψ(x)2 dx. (3.103b)

By a further integration we obtain

ψ(r)− ψ(0) = −r∫

0

1

y2

y∫

0

x2U(x)ψ(x) dx dy + C

r∫

0

1

y2

y∫

0

x2ψ(x)3 dx dy,

U(r)− U(0) = −8π

r∫

0

1

y2dy

r∫

0

x2ψ(x)2 dx.

The resulting double integral can be simplified via integration by parts (see ap-pendix A). This leads us finally to

ψ(r) = ψ(0) +

r∫

0

x(xr− 1

)ψ(x) dx

[U(x)− Cψ(x)2

], (3.104a)

U(r) = U(0) + 8π

r∫

0

x(xr− 1

)ψ(x)2 dx. (3.104b)

3.7 Asymptotic behavior of the wave function

Next we note two asymptotic forms of (3.52) that will help us later to reassessand interpret our numerical and variational solutions.

3.7. Asymptotic behavior of the wave function 37

3.7.1 Limit r → 0

We want to analyze the behavior of the functions ψ(r) and U(r) at the origin.Therefore we rewrite (3.101) as

∂2r (rψ) = −rUψ + Crψ3, (3.105a)

∂2r (rU) = −8πrψ2 (3.105b)

and omit henceforth the coordinate dependency in ψ(r) and U(r). We introducethe abbreviations

ψ =1

rψ, (3.106a)

U =1

rU (3.106b)

and insert (3.106) in (3.105), so that we finally obtain

∂2r ψ = −1

rUψ + C

1

r2ψ3, (3.107a)

∂2r U = −8π

rψ2. (3.107b)

Now we expand ψ and U in Taylor series

ψ(r) = ψ0 +(∂rψ0

) r1!

+(∂2r ψ0

) r2

2!+

(∂3r ψ0

) r3

3!+

(∂4r ψ0

) r4

4!+

(∂5r ψ0

) r5

5!+ . . . ,

U(r) = U0 +(∂rU0

) r1!

+(∂2r U0

) r2

2!+

(∂3r U0

) r3

3!+

(∂4r U0

) r4

4!+

(∂5r U0

) r5

5!+ . . . .

If we take the radial symmetry of the system into account (see subsection 3.1.2)the functions have to be even

ψ(x, 0, 0) = ψ(−x, 0, 0),

U(x, 0, 0) = U(−x, 0, 0),

and the even terms in r in the Taylor series of ψ and U drop out, so that the oddterms in ψ and U (compare (3.106)) vanish. Of course this is also valid for theother Cartesian coordinates leading us to

ψ(r) = ψ0 +(∂3r ψ0

) r3

3!+

(∂5r ψ0

) r5

5!+ . . . , (3.108a)

U(r) = U0 +(∂3r U0

) r3

3!+

(∂5r U0

) r5

5!+ . . . (3.108b)

with

ψ(0) = ψ0 = const.

V (0) = V0 = const.


Using the derivatives calculated in appendix B and with the back substitution of(3.106) in (3.108) we obtain

ψ(r) = ψ0 +1

6

(Cψ3

0 − U0ψ0

)r2

+1

120

(8πψ3

0 + ψ0U20 − 4CU0ψ

30 + 3C2ψ5

0

)r4 +O(r6), (3.109a)

U(r) = U0 − 4

3πψ2

0r2 +

2

15π

(U0ψ

20 − Cψ4

0

)r4 +O(r6). (3.109b)

The potential U(r) decreases proportional to −r2 for sufficiently small r. Fromthis follows that the potential Vu (3.51) increases proportional to r2 at the origin.

3.7.2 Limit r →∞First we consider the integral representation of the function (3.104b)

U(r) = U(0) + 8π

r∫

0

x(xr− 1

)ψ(x)2 dx, (3.110)

and rewrite (3.110) as

U(r) = U(0)− 8π

r∫

0

dx xψ(x)2 + 8π1

r

r∫

0

dx x2ψ(x)2. (3.111)

From (3.111) we get the asymptotic form of the potential

limr→∞

U(r) = A+B

r∼ A, (3.112)

where we use the abbreviations

A = U(0)− 8π

∞∫

0

dx xψ(x)2 (3.113)

B = 8π1

r

∞∫

0

dx x2ψ(x)2. (3.114)

By means of (3.51) we conclude from (3.111) that the potential Vu decreasesproportional to −1/r for large r. If we now insert (3.112) in (3.101a) we obtainthe asymptotic behavior of the wave function

limr→∞

ψ(r) ∼ e−κ(r), (3.115)

where κ is a function of r [Abramowitz and Stegun (1970)]. We conclude thatthe wave function decreases exponentially for r →∞.

3.8. Derivation of an analytical wave function 39

3.8 Derivation of an analytical wave function

There are two limiting regions for self-graviting BEC: The “Thomas-Fermi grav-itation” (TF-G) region, where the kinetic energy is negligible and the contactpotential balances the gravitational-like potential, and the purely “gravitational”(G) region, where the contact potential is negligible and the kinetic energy isbalanced by the gravitational-like potential [Giovanazzi et al. (2001)]. Neitherregion is sensitive to the trapping potential, so that we can turn it off (γ = 0)and select either the “G” or “TF-G” region.

For the“TF-G”region we can derive an analytic solution for the wave functionthat has to satisfy the GPTF−G equation

µψN(r) =

[8π

a

au|ψN(r)|2 − 2

∫d3r′

|ψN(r′)|2|r − r′|

]ψN(r). (3.116)

First we want to rewrite the last term in (3.116) that is the gravity-like potential.One possible method is to use the spherical harmonics expansion as described in[Jackson (2002)]:

1

|r − r′| = 4π∞∑

l=0

l∑m=−1

1

2l + 1

rl<rl+1>

Y ∗lm(θ′, φ′)Ylm(θ, φ). (3.117)

A drawback is the occurrence of a double sum. Another method that we followis the direct integration of the gravity-like potential using suitable substitutions,so that we can distinguish different cases. We assume that the r-axis is parallelto the z-axis, as shown in FIG. 3.1 and introduce spherical coordinates

r =

xyz

=

r sin θ cosφr sin θ sinφr cos θ

(3.118)

leading us to

− 2ψN(r)

∫d3r′

|ψN(r′)|2|r − r′|

= −2ψN(r)

2π∫

0

dφ′π∫

0

dθ′ sin θ′∞∫

0

dr′ r′2|ψN(r′)|2√

r2 + r′2 − 2rr′ cos θ′

= −2ψN(r) 2π

π∫

0

dθ′ sin θ′∞∫

0

dr′ r′2|ψN(r′)|2√

r2 + r′2 − 2rr′ cos θ′

= −4πψN(r)

π∫

0

∞∫

0

dr′ |ψN(r′)|2r′2 sin θ′√r2 + r′2 − 2rr′ cos θ′

dθ′. (3.119)


x

θ

r

r

r

r

z

yφ

FIG. 3.1: Orientation of the axes.

In (3.119) we use the Jacobian

∂(x, y, z)

∂(r, θ, φ)=

∣∣∣∣∣∣

sin θ cosφ r cos θ cosφ −r sin θ sinφsin θ sinφ r cos θ sinφ r sin θ cosφ

cos θ −r sin θ 0

∣∣∣∣∣∣= r2 cos2 θ sin θ cos2 φ+ r2 sin3 θ sin2 φ

+ r2 sin θ cos2 θ sin2 φ+ r2 sin3 θ cos2 φ

= r2 sin θ, (3.120)

as illustrated in [Nolting (2004a)]. Next we want to eliminate the angular compo-nents, so we have to find a suitable substitution. In the denominator of (3.119)stands a cosine which is the first derivative of a sine in the numerator. Thereforewe assume

a′(θ′) = cos θ′,

a′(θ′) =da′

dθ′= − sin θ′,

dθ′ =da′

a′.

With the new integration limits

θ′ = π → a′ = cos π = −1,

θ′ = 0 → a′ = cos 0 = 1


we obtain

− 4πψN(r)

−1∫

1

∞∫

0

dr′ |ψN(r′)|2r′2 a′√r2 + r′2 − 2rr′a′

da′

a′

= −4πψN(r)

−1∫

1

∞∫

0

dr′ |ψN(r′)|2r′2 1√r2 + r′2 − 2rr′a′

da′

= −4πψN(r)

∞∫

0

dr′ |ψN(r′)|2r′2[− 1

rr′√r2 + r′2 − 2rr′a′

]−1

1

= −4πψN(r)

∞∫

0

dr′ |ψN(r′)|2r′2[− 1

rr′√r2 + r′2 − 2rr′ +

1

rr′√r2 + r′2 + 2rr′

]

= −4πψN(r)

∞∫

0

dr′ |ψN(r′)|2r′2 1

rr′

[−√r2 + r′2 − 2rr′ +

√r2 + r′2 + 2rr′

].

Now multiplying by −1 and we get

4πψN(r)

∞∫

0

dr′ |ψN(r′)|2r′2 1

rr′

[√r2 + r′2 − 2rr′ −

√r2 + r′2 + 2rr′

].

In the square bracket stands a binomial of the form

(r − r′)2 = r2 + r′2 − 2rr′,

(r + r′)2 = r2 + r′2 + 2rr′.

Thus we obtain

4πψN(r)1

r

∞∫

0

dr′ |ψN(r′)|2r′[√

(r − r′)2 −√

(r + r′)2]

= 4πψN(r)1

r

∞∫

0

dr′ |ψN(r′)|2r′ [(r − r′)− (r + r′)] . (3.121)

Since the wave function is nonzero in the region 0 ≤ r ≤ R0 (see FIG. 3.2)the upper integration limit for r is set to R0. Due to the absolute value in thedenominator in the last term of (3.116) the parentheses in (3.121) have to be


00

rR

0

ψΝ

ψΝ0

FIG. 3.2: Wave function

replaced by absolute value bars and a casewise differentiation has to be made5.

4πψN(r)1

r

R0∫

0

dr′ |ψN(r′)|2r′ [|r − r′| − |r + r′|]

= −4πψN(r)1

r

R0∫

0

dr′ |ψN(r′)|2

2rr′ , if r < r′

2r′2 , if r ≥ r′

= −8πψN(r)1

r

R0∫0

dr′ |ψN(r′)|2r′2 , if r > R0

r∫0

dr′ |ψN(r′)|2r′2 +R0∫0

dr′ |ψN(r′)|2rr′ , if r ≤ R0

. (3.122)

For the case r > R0 the wave function ψN(r) vanishes, as one can see easily fromFIG. 3.2, so that we omit this case from (3.122) and we finally get

− 8πψN(r)

1

r

r∫

0

dr′ |ψN(r′)|2r′2 +1

r

R0∫

r

dr′ |ψN(r′)|2rr′

= −8πψN(r)

1

r

r∫

0

dr′ r′2|ψN(r′)|2 +

R0∫

r

dr′ r′|ψN(r′)|2 . (3.123)

5A similar casewise differentiation has to made by calculating the potential of a sphere ofhomogeneous fixed surface charge [Nolting (2004a)].


That means that the last term of (3.116) equals to (3.123):

∫d3r′

|ψN(r′)|2|r − r′| = 4π

1

r

r∫

0

dr′ r′2|ψN(r′)|2 +

R0∫

r

dr′ r′|ψN(r′)|2 . (3.124)

Using (3.124) we finally obtain

µψN(r) =

[8π

a

au|ψN(r)|2 − 2

∫d3r′

|ψN(r′)|2|r − r′|

]ψN(r)

−µψN(r) + 8πa

au|ψN(r)|3 = 8πψN(r)

1

r

r∫

0

dr′r′2|ψN(r′)|2 +

R0∫

r

dr′r′|ψN(r′)|2

−µ+ 8πa

au|ψN(r)|2 = 8π

1

r

r∫

0

dr′r′2|ψN(r′)|2 +

R0∫

r


− µ

8π+

a

au|ψN(r)|2 =

1

r

r∫

0

dr′r′2|ψN(r′)|2 +

R0∫

r


− µ

8πr +

a

aur|ψN(r)|2 =

r∫

0

dr′r′2|ψN(r′)|2 − r

r∫

R0

dr′r′|ψN(r′)|2. (3.125)

Equation (3.125) is multiplied from the left by ddr

(noted by a dot) and with thesubstitution

r|ψN(r)|2 = φN(r) (3.126)

we obtain6

− µ

8π+

a

auφN = rφN −

r∫

∞

dr′ φN(r′)− rφN . (3.127)

Now we differentiate once more with respect to the local coordinate r and obtain

a

auφN = −φN ,

φN = −auaφN . (3.128)

The solution of equation (3.128) is proportional to

φN ∼ sin

(r

√aua

), (3.129)

6 ddr

b∫a

f(r, t) dt =b∫

a

∂f(r,t)∂r dt [Bronstein et al. (2001)]


and with the back substitution (3.126) we arrive to

|ψN(r)|2 = A2sin

(r√

au

a

)

r, (3.130)

where A is the normalization factor. The first zero of (3.130) is

r0

√aua

= π,

r0 =π√au

a

= π

√a

au, (3.131)

so that (3.130) is equivalent to

|ψN(r)|2 = A2sin

(π rr0

)

r. (3.132)

From the normalization condition we calculate the normalization factor A∫dΩ r2dr |ψN(r)|2 !

= N,

4πA2

r0∫

0

sin(π rr0

)

rr2dr = N,

4πA2

r0∫

0

sin

(πr

r0

)rdr = N,

4πA2

sin

(π rr0

)

π2

r20

−r cos

(π rr0

)

πr0

r0

0

= N,

4πA2

[r0 · 1

πr0

− 0

]= N,

A2 =N

4r20

,

A =

√N

2r0,

so that we finally obtain the wave function for the “TF-G” region

|ψN(r)|2 =

√N

2r0

sin(π rr0

)

r. (3.133)


Rewriting with (3.131) and omitting the absolute value bars at ψ leads us to

ψN(r) =

√N

2π√

aau

√√√√√sin

(r√

aau

)

r. (3.134)

Of course we can replace ψN(r) by (2.48) so that we obtain

ψ(r) =1

2π√

aau

√√√√√sin

(r√

aau

)

r, (3.135)

that is normalized to unity.

Chapter 4

Approximation method forstationary states

In contrast with the quantum mechanical problem of an ideal Bose gas which canbe solved exactly (cf. subsection 2.3.1), an exact solution is not possible in thecase of weakly-interacting Bose gases (cf. subsection 2.3.2). In that case one has toresort to numerical solutions. There are, however, approximation methods, suchas perturbation theory [Cohen-Tannoudji et al. (1999)], the WKB approximation[Friedrich (2005)] and the variational method [Nolting (2004c)], that allow us inparticular cases to find analytical approximations for the solution of the underly-ing Schrodinger equation. Thus we can calculate approximate stationary statesand energy eigenvalues for nonlinear quantum mechanical systems. Perturbationtheory is applicable usually to systems where the problem differs from an exactlysolvable problem by a “small” amount. The WKB approximation is applicablein the nearly classical limit1, and the variational or “trial” function method issuitable for the calculation of the ground state energy if one has a qualitativeidea of the form of the wave function [Schwabl (2002)]. In our case of a weakly-interacting Bose gas we can assume the form of the wave function, and make useof the variational method in this chapter.

4.1 Variational principle

One important theorem in quantum mechanics says that from all normalizablewave functions describing a system only one wave function is realized by nature,namely that wave function which minimizes the energy functional of the system.This wave function is simultaneously the solution of the Schrodinger equationdescribing the system. All functions that differ from this wave function lead to a

1For the Newton-Schrodinger equation mentioned in subsection 2.4.2 see [Greiner (2005)].

47

48 Chapter 4. Approximation method for stationary states

higher energy. Mathematically this can be described by [Nolting (2004c)]

〈H〉ψ =〈ψ|H|ψ〉〈ψ|ψ〉 ≥ E, (4.1)

with the stationary Schrodinger equation

Hφ = Eφ (4.2)

which hold for the function φ which minimizes the energy functional of the system.For all other functions ψ 6= φ the expectation value 〈H〉ψ is greater than E. Only

in the case of ψ = φ we get 〈H〉ψ = E. In other words, the lower the variationalenergy of the system the better is the corresponding wave function. Thus theexpectation value of the energy is a measure for the quality of the wave function.

4.1.1 STO trial wave function

In section 3.7 we derived the asymptotic form of the wave function: In the limitr → ∞ the wave function has to decrease exponentially. Therefore we assumefirst of all a radially symmetric “Slater Type Orbital” (STO) for our variationalcalculation

ψ(r) = A exp

(−kr

2

). (4.3)

We can fix the normalization factor A such that ψ(r) is normalized to unity∫

R3

d3r |ψ(r)|2 != 1,

4πA2

∫dr r2 exp(−kr) = 1,

4πA2 2!

k2+1= 1,

A2 =k3

8π,

A =k

32√8π, (4.4)

From now on all analytical integrations are relegated to appendix C. We obtainthe normalized wave function by inserting (4.4) in (4.3)

ψ(r) =k

32√8π

exp

(−kr

2

). (4.5)

Our goal is now to determine the parameter2 k, so that the wave function (4.5)minimizes the energy (3.58) of the system with the Schrodinger equation (3.46).

2One can easily see from (3.46) that the parameter k depends only on the s-wave scatteringlength a

auand the particle number N of the condensate if the external trap is switched off.

4.1. Variational principle 49

The choice for STO allows us to derive an analytical form of the gravity-likepotential Vu(r) by means of (3.124). For this we need the square of the wavefunction

|ψ(r)|2 =k3

8πexp (−kr) , (4.6)

too. We insert (4.6) and (4.5) in (3.124) and obtain∫d3r′

|ψ(r′)|2|r − r′|

=k3

2

1

r

r∫

0

dr′ r′2e−kr′+

∞∫

r

dr′ r′e−kr′

=k3

2

[1

re−kr

′(−r

′2

k− 2r′

k2− 2

k3

)∣∣∣∣r

0

+1

k2e−kr

′(−kr′ − 1)|∞r

]

=k3

2

[1

r

(e−kr

(−r

2

k− 2r

k2− 2

k3

)+

2

k3

)+

1

k2

(0− e−kr (−kr − 1)

)]

=k3

2

[1

r

(2

k3+ e−kr

(−r

2

k− 2r

k2− 2

k3

))+r

r

e−kr

k2(kr + 1)

]

=k3

2

[1

r

(2

k3+ e−kr

(−r

2

k− 2r

k2− 2

k3+r2

k+

r

k2

))]

=k3

2

1

r

[2

k3+ e−kr

(− r

k2− 2

k3

)]

=k3

2

1

r

[2

k3− e−kr

(r

k2+

2

k3

)]

=1

r

[1− e−kr

(1 +

kr

2

)]. (4.7)

The gravity-like potential thus reads

Vu(r) = −2

∫d3r′

|ψ(r′)|2|r − r′| = −2

r

[1− exp(−kr)

(1 +

kr

2

)]. (4.8)

The limiting cases are

kr À 1 : Vu(r) = −2

r, (4.9)

kr ¿ 1 : Vu(r) = −2

r

[1−

(1− kr +

k2r2

2− k3r3

6

)(1 +

kr

2

)]

= −k +1

6k3r2 +O(k4r3), (4.10)

where in (4.10) we make use of the series expansion of the exponential function[Bronstein et al. (2001)]

exp(z) ≈ 1 +z

1!+z2

2!+z3

3!+ . . . . (4.11)


0r

0

Vu(r

)

-κ+1/6κ3r2

-2/r

-κ

FIG. 4.1: Effective gravity-like potential Vu(r) for a STO (black curve). The bluelines show the asymptotic behaviour of Vu(r) for r → 0 and r →∞, respectively.

The asymptotic limits are shown in Fig. 4.1 together with Vu(r). Next wecalculate the kinetic energy (3.42) using the abbreviations ∂2

∂r2ψ = ψ and ∂

∂rψ = ψ

〈T (r)〉 = 〈−∆r〉

=

⟨− ∂2

∂r2− 2

r

∂

∂r

⟩

= −∫

R3

d3r

(ψ(r)ψ(r) +

2

rψ(r)ψ(r)

)

= −4π

∞∫

0

dr r2

(ψ(r)ψ(r) +

2

rψ(r)ψ(r)

)

= −4πk3

8π

∞∫

0

dr r2

(k2

4e−kr − k

re−kr

)

= −4πk3

8π

(k2

4

2

k3− k

1

k2

)

=k2

4. (4.12)


The expectation value of the kinetic energy is proportional to the square of thewidth parameter k. Furthermore we compute (3.44) for the harmonic trap po-tential

〈Vext(r)〉 =⟨γ2r2

⟩

=

∫

R3

d3r γ2r2ψ(r)2

= 4πk3

8πγ2

∞∫

0

dr r4e−kr

= 4πk3

8πγ2 24

k5

= 12γ2

k2, (4.13)

which is proportional to the inverse square of the parameter k. Then we calculatethe contact potential (3.47)

〈Vc(r)〉 =

⟨8π

a

au|ψ(r)|2

⟩

=

∫

R3

d3r 8πa

auψ(r)4

= 4π8πa

au

k6

64π2

∞∫

0

dr r2e−2kr

=a

au

k6

2

2

8k3

=1

8

a

auk3, (4.14)

which is proportional to the cube of k. Finally we calculate the gravity-likepotential (3.48)

〈Vu(r)〉 =

⟨∫d3r′

|ψ(r′)|2|r − r′|

⟩

=

∫

R3

d3r

∫

R3

d3r′|ψ(r′)|2|r − r′|︸︷︷︸

use equation (3.124)

ψ(r)2. (4.15)

In (4.15) appears a double integral over the variables r and r′. To go on withthe calculation, we make use of (3.124), where we derived the analytical form ofthe gravity-like potential. We set the limit of the second integral in (3.124) to


R0 = ∞, so that we obtain

〈Vu(r)〉 =

∫

R3

d3r 4π

1

r

r∫

0

dr′ r′2|ψ(r′)|2 +

∞∫

r

dr′ r′|ψ(r′)|2ψ(r)2

= 4π

∞∫

0

dr r24π

1

r

r∫

0

dr′ r′2|ψ(r′)|2 +

∞∫

r

dr′ r′|ψ(r′)|2ψ(r)2

= 4π

∞∫

0

dr r2 k3

2

1

r

r∫

0

dr′ r′2e−kr′+

∞∫

r

dr′ r′e−kr′

︸︷︷︸use equation (4.7)

k3

8πe−kr (4.16)

and by means of (4.7) we finally have

〈Vu(r)〉 = 4π

∞∫

0

dr r2 1

r

[1− e−kr

(1 +

kr

2

)]k3

8πe−kr

=k3

2

∞∫

0

dr r2e−kr1

r

[1− e−kr

(1 +

kr

2

)]

=k3

2

∞∫

0

dr re−kr −∞∫

0

dr re−2kr − k

2

∞∫

0

dr r2e−2kr

=k3

2

[1

k2− 1

(2k)2− k

2

2

(2k)3

]

=k3

2

[1

k2− 1

4k2− 1

8k2

]

=k

2

[1− 1

4− 1

8

]=k

2

8− 2− 1

8=k

2

5

8

=5

16k. (4.17)

The expectation value of the gravity-like potential then takes the form

〈Vu(r)〉 =

⟨−2

∫

R3

d3r′|ψ(r′)|2|r − r′|

⟩

= −2k6

64π4

∫

R3

d3r

∫

R3

d3r′e−k(r+r′)

|r − r′|= −5

8k. (4.18)


Next we insert (4.12),(4.13),(4.14) and (4.18) into (3.58) and obtain the mean-field energy of the system

E(k) = N

[k2

4+ 12

γ2

k2+

1

16

a

auNk3 − 5

16Nk

]

=N

4

[k2 + 48

γ2

k2+

1

4

a

auNk3 − 5

4Nk

], (4.19)

from which we can determine the parameter k by minimizing E with respect tok. We also give the expressions for the root-mean-square radius (3.95) with thegiven “STO”

〈rrms〉 =

√√√√⟨

N∑i=1

r2i

⟩

=

√N

∫dr r2r2ψ2

=

√√√√√4πk3

8πN

∞∫

0

dr r4e−kr

=

√4πk3

8πN

4!

k5

=

√12

k2N, (4.20)

and the peak density at the center of the condensate (3.98)

〈ρmax〉 =

⟨N∑i=1

δ(ri)

⟩= N |ψ(0)|2 = NA2 =

k3

8πN. (4.21)

If the harmonic trap potential is switched off, viz. γ = 0, the second term in(4.19) can be omitted and we obtain

E(k) =N

4

[k2 +

1

4

a

auNk3 − 5

4Nk

]. (4.22)

Now we have to minimize the energy to find the parameter k for the STO trialwave function in order to calculate the energy for the new two physical regions,“TF-G” and “G”, that are described in 3.8. For the minimum of (4.22) we differ-entiate it with respect to k and set it equal to zero:

E ′(k) =N

4

[2k +

3

4

a

auNk2 − 5

4N

]= 0. (4.23)


This condition is fulfilled if

2k +3

4

a

auNk2 − 5

4N = 0,

k +3

8

a

auNk2 − 5

8N = 0,

k2 +8

3N aau

k − 5

3 aau

= 0. (4.24)

We can solve the quadratic equation (4.24) explicitly and obtain

kmin =1

2

− 8

3N aau

±

√√√√(

8

3N aau

)2

+20

3 aau

=1

2

(− 8

3N aau

± 8

3N aau

√1 +

20

3 aau

(3N a

au

8

)2)

=4

3N aau

(±

√1 +

15

16N2

a

au− 1

). (4.25)

In (4.25) the negative sign is omitted3, so that we finally obtain

kmin ≡ k+ =4

3N aau

(√1 +

15

16N2

a

au− 1

), (4.26)

for the quantity kmin which minimizes the energy of the system.We analyze the limiting cases where the new two physical regions, “TF-G”

and “G” of self-trapped BEC arise and calcute all quantities derived above.

1) Case: “G”→ contact potential negligible

In this case the gravity-like potential is dominant, whereas the contact potentialcan be neglected. Therefore a

au→ 0 and the energy (4.22) in the “G” region reads

E(k) =N

4

[k2 − 5

4Nk

]. (4.27)

Next we use the Taylor expansion [Bronstein et al. (2001)]

(1 + x)α = 1 +

(α

1

)x+

(α

2

)x2 + . . .+

(α

α− 1

)xα−1 + xα = 1 +αx+ . . . (4.28)

3The solutions corresponding to the negative sign leads to a tangent bifurcation describedin chapter 5.7. The solutions have higher energy than the groundstate.


so that for the parameter kmin of (4.26) we have with α = 12

in first order

kmin ≈ 4

3N aau

(1 +

15

32N2 a

au− 1

)=

5

8N = 6.25 · 10−1N. (4.29)

Now we insert (4.29) in (4.12), (4.18), (4.20), (4.21) and (4.27), obtaining allquantities in the “G” region

〈T 〉 =k2

4=

(58N

)2

4=

25

256N2

≈ 9.7656 · 10−2N2, (4.30)

〈Vu〉 = −5

8Nk = −5

8· 5

8N2 = −25

64N2

≈ −3.9063 · 10−1N2, (4.31)

〈rrms〉 =

√12

k2N =

√12 · 64

25N2N =

√768

25

1

N

≈ 5.54261√N, (4.32)

〈ρmax〉 =k3

8πN = N

125N3

512 · 8π = N4 125

4096π≈ 9.7140 · 10−3N4, (4.33)

E =N

4

[k2 − 5

4Nk

]=N

4

[(5

8N

)2

− 5

4N

(5

8N

)]

=N

4

[25

64N2 − 25

32N2

]= − 25

256N3

≈ −9.7656 · 10−2N3, (4.34)

respectively. This will allow us to compare the energy E of the “G” region (4.34)with the numerical value of the energy E calculated by accurate numerical in-tegration of the nonlinear Schrodinger equation at a later stage (cf. TAB. 5.28,where the results are given in red.).

Now we ask for the condition, under which the s-wave scattering length aau

andthe parameter γ are sufficiently small in order to neglect the contact potentialand the external potential, respectively. The neglected terms of (4.19) are bymeans of (4.29)

1

2Vc =

1

4

a

auNk3 =

1

4

a

auN

(5

3N

)3

=125

2048N4 a

au≈ 6.1035 · 10−2N4 a

au

4Vext = 48γ2

k2= 48

64

25

1

N2γ2 =

3072

25

1

N2γ2 ≈ 122.88

1

N2γ2.


Finally we compare with the kinetic energy (4.27). First we obtain for the contactpotential

1

2Vc =

125

2048N4 a

au¿ 4T = k2 =

25

64N2,

a

au¿ 32

5

1

N2= 6.4

1

N2(4.35)

and for the external potential

4Vext =3072

25

1

N2γ2 ¿ 4T = k2 =

25

64N2,

γ2 ¿ 625

196608N4 ≈ 3.1789 · 10−3N4 ≈

(N

4.2114

)4

. (4.36)

Hence neglecting the contact potential is no longer justified if

a

au& 32

5

1

N2= 6.4

1

N2. (4.37)

The external potential cannot be neglected if

γ2 & 625

196608N4,

γ & 25

256√

3N2 & 5.6382 · 10−2N2 &

(N

4.2114

)2

. (4.38)

2) Case: “TF-G”→ kinetic energy negligible

In the second case, where the kinetic energy is negligible, the contact potentialbalances the gravity-like potential. Therefore we can omit the first term in (4.22)representing the kinetic energy. The mean-field energy of the system in theThomas-Fermi limit then reads

E(k) =N

4

[1

4

a

auNk3 − 5

4Nk

]=N2

16

[a

auk3 − 5k

]. (4.39)

Next we have to calculate the parameter kmin for this region minimizing theenergy. For the minimum of (4.39) we have

E ′(k) =N2

16

[3a

auk2 − 5

]= 0. (4.40)


3a

auk2min = 5, (4.41)

k2min =

5

3

1aau

,

kmin =

√5

3

1√aau

≈ 1.29101√aau

. (4.42)


We insert (4.41) and (4.42) into (4.39) and obtain the mean-field energy in the“TF-G” region

E =N2

16k

[a

auk2 − 3k2 a

au

]=N2

16k

(−2k2 a

au

)= −10

3

√5

3

N2

16

1√aau

≈ −2.6896 · 10−1 1√aau

N2. (4.43)

Again we shall later compare this energy with the accurate energy calculatednumerically for large scattering lengths, e.g. a

au= 1 · 10+04 (cf. TAB. 5.28 where

these results are given in blue.).All other quantities are calculated by inserting (4.42) in (4.14), (4.18), (4.20)

and (4.21), so that we finally get

〈Vc〉 =1

8

a

auNk3 =

1

8

a

auN

(√5

3

1√aau

)3

=5

24N

√5

3

1√aau

≈ 2.6896 · 10−1 1√aau

N, (4.44)

〈Vu〉 = −5

8Nk = −5

8

√5

3

1√aau

N

≈ −8.0687 · 10−1 1√aau

N, (4.45)

〈rrms〉 =

√12

k2N =

√12

3

5Na

au=

6√5

√N

√a

au

≈ 2.6833√N

√a

au, (4.46)

〈ρmax〉 =k3

8πN =

1

8π

5√

5

3√

3

(a

au

)− 32

N

≈ 8.5612 · 10−2

(a

au

)− 32

N. (4.47)

We ask again for the validity of (4.39), viz. when the kinetic energy and theexternal potential can be omitted in the “TF-G” region. For that we compute theneglected terms via (4.42) and obtain

4T = k2 =5

3

1aau

, (4.48)

4Vext = 48γ2

k2=

48 · 35

a

auγ2 =

144

5

a

auγ2. (4.49)


Next we compare them with the contact potential from (4.39). First we get forthe kinetic energy

4T = k2 ¿ 1

2Vc =

1

4

a

auNk3,

1

N aau

¿ 1

4k =

1

4

√5

3

1√aau

,

1

N√

aau

¿ 1

4

√5

3,

a

auÀ 48

5

1

N2≈ 9.6

1

N2(4.50)

and for the external potential

4Vext =144

5

a

auγ2 ¿ 1

4Na

auk3 =

1

4Na

au

(5

3

) 32(a

au

)− 32

=1

4N

(5

3

) 32(a

au

)− 12

,

γ2 ¿ 1

4N

(5

3

) 32 5

144

(a

au

)− 32

≈ 1.8678 · 10−2N

(a

au

)− 32

.

(4.51)

The kinetic energy is negligible if

a

auÀ 48

5

1

N2≈ 9.6

1

N2(4.52)

or rewritten

N À 4

√3

5

1√aau

≈ 3.09841√aau

. (4.53)

For a typical BEC with gravity-like interaction we assume the values given by[O’Dell et al. (2000)]

a ≈ 3 · 10−9 m,

au ≈ 10 cm = 1 · 10−1 m

⇒ a

au≈ 3 · 10−8,

√a

au≈ 1.7321 · 10−4 (4.54)

to give an estimate of the order of magnitude for the particle number N andthe parameter γ in the “TF-G” region. We insert (4.54) in (4.53) and (4.51) and


obtain

N À 3.0984

1.7321 · 10−4≈ 16970.5628 ∼ 1.7 · 104, (4.55)

γ2 ¿ 1.8678 · 10−2N(3 · 10−8

)− 32 ≈ 3.5946 · 109N,

γ ¿ 5.9955 · 104√N. (4.56)

3) Case: N2 aauÀ 1

In this limit there is no need for the Taylor expansion (4.28) of (4.26). We canjust omit the factors 1 in the radicand and −1 in the brackets. This leads us to

kmin =4

3N aau

√15

16N2

a

au=

√5

3

1√aau

. (4.57)

This limiting case is similar to the case, where the kinetic energy is neglected, as(4.57) is identical to (4.42).

4) Case: negative scattering length

For a negative scattering length we have to assume that the radicand of (4.26)must be greater than or equal to zero

1 +15

16N2 a

au≥ 0,

1 ≥ 15

16N2

∣∣∣∣a

au

∣∣∣∣ ,

N2 = N2cr ≤

16

15

1∣∣∣ aau

∣∣∣, (4.58)

leading us to

Ncr =

√16

15

1√∣∣∣ aau

∣∣∣≈ 1.0328

1√∣∣∣ aau

∣∣∣. (4.59)

Rewriting (4.58) we have

∣∣∣∣a

au

∣∣∣∣cr

=

∣∣∣∣a

au

∣∣∣∣ =16

15

1

N2≈ 1.0667

1

N2. (4.60)

Again we can insert the values of (4.54) in (4.59) and obtain

Ncr ≈ 5.6569 · 103 ≈ 6 · 103. (4.61)


A “collapse” of the condensate occurs for negative scattering lengths, when theparticle number N exceeds a critical value Ncr because the energy due to thecontact potential is dominant at small radii. For positive scattering lengths thegravitational-like attraction between particles does not show this kind of insta-bility, since, at short radii, it is always weaker than the kinetic energy. The twonew physical regions are controlled by the balance of the gravity-like potentialwith either the kinetic energy (G) or the s-wave scattering length (TF-G) [O’Dellet al. (2000)].

4.1.2 GTO trial wave function

The advantage of the STO is that the gravity-like potential Vu can be evaluatedanalytically. The disadvantage is that the STO has a cusp-like behaviour forr → 0, while the correct wave function behaves smoothly as r → 0, i.e. has azero gradient. Therefore we assume “Gaussian Type Orbitals” (GTO) for ourvariational calculations. At the end we take the opportunity to compare theenergy of the system calculated on the one hand with STO and on the otherhand with GTO. Either of them will fulfill (4.1), so that the calculated energy isvery closely to the “real” energy of the system.

We start with the GTO trial wave function

ψ(r) = A exp

(−k

2r2

2

). (4.62)

The normalization factor A can be calculated by requiring the wave function(4.62) to be normalized to unity

∫

R3

d3r ψ2(r)!= 1,

4πA2

∫dr r2e−k

2r2 = 1,

4πA2

√π

4k3= 1,

A2 =k3

π32

,

A =k

32

π34

, (4.63)

where we make use of the well known Gaussian integral (C.3). If we now insert(4.63) in (4.62) we obtain the normalized wave function

ψ(r) =k

32

π34

exp

(−k

2r2

2

). (4.64)


In the following we calculate all quantities as we did above with the STO. Aftercalculating the kinetic energy, the external potential, the contact potential, thegravity-like potential, the root-mean-square radius, the peak density, and themean-field energy of the system we consider the two limit regions. Minimizingthe energy leads to the “best” parameter k of the wave function (4.64). Alsowe find a critical number of particle Ncr for negative s-wave scattering lengths,so that a “collapse” of the condensate can occur, because the balance betweenthe gravity-like potential and either the kinetic energy or the s-wave scatteringlength is disturbed. First of all we start with the kinetic energy (3.42), in whichwe insert (4.64) and obtain

〈T (r)〉 = 〈−∆r〉

=

⟨− ∂2

∂r2− 2

r

∂

∂r

⟩

= −∫

R3

d3r

(ψ(r)ψ(r) +

2

rψ(r)ψ(r)

)

= −4π

∞∫

0

dr r2

(ψ(r)ψ(r) +

2

rψ(r)ψ(r)

)

= −4πk3

π32

∞∫

0

dr r2(r2k4e−k

2r2 − k2e−k2r2 − 2k2e−k

2r2)

= −4k3

√π

(3√πk4

8k5−√πk2

4k3−√π2k2

4k3

)

=3

2k2, (4.65)

using the abbreviations

∂2

∂r2ψ = ψ,

∂

∂rψ = ψ.

The kinetic energy is proportional to the square of the parameter k. Next we


compute the harmonic trap potential (3.44)

〈Vext(r)〉 =⟨γ2r2

⟩

=

∫

R3

d3r γ2r2ψ(r)2

= 4πk3

π32

γ2

∞∫

0

dr r4e−k2r2

= 4πk3

π32

γ2 3√π

8k5

=3

2

γ2

k2, (4.66)

which is proportional to the inverse square of the parameter k. In our case of aself-bound BEC this trapping potential is switched off. For the contact potential(3.47) we get

〈Vc(r)〉 =

⟨8π

a

au|ψ(r)|2

⟩

=

∫

R3

d3r 8πa

auψ(r)4

= 4π8πa

au

k6

π3

∞∫

0

dr r2e−2k2r2 (4.67)

= 32π2 a

au

k6

π3

√π

4(√

2k)3(4.68)

= 2

√2

π

a

auk3, (4.69)

which is proportional to the cube of the parameter k. The gravity-like potential(3.48) can not be transformed in an analytical form as in (4.8) because we nowhave a Gaussian ansatz for the wave function. To calculate its expectation value,we introduce center of mass coordinates4

R =1

2(r1 + r2), (4.70)

X =1

2(x1 + x2), etc. (4.71)

and relative coordinates

r = (r2 − r1), (4.72)

x = (x2 − x1), etc.. (4.73)

4The center of mass coordinate system is a coordinate system in which the center of massof the system remains still.


The Jacobian reads ∣∣∣∣12

12

−1 1

∣∣∣∣ =1

2−

(−1

2

)= 1, (4.74)

viz.dx1 dx2 = dX dx, (4.75)

and the same for the y and z-components. From this follows, that

d3r1 d3r2 = d3R d3r. (4.76)

We make use of the following transformation

4R2 + r2 = r21 + r2

2 + 2r1r2 + r21 + r2

2 − 2r1r2 = 2(r21 + r2

2), (4.77)

and dividing (4.77) by 2 we obtain

2R2 +r2

2= r2

1 + r22. (4.78)

The expectation value of the gravity-like potential for (4.64) reads

〈Vu(r)〉 =

⟨∫d3 r1

|ψ(r1)|2|r2 − r1|

⟩

=

∫

R3

d3r2

∫

R3

d3r1|ψ(r1)|2|r2 − r1|ψ(r2)

2

=k6

π3

∫d3r1

∫d3r2

e−k2(r2

1+r22)

|r2 − r1| . (4.79)

We insert (4.76) and (4.78) in (4.79) and we finally obtain

〈Vu(r)〉 =k6

π3

∫d3R

∫d3r

e−k2(2R2+ r2

2)

|r|

=k6

π3

4π

∞∫

0

dRR2e−2k2R2

4π

∞∫

0

dr r2e−12k2r2 1

r

=k6

π3

4π

∞∫

0

dRR2e−2k2R2

4π

∞∫

0

dr re−12k2r2

=k6

π3

(4π

√π

4

1

(√

2k)3

)(4π

1

k2

)

=k6

π3π√π

1

2√

2k34π

1

k2

=k6

π3

2π2√π√

2

1

k5=k6

π3π2√

2π1

k5

=

√2

πk, (4.80)


whence in “atomic” units (3.48)

〈Vu(r)〉 =

⟨−2

∫d3r′

|ψ(r′)|2|r − r′|

⟩= −2A4

∫d3r

∫d3r′

e−k2(r2+r′2)

|r − r′|= −2A4π2

√π√

21

k5

= −2

√2

πk. (4.81)

The gravity-like potential is proportional to k. We insert (4.65),(4.66),(4.69) and(4.81) into (3.58) and obtain the mean-field energy of the system as a function ofk

E(k) = N

[3

2k2 +

3

2

γ2

k2+

√2

π

a

auNk3 −

√2

πNk

]

=3

2N

[k2 +

γ2

k2+

2

3

√2

π

a

auNk3 − 2

3

√2

πNk

], (4.82)

which we can minimize in order to obtain the optimum width parameter k. Wecalculate the expressions for the root-mean-square radius

〈rrms〉 =

√√√√⟨

N∑i=1

r2i

⟩

=

√N

∫dr r2r2ψ2

=

√√√√√4πk3

π32

N

∞∫

0

dr r4e−k2r2

=

√4π

k3

π32

N3√π

8k5

=

√3

2

1

k2N (4.83)

and the peak density at the center of the condensate

〈ρmax〉 =

⟨N∑i=1

δ(ri)

⟩= N |ψ(0)|2 = NA2 =

k3

π32

N. (4.84)

Now we switch off the harmonic trap potential, viz. γ = 0, because the condensateis assumed to be self-bound. Thus the second term in (4.82) can be omitted and


we obtain

E(k) =3

2N

[k2 +

2

3

√2

π

a

auNk3 − 2

3

√2

πNk

]

=3

2

[Nk2 +

2

3

√2

π

a

auN2k3 − 2

3

√2

πN2k

]. (4.85)

For the minimum of (4.85), we differentiate (4.85) with respect to k and obtain

E ′(k) =3

2

[2Nk + 2

√2

π

a

auN2k2 − 2

3N2

√2

π

]= 0. (4.86)


2Nk + 2

√2

π

a

auN2k2 − 2

3N2

√2

π= 0

k +

√2

π

a

auNk2 − 1

3N

√2

π= 0

k2 +

√π

2

1

N aau

k − 1

3 aau

= 0. (4.87)

Now the quadratic equation (4.87) is solved with the formula

kmin =1

2

−

√π

2

1

N aau

±

√√√√(√

π

2

1

N aau

)2

+4

3 aau

=1

2

−

√π

2

1

N aau

±√π

2

1

N aau

√√√√1 +4

3 aau

(√2

πNa

au

)2

=1

2

√π

2

1

N aau

(±

√1 +

8

3πN2

a

au− 1

). (4.88)

In (4.88) the negative sign is omitted5 so that we finally obtain

kmin ≡ k+ =1

2

√π

2

1

N aau

(√1 +

8

3πN2

a

au− 1

), (4.89)

where k+ denotes the use of the positive sign in (4.88). We now analyze the twophysical regions “TF-G” and “G”. We derive the parameter k starting with the“G” region.

5For the variational solutions including the negative sign we obtain the tangent bifurcationdescribed in chapter 5.7. There we calculate also some numerical values and compare themwith the results obtained with k−.


1) Case: “G”→ contact potential negligible

The s-wave scattering length aau

is negligible so that we can just omit the secondterm in (4.85) and have

E(k) =3

2N

[k2 − 2

3

√2

πNk

]. (4.90)

By means of (4.28) and with α = 12

we get from (4.89)

kmin =1

2

√π

2

1

N aau

(1 +

4

3πN2 a

au− 1

)=

2

3√

2πN

=1

3

√2

πN ≈ 2.6596 · 10−1N. (4.91)

Now we insert (4.91) in (4.65), (4.81), (4.83), (4.84) and (4.90) leading us to

〈T 〉 =3

2k2 =

3

2

(N

3

√2

π

)2

=1

3πN2

≈ 1.0610 · 10−1N2, (4.92)

〈Vu〉 = −2N

√2

πk = −2N

√2

π

N

3

√2

π= − 4

3πN2

≈ −4.2441 · 10−1N2, (4.93)

〈rrms〉 =

√3

2

1

k2N =

√3

2N

9π

2N2=

3

2

√3π

1√N

≈ 4.60501√N, (4.94)

〈ρmax〉 =k3

π32

N =N

π32

N32√

2

27π32

=2√

2

27π3N4

≈ 3.3786 · 10−3N4, (4.95)

E =3

2N

[k2 − 2k2

]= −3

2Nk2 = −3

2N2 1

9

2

πN = − 1

3πN3

≈ −1.0610 · 10−1N3. (4.96)

The energy (4.96) can later be compared with the accurate numerical calculationof the energy in the “G” region (cf. TAB. 5.28, where the results are given inred). Now we ask for the applicability of (4.90). What are the conditions for thescattering length a

auin the contact potential and the parameter γ in the external


potential so that we can omit them. The neglected terms are

2

3Vc =

2

3

√2

π

a

auNk3 =

2

3

√2

π

a

auNN32

√2

27π32

=8

81π2N4 a

au≈ 1.0007 · 10−2N4 a

au,

2

3Vext =

γ2

k2=

9π

2

1

N2γ2 ≈ 14.1372

1

N2γ2.

They are compared now with the kinetic energy from (4.90). For the contactpotential we obtain

2

3Vc =

8

81π2N4 a

au¿ 2

3T = k2 =

2

9πN2,

a

au¿ 9π

4

1

N2≈ 7.0689

1

N2(4.97)

and for the trapping potential

2

3Vext =

9π

2

1

N2γ2 ¿ 2

3T = k2 =

2

9πN2,

γ2 ¿ 4

81π2N4 ≈ 5.0035 · 10−3N4 ≈

(N

3.7599

)4

. (4.98)

Finally we conclude from (4.97) that the contact potential may not be omitted if

a

au& 9π

4

1

N2' 7.0689

1

N2, (4.99)

and from (4.98) that the neglect of the trapping potential is not justified if

γ2 & 4

81π2N4,

γ & 2

9πN2 & 7.0736 · 10−2N2 &

(N

3.7599

)2

. (4.100)

2) Case: “TF-G”→ kinetic energy negligible

In the Thomas-Fermi limit we just omit the kinetic energy of the mean-fieldenergy in (4.85) and obtain

E(k) =3

2N

[2

3

√2

π

a

auNk3 − 2

3

√2

πNk

]= N2

√2

π

(a

auk3 − k

). (4.101)

For the minimum of (4.101) we get

E ′(k) = N2

√2

π

(3a

auk2 − 1

)= 0. (4.102)



3a

auk2min = 1,

k2min =

1

3 aau

,

kmin =1√3 aau

≈ 5.7735 · 10−1 1√aau

. (4.103)

We insert (4.103) into (4.101) and obtain the energy of the system in the “TF-G”region

E = N2

√2

πk

(a

auk2 − 1

)= N2

√2

πk

(1

3− 1

)= −2

3

√2

πN2 1√

3 aau

= − 2√

2

3√

3π

1√aau

N2 ≈ −3.0711 · 10−1 1√aau

N2, (4.104)

which again will be compared with the numerically accurate energy (cf. TAB.5.28, where the results are given in blue). Next we calculate all other quantitiesinserting (4.103) in (4.69), (4.81), (4.83) and (4.84)

〈Vc〉 = 2Na

auk3

√2

π= 2N

a

au

(1√3 aau

)3 √2

π=

2

3

√2

3π

1√aau

N

≈ 3.0711 · 10−1 1√aau

N, (4.105)

〈Vu〉 = −2N

√2

πk = −2N

√2

π

1√3 aau

= −2N

√2

3π

1√aau

≈ −9.2132 · 10−1 1√aau

N, (4.106)

〈rrms〉 =

√N

3

2

1

k2=

√9

2Na

au=

3√2

√N

√a

au

≈ 2.1213√N

√a

au, (4.107)

〈ρmax〉 = Nk3

π32

= N1

π32

(1

3 aau

) 32

=

(1

3π

) 32(a

au

)− 32

N

≈ 3.4562 · 10−2

(a

au

)− 32

N. (4.108)

Now we give an estimate about the scattering length aau

and the parameter γ inorder to be able to make a statement when the contact potential and the external


potential can be omitted in the energy (4.101). For that we compare (4.82) and(4.101) and obtain for the neglected terms

2

3T = k2 =

1

3 aau

, (4.109)

2

3Vext =

γ2

k2= 3

a

auγ2. (4.110)

The contact potential is compared with the kinetic energy

2

3T = k2 ¿ 2

9

√2

3π

1√aau

N, (4.111)

1

3 aau

¿ 2

9

√2

3π

1√aau

N,

1

N√

aau

¿ 2

3

√2

3π,

a

auÀ 27π

8

1

N2≈ 10.6029

1

N2(4.112)

or rewritten

N À√

27π

8

1√aau

≈ 3.2562√aau

(4.113)

and with the external potential

2

3Vext = 3

a

auγ2 ¿ 2

9

√2

3π

1√aau

N,

γ2 ¿ 2

27

√2

3π

(a

au

)− 32

N ≈ 3.4123 · 10−2

(a

au

)− 32

N. (4.114)

We have to include the kinetic energy if

a

au. 27π

8

1

N2≈ 10.6029

1

N2(4.115)

and the external potential if

γ2 & 2

27

√2

3π

(a

au

)− 32

N,

γ &(

2

27

) 12(

2

3π

) 14(a

au

)− 34 √

N ' 1.8472 · 10−1

(a

au

)− 34 √

N (4.116)


in the mean-field energy (4.101), otherwise we can neglect them. For the examplevalues (4.54) we obtain for the particle number

N & 3.2562

1.7321 · 10−4≈ 18799.7120 ∼ 1.9 · 104 (4.117)

and for the parameter γ

γ2 . 3.4123 · 10−2N(3 · 10−8

)− 32 ≈ 6.5670 · 109N, (4.118)

γ . 8.1037 · 104√N. (4.119)

3) Case: N2 aauÀ 1

In this limit we can just omit the factors 1 and −1 in (4.89), which leads us to

k =1

2

√π

2

1

N aau

(√8

3πN2

a

au

)=

√1

3

1√aau

=1√3 aau

. (4.120)

This limiting case is comparable with the case, where the kinetic energy is ne-glected, as (4.120) is identical with (4.103).

4) Case: negative scattering length

We now consider a negative scattering length with the condition that the radicandin (4.89) is still greater than or equal to zero

1 +8

3πN2 a

au≥ 0, (4.121)

1 ≥ − 8

3πN2 a

au, (4.122)

−3π

8≤ N2 a

au(4.123)

or, rewritten by introducing absolute value bars,

1 ≥ 8

3πN2

∣∣∣∣a

au

∣∣∣∣ , (4.124)

3π

8≤ N2

∣∣∣∣a

au

∣∣∣∣ , (4.125)

N2 ≤ 3π

8

1∣∣∣ aau

∣∣∣= N2

cr. (4.126)

This lead us to the critical number of particles

Ncr =

√3π

8

1√∣∣∣ aau

∣∣∣≈ 1.0854

1√∣∣∣ aau

∣∣∣, (4.127)

4.2. Summary 71

and to a critical scattering length by rewriting (4.127)

∣∣∣∣a

au

∣∣∣∣ =3π

8

1

N2cr

≈ 1.17811

N2cr

. (4.128)

Ncr and∣∣∣ aau

∣∣∣ describes the“collapse”of the condensate as mentioned at the end of

subsection 4.1.1. If we insert the example (4.54) in (4.127) we obtain an estimatefor the critical number

Ncr ≈ 6266.5707 ≈ 6 · 103 (4.129)

which is of the same order of magnitude as (4.61).

4.2 Summary

We summarize the variational results of this chapter by comparing the STO andGTO results in the two new physical regions “G” and “TF-G”.

“G” region: N2 aau¿ 1

In the “G” region the contact interaction is negligible.

STO GTO〈T 〉 9.7656 · 10−2N2 1.0610 · 10−1N2

〈Vu〉 −3.9063 · 10−1N2 −4.2441 · 10−1N2

〈rrms〉 5.5426 1√N

4.6050 1√N

〈ρmax〉 9.7140 · 10−3N4 3.3786 · 10−3N4

E −9.7656 · 10−2N3 −1.0610 · 10−1N3

kmin 6.25 · 10−1N 2.6596 · 10−1N

TAB. 4.1: Comparison between STO and GTO

“TF-G” region: N2 aauÀ 1

In the “TF-G” region the kinetic energy is negligible.

STO GTO〈Vc〉 2.6896 · 10−1 1√

aau

N 3.0711 · 10−1 1√a

au

N

〈Vu〉 −8.0687 · 10−1 1√a

au

N −9.2132 · 10−1 1√a

au

N

〈rrms〉 2.6833√N

√aau

2.1213√N

√aau

〈ρmax〉 8.5612 · 10−2(aau

)− 32N 3.4562 · 10−2

(aau

)− 32N

continued on next page


continued from previous page

STO GTOE −2.6896 · 10−1 1√

aau

N2 −3.0711 · 10−1 1√a

au

N2

kmin 1.291 1√a

au

5.7735 · 10−1 1√a

au

TAB. 4.2: Comparison between STO and GTO

If we compare the energies E in the two limit regions, we see that the energy ofthe GTO is lower than the energy of the STO. From this it follows that GTO wavefunction is a better variational approximation. For the full variational calculationin chapter 5 we therefore make use of the GTO trial wave function (4.64).

Chapter 5

Variational and numericalcalculations

In this chapter we introduce the numerical procedures for determining the vari-ational and numerical results. We give a brief overview of the numerical methodwe employ to solve the nonlinear radially symmetric Schrodinger equation andpresent our results in table form. Finally the results are analyzed and physicallyinterpretations are given.

5.1 The numerical procedure

The radial GP system

∆rψN(r) = −U(r)ψN(r) + 8πa

auψN(r)3, (5.1a)

∆rU(r) = −8π|ψN(r)|2, (5.1b)

is transformed for the numerical calculations into

∆rS(r) = −U(r)S(r) + bS(r)3, (5.2a)

∆rU(r) = −|S(r)|2, (5.2b)

using the abbreviations

ψN(r) =1√8πS(r) (5.3)

for the wave function and

b =a

au(5.4)

73

74 Chapter 5. Variational and numerical calculations

for the unscaled s-wave scattering length. The two equations (5.2), which arenon-linear second-order differential equations, are converted into four nonlinearfirst-order differential equations

f [0] = ∂rU, (5.5a)

f [1] = f ′[0] = −|S|2 − 2

r∂rU, (5.5b)

f [2] = ∂rS, (5.5c)

f [3] = f ′[2] = −US − 2

r∂rS + bS3. (5.5d)

These equations (5.5) are solved by means of a fourth-order Runge-Kutta routinewith adaptive step-size control as it is provided by the Gnu Scientific Library(GSL) for ordinary differential equations1.

As a result of the radial symmetry discussed in subsection 3.1.1 we can fixinitial conditions namely that the derivatives ∂rS of the wave function and thepotential ∂rU at r = 0 vanish. Because of the scale invariance (3.28) we can alsochoose the initial value of the wave function S(r = 0) = S0 arbitrarily, so thatwe finally have to find only one initial value namely that for the potential U0.Furthermore we have to pay attention, that the radial part of the Laplacian

∆r =

(∂2

∂r2+

2

r

∂

∂r

)(5.6)

diverges for r → 0, so that we cannot start our calculations at r = 0. Thereforewe start the calculations at r = 10−19. The error arising from it can be neglected.

The correct initial value for the potential U(r = 0) is determined by means ofa nested intervals method. Below the accurate initial value of U(r = 0) the wavefunction S diverges to +∞ from a certain radius on, whereas above this accuratevalue it diverges to −∞. The closer the selected initial value is to the accurateinitial value the later the divergence arises.

5.1.1 Normalization of the numerical solutions

The numerical solutions have to be normalized to the number of particles N . Thenormalization has to be achieved by an adequate scaling of lengths because theHamiltonian is nonlinear. We assume∫

d3r |ψN (r) |2 != N, (5.7)

where the tilde denotes the normalized solutions. Now we replace ψN and r by

ψN = ν2ψN , (5.8a)

r =r

ν, (5.8b)

1http://www.gnu.org/software/gsl

5.1. The numerical procedure 75

as in section 3.2, so that we obtain

ν

∫d3r |ψN (r) |2 !

= N, (5.9)

viz.

ν =N∫

d3r |ψN (r) |2 ≡N

‖ ψN ‖2. (5.10)

Using (3.27) we can now calculate the normalized numerical solutions. Startingwith different initial values of U0, we obtain different non-normalized solutions Sand U as is illustrated in TAB. 5.1 for b = 0

U(r = 0) U0 S0 ν/N S0 = S0ν2

0.2296 1/100 1/4 1.1533 0.33250.2296 2 1/4 1.1533 0.33250.2296 100 1/4 1.1533 0.3325

0.4592 1/100 1/2 0.8155 0.33250.4592 2 1/2 0.8155 0.33250.4592 100 1/2 0.8155 0.3325

0.9185 1/100 1 0.5766 0.33250.9185 2 1 0.5766 0.33250.9185 100 1 0.5766 0.3325

1.8371 1/100 2 0.4077 0.33251.8371 2 2 0.4077 0.33251.8371 100 2 0.4077 0.3325

TAB. 5.1: Numerical calculations for b = 0. U0 and S0 are the initial values,U(r = 0) is the correct final value, ν is the parameter for the scaling, and S0 isthe scaled value at r = 0.

and pictured in FIG. D.1 in appendix D. By scaling them adequately using (3.27)we get equal normalized solutions S and U for different initial values of U0 as onecan see in the last column of TAB. 5.1. The normalized solutions to unity forsome negative and positive scattering lengths are pictured2 in FIG. 5.1 and FIG.5.2. The smaller the scattering length a

auis the shallower is the effective potential

Veff of the condensates because of the dominance of the gravity-like attractionVu compared to the contact potential Vc. The prismatic colors of the two figuresexhibit this fact. The long-wave character of the prismatic colors denotes thedegree of lesser binding of the solutions.

2In appendix E more solutions for different scattering lengths are presented.


0 5 10 15 20 25 30rN

0

0.04

0.08

0.12

0.16

0.2ψ

/N3/

2

N2a = -1.02

N2a = -1

N2a = -0.3

N2a = -0.1

N2a = 0

N2a = 0.1

N2a = 0.3

N2a = 1

N2a = 10

FIG. 5.1: Wave functions at different scattering lengths.

0 5 10 15 20 25 30rN

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

Vef

f/N3

N2a = -1.02

N2a = -1

N2a = -0.3

N2a = -0.1

N2a = 0

N2a = 0.1

N2a = 0.3

N2a = 1

N2a = 10

-1/r

FIG. 5.2: Self-trapping potentials at different scattering lengths. The asymptotic−1/r behaviour is shown for comparison.

5.2. The variational procedure 77

5.2 The variational procedure

We evaluate the equations

T =3

2k2, (5.11)

Vc = 2

√2

π

a

auNk3, (5.12)

Vu = −2

√2

πNk (5.13)

for the kinetic energy, the contact potential and the gravity-like potential, respec-tively, and

rrms =

√3

2

1

k2N, (5.14)

ρmax =k3

π32

N (5.15)

for the root-mean-square radius and the peak density at the center of the con-densate, respectively, and the mean-field energy of the system

E(k) =3

2N

[k2 +

2

3

√2

π

a

auNk3 − 2

3

√2

πNk

](5.16)

by varying the scattering length aau

and the number of particles N in the optimumparameter k

kmin ≡ k+ =1

2

√π

2

1

N aau

(√1 +

8

3πN2

a

au− 1

). (5.17)

Obviously for the chemical potential applies

µ = T + Vc + Vu = T + Veff , (5.18)

whereVeff = Vc + Vu, (5.19)

is the effective self-binding potential.

5.3 Calculations for different number of parti-

cles N

In the following we tabulate results of our numerical and variational calculationsfor different numbers of particles N in“atomic”units and reveal the appearance oftwo new physical regions for self-trapped BEC, where the physics of self-trappingdepends only on the parameter N2 a

au.


5.3.1 Calculations for N = 102

b aau

N2 aau

rrms ρmax E

7.902 10+00 10+04 2.0082 · 10+01 2.4842 · 10+00 −3.1375 · 10+03

4.367 10−01 10+03 6.6002 · 10+00 7.5798 · 10+01 −9.4217 · 10+03

2.3 10−02 10+02 2.3089 · 10+00 2.0894 · 10+03 −2.6335 · 10+04

1 10−03 10+01 9.4600 · 10−01 3.9300 · 10+04 −6.0762 · 10+04

0.25 10−04 10+00 5.5099 · 10−01 2.4405 · 10+05 −9.5568 · 10+04

0.033 10−05 10−01 4.7396 · 10−01 4.0743 · 10+05 −1.0675 · 10+05

3.35 · 10−03 10−06 10−02 4.6445 · 10−01 4.3655 · 10+05 −1.0827 · 10+05

3.35 · 10−04 10−07 10−03 4.6350 · 10−01 4.3962 · 10+05 −1.0842 · 10+05

3.35 · 10−05 10−08 10−04 4.6340 · 10−01 4.3993 · 10+05 −1.0844 · 10+05

3.35 · 10−06 10−09 10−05 4.6339 · 10−01 4.3996 · 10+05 −1.0844 · 10+05

TAB. 5.2: Numerical calculations for N = 102

aau

T Vu Vc µ

10+00 6.7061 · 10−01 −9.4512 · 10+01 3.0361 · 10+01 −6.3451 · 10+01

10−01 5.6426 · 10+00 −2.8826 · 10+02 8.8320 · 10+01 −1.9418 · 10+02

10−02 4.2918 · 10+01 −8.3376 · 10+02 2.2034 · 10+02 −5.7006 · 10+02

10−03 2.5152 · 10+02 −2.0767 · 10+03 3.5589 · 10+02 −1.4680 · 10+03

10−04 7.5738 · 10+02 −3.6284 · 10+03 1.9781 · 10+02 −2.6710 · 10+03

10−05 1.0342 · 10+03 −4.2416 · 10+03 3.2674 · 10+01 −3.1720 · 10+03

10−06 1.0786 · 10+03 −4.3317 · 10+03 3.4188 · 10+00 −3.2469 · 10+03

10−07 1.0833 · 10+03 −4.3410 · 10+03 3.4293 · 10−01 −3.2546 · 10+03

10−08 1.0837 · 10+03 −4.3419 · 10+03 3.4304 · 10−02 −3.2554 · 10+03

10−09 1.0838 · 10+03 −4.3420 · 10+03 3.4305 · 10−03 −3.2554 · 10+03


N2 aau

kmin rrms ρmax E

10+04 5.7111 · 10−01 2.1444 · 10+01 3.3454 · 10+00 −3.0215 · 10+03

10+03 1.7641 · 10+00 6.9424 · 10+00 9.8601 · 10+01 −9.2283 · 10+03

10+02 5.1807 · 10+00 2.3640 · 10+00 2.4972 · 10+03 −2.6215 · 10+04

10+01 1.3036 · 10+00 9.3948 · 10−01 3.9787 · 10+04 −6.0846 · 10+04

10+00 2.2541 · 10+01 5.4332 · 10−01 2.0570 · 10+05 −9.4498 · 10+04

10−01 2.6054 · 10+01 4.7007 · 10−01 3.1763 · 10+05 −1.0464 · 10+05

10−02 2.6539 · 10+01 4.6147 · 10−01 3.3571 · 10+05 −1.0595 · 10+05


5.3. Calculations for different number of particles N 79


N2 aau

kmin rrms ρmax E

10−03 2.6590 · 10+01 4.6059 · 10−01 3.3764 · 10+05 −1.0608 · 10+05

10−04 2.6595 · 10+01 4.6050 · 10−01 3.3783 · 10+05 −1.0610 · 10+05

10−05 2.6596 · 10+01 4.6049 · 10−01 3.3785 · 10+05 −1.0610 · 10+05

TAB. 5.4: Variational calculations with k+ for N = 102

aau

T Vu Vc µ

10+00 4.8926 · 10−01 −9.1137 · 10+01 2.9726 · 10+01 −6.0921 · 10+01

10−01 4.6683 · 10+00 −2.8151 · 10+02 8.7614 · 10+01 −1.8923 · 10+02

10−02 4.0260 · 10+01 −8.2672 · 10+02 2.2189 · 10+02 −5.6457 · 10+02

10−03 2.5492 · 10+02 −2.0803 · 10+03 3.5354 · 10+02 −1.4718 · 10+03

10−04 7.6220 · 10+02 −3.5971 · 10+03 1.8278 · 10+02 −2.6521 · 10+03

10−05 1.0182 · 10+03 −4.1576 · 10+03 2.8224 · 10+01 −3.1112 · 10+03

10−06 1.0565 · 10+03 −4.2351 · 10+03 2.9831 · 10+00 −3.1756 · 10+03

10−07 1.0605 · 10+03 −4.2432 · 10+03 3.0001 · 10−01 −3.1823 · 10+03

10−08 1.0609 · 10+03 −4.2440 · 10+03 3.0019 · 10−02 −3.1830 · 10+03

10−09 1.0610 · 10+03 −4.2441 · 10+03 3.0020 · 10−03 −3.1830 · 10+03



b aau

N2 aau

rrms ρmax E

7.902 10−04 10+04 2.0082 · 10+00 2.4842 · 10+08 −3.1375 · 10+09

4.367 10−05 10+03 6.6002 · 10−01 7.5798 · 10+09 −9.4217 · 10+09

2.3 10−06 10+02 2.3089 · 10−01 2.0894 · 10+11 −2.6335 · 10+10

1 10−07 10+01 9.4600 · 10−02 3.9300 · 10+12 −6.0762 · 10+10

0.25 10−08 10+00 5.5099 · 10−02 2.4405 · 10+13 −9.5568 · 10+10

0.033 10−09 10−01 4.7396 · 10−02 4.0743 · 10+13 −1.0675 · 10+11

3.35 · 10−03 10−10 10−02 4.6445 · 10−02 4.3655 · 10+13 −1.0827 · 10+11

3.35 · 10−04 10−11 10−03 4.6350 · 10−02 4.3962 · 10+13 −1.0842 · 10+11

3.35 · 10−05 10−12 10−04 4.6340 · 10−02 4.3993 · 10+13 −1.0844 · 10+11

3.35 · 10−06 10−13 10−05 4.6339 · 10−02 4.3996 · 10+13 −1.0844 · 10+11



aau

T Vu Vc µ

10−04 6.7061 · 10+03 −9.4512 · 10+05 3.0361 · 10+05 −6.3451 · 10+05

10−05 5.6426 · 10+04 −2.8826 · 10+06 8.8320 · 10+05 −1.9418 · 10+06

10−06 4.2918 · 10+05 −8.3376 · 10+06 2.2034 · 10+06 −5.7006 · 10+06

10−07 2.5152 · 10+06 −2.0767 · 10+06 3.5589 · 10+06 −1.4680 · 10+07

10−08 7.5738 · 10+06 −3.6284 · 10+07 1.9781 · 10+06 −2.6710 · 10+07

10−09 1.0342 · 10+07 −4.2416 · 10+07 3.2674 · 10+05 −3.1720 · 10+07

10−10 1.0786 · 10+07 −4.3317 · 10+07 3.4188 · 10+04 −3.2469 · 10+07

10−11 1.0833 · 10+07 −4.3410 · 10+07 3.4293 · 10+03 −3.2469 · 10+07

10−12 1.0837 · 10+07 −4.3419 · 10+07 3.4304 · 10+02 −3.2554 · 10+07

10−13 1.0838 · 10+07 −4.3420 · 10+07 3.4305 · 10+01 −3.2554 · 10+07


N2 aau

kmin rrms ρmax E

10+04 5.7111 · 10+01 2.1444 · 10+00 3.3454 · 10+08 −3.0215 · 10+09

10+03 1.7641 · 10+02 6.9424 · 10−01 9.8601 · 10+09 −9.2283 · 10+09

10+02 5.1807 · 10+02 2.3640 · 10−01 2.4972 · 10+11 −2.6215 · 10+10

10+01 1.3036 · 10+03 9.3948 · 10−02 3.9787 · 10+12 −6.0846 · 10+10

10+00 2.2541 · 10+03 5.4332 · 10−02 2.0570 · 10+13 −9.4498 · 10+10

10−01 2.6054 · 10+03 4.7007 · 10−02 3.1763 · 10+13 −1.0464 · 10+11

10−02 2.6539 · 10+03 4.6147 · 10−02 3.3571 · 10+13 −1.0595 · 10+11

10−03 2.6590 · 10+03 4.6059 · 10−02 3.3764 · 10+13 −1.0608 · 10+11

10−04 2.6595 · 10+03 4.6050 · 10−02 3.3783 · 10+13 −1.0610 · 10+11

10−05 2.6596 · 10+03 4.6049 · 10−02 3.3785 · 10+13 −1.0610 · 10+11


aau

T Vu Vc µ

10−04 4.8926 · 10+03 −9.1137 · 10+05 2.9726 · 10+05 −6.0921 · 10+05

10−05 4.6683 · 10+04 −2.8151 · 10+06 8.7614 · 10+05 −1.8923 · 10+06

10−06 4.0260 · 10+05 −8.2672 · 10+06 2.2189 · 10+06 −5.6457 · 10+06

10−07 2.5492 · 10+06 −2.0803 · 10+07 3.5354 · 10+06 −1.4718 · 10+07

10−08 7.6220 · 10+06 −3.5971 · 10+07 1.8278 · 10+06 −2.6521 · 10+07

10−09 1.0182 · 10+07 −4.1576 · 10+07 2.8224 · 10+05 −3.1112 · 10+07

10−10 1.0565 · 10+07 −4.2351 · 10+07 2.9831 · 10+04 −3.1756 · 10+07

10−11 1.0605 · 10+07 −4.2432 · 10+07 3.0001 · 10+03 −3.1823 · 10+07

10−12 1.0609 · 10+07 −4.2440 · 10+07 3.0019 · 10+02 −3.1830 · 10+07



continued from previous pageaau

T Vu Vc µ

10−13 1.0610 · 10+07 −4.2441 · 10+07 3.0020 · 10+01 −3.1830 · 10+07



b aau

N2 aau

rrms ρmax E

7.902 10−08 10+04 2.0082 · 10−01 2.4842 · 10+16 −3.1375 · 10+15

4.367 10−09 10+03 6.6002 · 10−02 7.5798 · 10+17 −9.4217 · 10+15

2.3 10−10 10+02 2.3089 · 10−02 2.0894 · 10+19 −2.6335 · 10+16

1 10−11 10+01 9.4600 · 10−03 3.9300 · 10+20 −6.0762 · 10+16

0.25 10−12 10+00 5.5099 · 10−03 2.4405 · 10+21 −9.5568 · 10+16

0.033 10−13 10−01 4.7396 · 10−03 4.0743 · 10+21 −1.0675 · 10+17

3.35 · 10−03 10−14 10−02 4.6445 · 10−03 4.3655 · 10+21 −1.0827 · 10+17

3.35 · 10−04 10−15 10−03 4.6350 · 10−03 4.3962 · 10+21 −1.0842 · 10+17

3.35 · 10−05 10−16 10−04 4.6340 · 10−03 4.3993 · 10+21 −1.0844 · 10+17

3.35 · 10−06 10−17 10−05 4.6339 · 10−03 4.3996 · 10+21 −1.0844 · 10+17


aau

T Vu Vc µ

10−08 6.7061 · 10+07 −9.4512 · 10+09 3.0361 · 10+09 −6.3451 · 10+09

10−09 5.6426 · 10+08 −2.8826 · 10+10 8.8320 · 10+09 −1.9418 · 10+10

10−10 4.2918 · 10+09 −8.3376 · 10+10 2.2034 · 10+10 −5.7006 · 10+10

10−11 2.5152 · 10+10 −2.0767 · 10+10 3.5589 · 10+10 −1.4680 · 10+11

10−12 7.5738 · 10+10 −3.6284 · 10+11 1.9781 · 10+10 −2.6710 · 10+11

10−13 1.0342 · 10+11 −4.2416 · 10+11 3.2674 · 10+09 −3.1720 · 10+11

10−14 1.0786 · 10+11 −4.3317 · 10+11 3.4188 · 10+08 −3.2469 · 10+11

10−15 1.0833 · 10+11 −4.3410 · 10+11 3.4293 · 10+07 −3.2469 · 10+11

10−16 1.0837 · 10+11 −4.3419 · 10+11 3.4304 · 10+06 −3.2554 · 10+11

10−17 1.0838 · 10+11 −4.3420 · 10+11 3.4305 · 10+05 −3.2554 · 10+11


N2 aau

kmin rrms ρmax E

10+04 5.7111 · 10+03 2.1444 · 10−01 3.3454 · 10+16 −3.0215 · 10+15




N2 aau

kmin rrms ρmax E

10+03 1.7641 · 10+04 6.9424 · 10−02 9.8601 · 10+17 −9.2283 · 10+15

10+02 5.1807 · 10+04 2.3640 · 10−02 2.4972 · 10+19 −2.6215 · 10+16

10+01 1.3036 · 10+05 9.3948 · 10−03 3.9787 · 10+20 −6.0846 · 10+16

10+00 2.2541 · 10+05 5.4332 · 10−03 2.0570 · 10+21 −9.4498 · 10+16

10−01 2.6054 · 10+05 4.7007 · 10−03 3.1763 · 10+21 −1.0464 · 10+17

10−02 2.6539 · 10+05 4.6147 · 10−03 3.3571 · 10+21 −1.0595 · 10+17

10−03 2.6590 · 10+05 4.6059 · 10−03 3.3764 · 10+21 −1.0608 · 10+17

10−04 2.6595 · 10+05 4.6050 · 10−03 3.3783 · 10+21 −1.0610 · 10+17

10−05 2.6596 · 10+05 4.6049 · 10−03 3.3785 · 10+21 −1.0610 · 10+17


aau

T Vu Vc µ

10−08 4.8926 · 10+07 −9.1137 · 10+09 2.9726 · 10+09 −6.0921 · 10+09

10−09 4.6683 · 10+08 −2.8151 · 10+10 8.7614 · 10+09 −1.8923 · 10+10

10−10 4.0260 · 10+09 −8.2672 · 10+10 2.2189 · 10+10 −5.6457 · 10+10

10−11 2.5492 · 10+10 −2.0803 · 10+11 3.5354 · 10+10 −1.4718 · 10+11

10−12 7.6220 · 10+10 −3.5971 · 10+11 1.8278 · 10+10 −2.6521 · 10+11

10−13 1.0182 · 10+11 −4.1576 · 10+11 2.8224 · 10+09 −3.1112 · 10+11

10−14 1.0565 · 10+11 −4.2351 · 10+11 2.9831 · 10+08 −3.1756 · 10+11

10−15 1.0605 · 10+11 −4.2432 · 10+11 3.0001 · 10+07 −3.1823 · 10+11

10−16 1.0609 · 10+11 −4.2440 · 10+11 3.0019 · 10+06 −3.1830 · 10+11

10−17 1.0610 · 10+11 −4.2441 · 10+11 3.0020 · 10+05 −3.1830 · 10+11



b aau

N2 aau

rrms ρmax E

7.902 10−12 10+04 2.0082 · 10−02 2.4842 · 10+24 −3.1375 · 10+21

4.367 10−13 10+03 6.6002 · 10−03 7.5798 · 10+25 −9.4217 · 10+21

2.3 10−14 10+02 2.3089 · 10−03 2.0894 · 10+27 −2.6335 · 10+22

1 10−15 10+01 9.4600 · 10−04 3.9300 · 10+28 −6.0762 · 10+22

0.25 10−16 10+00 5.5099 · 10−04 2.4405 · 10+29 −9.5568 · 10+22

0.033 10−17 10−01 4.7396 · 10−04 4.0743 · 10+29 −1.0675 · 10+23

3.35 · 10−03 10−18 10−02 4.6445 · 10−04 4.3655 · 10+29 −1.0827 · 10+23

3.35 · 10−04 10−19 10−03 4.6350 · 10−04 4.3962 · 10+29 −1.0842 · 10+23




b aau

N2 aau

rrms ρmax E

3.35 · 10−05 10−20 10−04 4.6340 · 10−04 4.3993 · 10+29 −1.0844 · 10+23

3.35 · 10−06 10−21 10−05 4.6339 · 10−04 4.3996 · 10+29 −1.0844 · 10+23


aau

T Vu Vc µ

10−12 6.7061 · 10+11 −9.4512 · 10+13 3.0361 · 10+13 −6.3451 · 10+13

10−13 5.6426 · 10+12 −2.8826 · 10+14 8.8320 · 10+13 −1.9418 · 10+14

10−14 4.2918 · 10+13 −8.3376 · 10+14 2.2034 · 10+14 −5.7006 · 10+14

10−15 2.5152 · 10+14 −2.0767 · 10+14 3.5589 · 10+14 −1.4680 · 10+15

10−16 7.5738 · 10+14 −3.6284 · 10+15 1.9781 · 10+14 −2.6710 · 10+15

10−17 1.0342 · 10+15 −4.2416 · 10+15 3.2674 · 10+13 −3.1720 · 10+15

10−18 1.0786 · 10+15 −4.3317 · 10+15 3.4188 · 10+12 −3.2469 · 10+15

10−19 1.0833 · 10+15 −4.3410 · 10+15 3.4293 · 10+11 −3.2469 · 10+15

10−20 1.0837 · 10+15 −4.3419 · 10+15 3.4304 · 10+10 −3.2554 · 10+15

10−21 1.0838 · 10+15 −4.3420 · 10+15 3.4305 · 10+09 −3.2554 · 10+15


N2 aau

kmin rrms ρmax E

10+04 5.7111 · 10+05 2.1444 · 10−02 3.3454 · 10+24 −3.0215 · 10+21

10+03 1.7641 · 10+06 6.9424 · 10−03 9.8601 · 10+25 −9.2283 · 10+21

10+02 5.1807 · 10+06 2.3640 · 10−03 2.4972 · 10+27 −2.6215 · 10+22

10+01 1.3036 · 10+07 9.3948 · 10−04 3.9787 · 10+28 −6.0846 · 10+22

10+00 2.2541 · 10+07 5.4332 · 10−04 2.0570 · 10+29 −9.4498 · 10+22

10−01 2.6054 · 10+07 4.7007 · 10−04 3.1763 · 10+29 −1.0464 · 10+23

10−02 2.6539 · 10+07 4.6147 · 10−04 3.3571 · 10+29 −1.0595 · 10+23

10−03 2.6590 · 10+07 4.6059 · 10−04 3.3764 · 10+29 −1.0608 · 10+23

10−04 2.6595 · 10+07 4.6050 · 10−04 3.3783 · 10+29 −1.0610 · 10+23

10−05 2.6596 · 10+07 4.6049 · 10−04 3.3785 · 10+29 −1.0610 · 10+23


aau

T Vu Vc µ

10−12 4.8926 · 10+11 −9.1137 · 10+13 2.9726 · 10+13 −6.0921 · 10+13




T Vu Vc µ

10−13 4.6683 · 10+12 −2.8151 · 10+14 8.7614 · 10+13 −1.8923 · 10+14

10−14 4.0260 · 10+13 −8.2672 · 10+14 2.2189 · 10+14 −5.6457 · 10+14

10−15 2.5492 · 10+14 −2.0803 · 10+15 3.5354 · 10+14 −1.4718 · 10+15

10−16 7.6220 · 10+14 −3.5971 · 10+15 1.8278 · 10+14 −2.6521 · 10+15

10−17 1.0182 · 10+15 −4.1576 · 10+15 2.8224 · 10+13 −3.1112 · 10+15

10−18 1.0565 · 10+15 −4.2351 · 10+15 2.9831 · 10+12 −3.1756 · 10+15

10−19 1.0605 · 10+15 −4.2432 · 10+15 3.0001 · 10+11 −3.1823 · 10+15

10−20 1.0609 · 10+15 −4.2440 · 10+15 3.0019 · 10+10 −3.1830 · 10+15

10−21 1.0610 · 10+15 −4.2441 · 10+15 3.0020 · 10+09 −3.1830 · 10+15


5.3.5 “TF-G” region: N 2 aau

= 10+04

For a large parameter N2 aau

we are well located in the “TF-G” region, where thekinetic energy T is negligible in respect of the effective self-binding potential Veff ,where the contact potential Vc balances the gravity-like potential Vu. This canbe related to the outcomes of TAB. 5.2 - TAB. 5.17. For the calculations in the“TF-G” region we fix therefore a representative parameter of N2 a

au= 10+04.

N aau

rrms ρmax E

10+01 10+02 2.0082 · 10+1.5 2.4842 · 10−04 −3.1375 · 10+00

10+02 10+00 2.0082 · 10+01 2.4842 · 10+00 −3.1375 · 10+03

10+03 10−02 2.0082 · 10+0.5 2.4842 · 10+04 −3.1375 · 10+06

10+04 10−04 2.0082 · 10+00 2.4842 · 10+08 −3.1375 · 10+09

10+05 10−06 2.0082 · 10−0.5 2.4842 · 10+12 −3.1375 · 10+12

10+06 10−08 2.0082 · 10−01 2.4842 · 10+16 −3.1375 · 10+15

10+07 10−10 2.0082 · 10−1.5 2.4842 · 10+20 −3.1375 · 10+18

10+08 10−12 2.0082 · 10−02 2.4842 · 10+24 −3.1375 · 10+21

10+09 10−14 2.0082 · 10−2.5 2.4842 · 10+28 −3.1375 · 10+24

TAB. 5.18: Numerical calculations for N2 aau

= 10+04

N T Vu Vc µ10+01 6.7061 · 10−03 −9.4512 · 10−01 3.0361 · 10−01 −6.3451 · 10−01

10+02 6.7061 · 10−01 −9.4512 · 10+01 3.0361 · 10+01 −6.3451 · 10+01

10+03 6.7061 · 10+01 −9.4512 · 10+03 3.0361 · 10+03 −6.3451 · 10+03

10+04 6.7061 · 10+03 −9.4512 · 10+05 3.0361 · 10+05 −6.3451 · 10+05




N T Vu Vc µ10+05 6.7061 · 10+05 −9.4512 · 10+07 3.0361 · 10+07 −6.3451 · 10+07

10+06 6.7061 · 10+07 −9.4512 · 10+09 3.0361 · 10+09 −6.3451 · 10+09

10+07 6.7061 · 10+09 −9.4512 · 10+11 3.0361 · 10+11 −6.3451 · 10+11

10+08 6.7061 · 10+11 −9.4512 · 10+13 3.0361 · 10+13 −6.3451 · 10+13

10+09 6.7061 · 10+13 −9.4512 · 10+15 3.0361 · 10+15 −6.3451 · 10+15


= 10+04

N aau

kmin rrms E

10+01 10+02 5.7111 · 10−02 2.1444 · 10+1.5 −3.0215 · 10+00

10+02 10+00 5.7111 · 10−01 2.1444 · 10+01 −3.0215 · 10+03

10+03 10−02 5.7111 · 10+00 2.1444 · 10+0.5 −3.0215 · 10+06

10+04 10−04 5.7111 · 10+01 2.1444 · 10+00 −3.0215 · 10+09

10+05 10−06 5.7111 · 10+02 2.1444 · 10−0.5 −3.0215 · 10+12

10+06 10−08 5.7111 · 10+03 2.1444 · 10−01 −3.0215 · 10+15

10+07 10−10 5.7111 · 10+04 2.1444 · 10−1.5 −3.0215 · 10+18

10+08 10−12 5.7111 · 10+05 2.1444 · 10−02 −3.0215 · 10+21

10+09 10−14 5.7111 · 10+06 2.1444 · 10−2.5 −3.0215 · 10+24

TAB. 5.20: Variational calculations with k+ for N2 aau

= 10+04

N Veff ρmax10+01 −6.1411 · 10−01 3.3454 · 10−04

10+02 −6.1411 · 10+01 3.3454 · 10+00

10+03 −6.1411 · 10+03 3.3454 · 10+04

10+04 −6.1411 · 10+05 3.3454 · 10+08

10+05 −6.1411 · 10+07 3.3454 · 10+12

10+06 −6.1411 · 10+09 3.3454 · 10+16

10+07 −6.1411 · 10+11 3.3454 · 10+20

10+08 −6.1411 · 10+13 3.3454 · 10+24

10+09 −6.1411 · 10+15 3.3454 · 10+28


= 10+04

N T Vu Vc µ10+01 4.8926 · 10−03 −9.1137 · 10−01 2.9726 · 10−01 −6.0921 · 10−01




N T Vu Vc µ10+02 4.8926 · 10−01 −9.1137 · 10+01 2.9726 · 10+01 −6.0921 · 10+01

10+03 4.8926 · 10+01 −9.1137 · 10+03 2.9726 · 10+03 −6.0921 · 10+03

10+04 4.8926 · 10+03 −9.1137 · 10+05 2.9726 · 10+05 −6.0921 · 10+05

10+05 4.8926 · 10+05 −9.1137 · 10+07 2.9726 · 10+07 −6.0921 · 10+07

10+06 4.8926 · 10+07 −9.1137 · 10+09 2.9726 · 10+09 −6.0921 · 10+09

10+07 4.8926 · 10+09 −9.1137 · 10+11 2.9726 · 10+11 −6.0921 · 10+11

10+08 4.8926 · 10+11 −9.1137 · 10+13 2.9726 · 10+13 −6.0921 · 10+13

10+09 4.8926 · 10+13 −9.1137 · 10+15 2.9726 · 10+15 −6.0921 · 10+15


= 10+04

5.3.6 “G” region: N 2 aau

= 10−05

For a small parameter N2 aau

we can neglect the contact potential Vc, which leadsus to the “G” region for self-trapped BEC. The kinetic energy T balances thegravity-like potential Vu. From TAB. 5.2 - TAB. 5.17 we choose the small pa-rameter N2 a

au= 10−05 performing the comparison between the results of the

numerical and variational calculations.

N aau

rrms ρmax E

10+01 10−07 4.6339 · 10−0.5 4.3996 · 10+01 −1.0844 · 10+02

10+02 10−09 4.6339 · 10−01 4.3996 · 10+05 −1.0844 · 10+05

10+03 10−11 4.6339 · 10−1.5 4.3996 · 10+09 −1.0844 · 10+08

10+04 10−13 4.6339 · 10−02 4.3996 · 10+13 −1.0844 · 10+11

10+05 10−15 4.6339 · 10−2.5 4.3996 · 10+17 −1.0844 · 10+14

10+06 10−17 4.6339 · 10−03 4.3996 · 10+21 −1.0844 · 10+17

10+07 10−19 4.6339 · 10−3.5 4.3996 · 10+25 −1.0844 · 10+20

10+08 10−21 4.6339 · 10−04 4.3996 · 10+29 −1.0844 · 10+23

10+09 10−23 4.6339 · 10−4.5 4.3996 · 10+33 −1.0844 · 10+26


= 10−05

N T Vu Vc µ10+01 1.0838 · 10+01 −4.3420 · 10+01 3.4305 · 10−05 −3.2554 · 10+01

10+02 1.0838 · 10+03 −4.3420 · 10+03 3.4305 · 10−03 −3.2554 · 10+03

10+03 1.0838 · 10+05 −4.3420 · 10+05 3.4305 · 10−01 −3.2554 · 10+05

10+04 1.0838 · 10+07 −4.3420 · 10+07 3.4305 · 10+01 −3.2554 · 10+07

10+05 1.0838 · 10+09 −4.3420 · 10+09 3.4305 · 10+03 −3.2554 · 10+09




N T Vu Vc µ10+06 1.0838 · 10+11 −4.3420 · 10+11 3.4305 · 10+05 −3.2554 · 10+11

10+07 1.0838 · 10+13 −4.3420 · 10+13 3.4305 · 10+06 −3.2554 · 10+13

10+08 1.0838 · 10+15 −4.3420 · 10+15 3.4305 · 10+07 −3.2554 · 10+15

10+09 1.0838 · 10+17 −4.3420 · 10+17 3.4305 · 10+08 −3.2554 · 10+17


= 10−05

N aau

kmin rrms E

10+01 10−07 2.6596 · 10+00 4.6049 · 10−0.5 −1.0610 · 10+02

10+02 10−09 2.6596 · 10+01 4.6049 · 10−01 −1.0610 · 10+05

10+03 10−11 2.6596 · 10+02 4.6049 · 10−1.5 −1.0610 · 10+08

10+04 10−13 2.6596 · 10+03 4.6049 · 10−02 −1.0610 · 10+11

10+05 10−15 2.6596 · 10+04 4.6049 · 10−2.5 −1.0610 · 10+14

10+06 10−17 2.6596 · 10+05 4.6049 · 10−03 −1.0610 · 10+17

10+07 10−19 2.6596 · 10+06 4.6049 · 10−3.5 −1.0610 · 10+20

10+08 10−21 2.6596 · 10+07 4.6049 · 10−04 −1.0610 · 10+23

10+09 10−23 2.6596 · 10+08 4.6049 · 10−4.5 −1.0610 · 10+26


= 10−05

N Veff ρmax10+01 −4.2441 · 10+01 3.3785 · 10+01

10+02 −4.2441 · 10+03 3.3785 · 10+05

10+03 −4.2441 · 10+05 3.3785 · 10+09

10+04 −4.2441 · 10+07 3.3785 · 10+13

10+05 −4.2441 · 10+09 3.3785 · 10+17

10+06 −4.2441 · 10+11 3.3785 · 10+21

10+07 −4.2441 · 10+13 3.3785 · 10+25

10+08 −4.2441 · 10+15 3.3785 · 10+29

10+09 −4.2441 · 10+17 3.3785 · 10+33


= 10−05

N T Vu Vc µ10+01 1.0610 · 10+01 −4.2441 · 10+01 3.0020 · 10−05 −3.1830 · 10+01

10+02 1.0610 · 10+03 −4.2441 · 10+03 3.0020 · 10−03 −3.1830 · 10+03




N T Vu Vc µ10+03 1.0610 · 10+05 −4.2441 · 10+05 3.0020 · 10−01 −3.1830 · 10+05

10+04 1.0610 · 10+07 −4.2441 · 10+07 3.0020 · 10+01 −3.1830 · 10+07

10+05 1.0610 · 10+09 −4.2441 · 10+09 3.0020 · 10+03 −3.1830 · 10+09

10+06 1.0610 · 10+11 −4.2441 · 10+11 3.0020 · 10+05 −3.1830 · 10+11

10+07 1.0610 · 10+13 −4.2441 · 10+13 3.0020 · 10+06 −3.1830 · 10+13

10+08 1.0610 · 10+15 −4.2441 · 10+15 3.0020 · 10+08 −3.1830 · 10+15

10+09 1.0610 · 10+17 −4.2441 · 10+17 3.0020 · 10+09 −3.1830 · 10+17


= 10−05

The two asymptotic regions are characterized by one relevant parameter,namely N2 a

au. For N2 a

auÀ 1 (“TF-G” region) the kinetic energy is negligi-

ble and self-trapping results from the balance between repulsive scattering andgravity-like attraction. For N2 a

au¿ 1 (“G” region) the scattering is negligible

and self-trapping arises by a balance between kinetic energy and gravity-like at-traction. In FIG. 5.3 we can see the phase diagram of self-trapped BEC. The

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

a/au

102

103

104

105

106

N

"TF-G"

"G"

kinetic energy

negligible"gravity" dominates

N 2a/a

u = 10.000

1

10

100

1.00010 -1

10 -2

10 -3

FIG. 5.3: Phase diagram: The total number N of atoms as a function of thes-wave scattering length a

au. The two asymptotic regions are separated by the

line N2 aau

= 1.

5.4. Energies, sizes and peak densities for different scattering lengths 89

physics on each sloping line with N2 aau

= const. is identical because of the scalingproperty derived in section 3.2 and subsection 3.4.1.

5.4 Energies, sizes and peak densities for differ-

ent scattering lengths

Due to the remarkable scaling property of our system we can formally assumeN = 1 and calculate all energies, sizes and peak densities for all numbers ofparticles N and different scattering lengths from these. Graphic representationsare given in appendix F. We present our results in tabular form.

The numerical calculations can be made up to a value of N2 aau

= −1.0251as one can see in TAB. 5.28. For smaller values the numerics fails because weobtain ambiguous solutions for the same scattering length arising from a tangentbifurcation3 (see section 5.7). For the variational calculations we refer to the endof subsection 4.1.2 where we have derived the limit of the negative scatteringlength in the fourth limiting case that is N2 a

au= −3π

8≈ −1.1781.

b N2 aau

rrms√N E/N3

−1.02516 −1.0251 · 10+00 2.4261 · 10+00 −1.4305 · 10−01

−1.0 −1.0250 · 10+00 2.4618 · 10+00 −1.4304 · 10−01

−0.9 −1.0189 · 10+00 2.6105 · 10+00 −1.4235 · 10−01

−0.8 −1.0023 · 10+00 2.7710 · 10+00 −1.4075 · 10−01

−0.7 −9.7234 · 10−01 2.9444 · 10+00 −1.3836 · 10−01

−0.6 −9.2571 · 10−01 3.1319 · 10+00 −1.3530 · 10−01

−0.5 −8.5841 · 10−01 3.3348 · 10+00 −1.3168 · 10−01

−0.4 −7.6556 · 10−01 3.5545 · 10+00 −1.2761 · 10−01

−0.3 −6.4122 · 10−01 3.7926 · 10+00 −1.2317 · 10−01

−0.2 −4.7823 · 10−01 4.0506 · 10+00 −1.1844 · 10−01

−0.114 −3 · 10−01 4.2899 · 10+00 −1.1421 · 10−01

−3.47 · 10−02 −1 · 10−01 4.5258 · 10+00 −1.1021 · 10−01

−3.35 · 10−03 −1 · 10−02 4.6234 · 10+00 −1.0861 · 10−01

−3.35 · 10−04 −1 · 10−03 4.6329 · 10+00 −1.0846 · 10−01

−3.35 · 10−05 −1 · 10−04 4.6338 · 10+00 −1.0844 · 10−01

−3.35 · 10−06 −1 · 10−05 4.6339 · 10+00 −1.0844 · 10−01

−3.35 · 10−07 −1 · 10−06 4.6339 · 10+00 −1.0844 · 10−01

−3.35 · 10−08 −1 · 10−07 4.6339 · 10+00 −1.0844 · 10−01

0 0 4.6339 · 10+00 −1.0844 · 10−01

3.35 · 10−08 1 · 10−07 4.6339 · 10+00 −1.0844 · 10−01


3For a general introduction into bifurcation theory see [Sastry (2000); Kielhofer (2004)].



b N2 aau

rrms√N E/N3

3.35 · 10−07 1 · 10−06 4.6339 · 10+00 −1.0844 · 10−01

3.35 · 10−06 1 · 10−05 4.6339 · 10+00 −1.0844 · 10−01

3.35 · 10−05 1 · 10−04 4.6340 · 10+00 −1.0844 · 10−01

3.35 · 10−04 1 · 10−03 4.6350 · 10+00 −1.0842 · 10−01

3.35 · 10−03 1 · 10−02 4.6445 · 10+00 −1.0827 · 10−01

3.22 · 10−02 1 · 10−01 4.7370 · 10+00 −1.0679 · 10−01

9.00 · 10−02 3 · 10−01 4.9290 · 10+00 −1.0381 · 10−01

0.25 1 · 10+00 5.5099 · 10+00 −9.5568 · 10−02

1 1 · 10+01 9.4600 · 10+00 −6.0762 · 10−02

1.55 3 · 10+01 1.4036 · 10+01 −4.2318 · 10−02

1.84625 5 · 10+01 1.7207 · 10+01 −3.4912 · 10−02

2.146 8 · 10+01 2.0952 · 10+01 −2.8920 · 10−02

2.2975 1 · 10+02 2.3054 · 10+01 −2.6374 · 10−02

3.1554 3 · 10+02 3.7634 · 10+01 −1.6364 · 10−02

3.629 5 · 10+02 4.7637 · 10+01 −1.2981 · 10−02

4.1165 8 · 10+02 5.9386 · 10+01 −1.0448 · 10−02

4.3668 1 · 10+03 6.5996 · 10+01 −9.4226 · 10−03

5.81211 3 · 10+03 1.1173 · 10+02 −5.5996 · 10−03

6.6255 5 · 10+03 1.4316 · 10+02 −4.3802 · 10−03

7.46803 8 · 10+03 1.8006 · 10+02 −3.4926 · 10−03

7.90277 1 · 10+04 2.0084 · 10+02 −3.1346 · 10−03

TAB. 5.28: Numerical calculations

N2 aau

Veff/N2 µ/N2 ρmax/N

4

−1.0251 · 10+00 −1.1293 −7.0774 · 10−01 3.9789 · 10−02

−1.0250 · 10+00 −1.1037 −6.9492 · 10−01 3.7869 · 10−02

−1.0189 · 10+00 −1.0072 −6.4596 · 10−01 3.1038 · 10−02

−1.0023 · 10+00 −9.1862 · 10−01 −6.0006 · 10−01 2.5346 · 10−02

−9.7234 · 10−01 −8.3742 · 10−01 −5.5708 · 10−01 2.0621 · 10−02

−9.2571 · 10−01 −7.6305 · 10−01 −5.1684 · 10−01 1.6715 · 10−02

−8.5841 · 10−01 −6.9503 · 10−01 −4.7920 · 10−01 1.3498 · 10−02

−7.6556 · 10−01 −6.3288 · 10−01 −4.4406 · 10−01 1.0862 · 10−02

−6.4122 · 10−01 −5.7612 · 10−01 −4.1122 · 10−01 8.7091 · 10−03

−4.7823 · 10−01 −5.2434 · 10−01 −3.8062 · 10−01 6.9589 · 10−03

−3 · 10−01 −4.8350 · 10−01 −3.5596 · 10−01 5.7225 · 10−03

−1 · 10−01 −4.4864 · 10−01 −3.3454 · 10−01 4.7682 · 10−03




N2 aau


4

−1 · 10−02 −4.3556 · 10−01 −3.2640 · 10−01 4.4340 · 10−03

−1 · 10−03 −4.3433 · 10−01 −3.2562 · 10−01 4.4031 · 10−03

−1 · 10−04 −4.3420 · 10−01 −3.2554 · 10−01 4.4000 · 10−03

−1 · 10−05 −4.3420 · 10−01 −3.2554 · 10−01 4.3997 · 10−03

−1 · 10−06 −4.3420 · 10−01 −3.2554 · 10−01 4.3997 · 10−03

−1 · 10−07 −4.3420 · 10−01 −3.2554 · 10−01 4.3997 · 10−03

0 −4.3420 · 10−01 −3.2554 · 10−01 4.3997 · 10−03

1 · 10−07 −4.3419 · 10−01 −3.2555 · 10−01 4.3997 · 10−03

1 · 10−06 −4.3419 · 10−01 −3.2554 · 10−01 4.3996 · 10−03

1 · 10−05 −4.3419 · 10−01 −3.2554 · 10−01 4.3996 · 10−03

1 · 10−04 −4.3417 · 10−01 −3.2554 · 10−01 4.3993 · 10−03

1 · 10−03 −4.3406 · 10−01 −3.2546 · 10−01 4.3962 · 10−03

1 · 10−02 −4.3281 · 10−01 −3.2468 · 10−01 4.3655 · 10−03

1 · 10−01 −4.2120 · 10−01 −3.1740 · 10−01 4.0819 · 10−03

3 · 10−01 −3.9885 · 10−01 −3.0324 · 10−01 3.5654 · 10−03

1 · 10+00 −3.4305 · 10−01 −2.6710 · 10−01 2.4405 · 10−03

1 · 10+01 −1.7208 · 10−01 −1.4680 · 10−01 3.9300 · 10−04

3 · 10+01 −1.0765 · 10−01 −9.6144 · 10−02 1.0583 · 10−04

5 · 10+01 −8.5224 · 10−02 −7.7528 · 10−02 5.4244 · 10−05

8 · 10+01 −6.8320 · 10−02 −6.3078 · 10−02 2.8595 · 10−05

1 · 10+02 −6.1446 · 10−02 −5.7098 · 10−02 2.0998 · 10−05

3 · 10+02 −3.6101 · 10−02 −3.4414 · 10−02 4.3989 · 10−06

5 · 10+02 −2.8108 · 10−02 −2.7036 · 10−02 2.0959 · 10−06

8 · 10+02 −2.2302 · 10−02 −2.1598 · 10−02 1.0530 · 10−06

1 · 10+03 −1.9995 · 10−02 −1.9420 · 10−02 7.5822 · 10−07

3 · 10+03 −1.1622 · 10−02 −1.1411 · 10−02 1.4930 · 10−07

5 · 10+03 −9.0252 · 10−03 −8.8932 · 10−03 6.9852 · 10−08

8 · 10+03 −7.1566 · 10−03 −7.0712 · 10−03 3.4669 · 10−08

1 · 10+04 −6.4094 · 10−03 −6.3394 · 10−03 2.4846 · 10−08


N2 aau

T/N2 Vu/N2 Vc/N

2

−1.0251 · 10+00 4.2104 · 10−01 −8.5120 · 10−01 −2.7818 · 10−01

−1.0250 · 10+00 4.0826 · 10−01 −8.3836 · 10−01 −2.6540 · 10−01

−1.0189 · 10+00 3.6072 · 10−01 −7.8868 · 10−01 −2.1852 · 10−01

−1.0023 · 10+00 3.1806 · 10−01 −7.4116 · 10−01 −1.7746 · 10−01

−9.7234 · 10−01 2.7988 · 10−01 −6.9576 · 10−01 −1.4166 · 10−01




N2 aau

T/N2 Vu/N2 Vc/N

2

−9.2571 · 10−01 2.4580 · 10−01 −6.5244 · 10−01 −1.1061 · 10−01

−8.5841 · 10−01 2.1542 · 10−01 −6.1118 · 10−01 −8.3856 · 10−02

−7.6556 · 10−01 1.8846 · 10−01 −5.7194 · 10−01 −6.0942 · 10−02

−6.4122 · 10−01 1.6455 · 10−01 −5.3466 · 10−01 −4.1464 · 10−02

−4.7823 · 10−01 1.4341 · 10−01 −4.9930 · 10−01 −2.5044 · 10−02

−3 · 10−01 1.2724 · 10−01 −4.7040 · 10−01 −1.3100 · 10−02

−1 · 10−01 1.1383 · 10−01 −4.4496 · 10−01 −3.6808 · 10−03

−1 · 10−02 1.0889 · 10−01 −4.3522 · 10−01 −3.4422 · 10−04

−1 · 10−03 1.0843 · 10−01 −4.3430 · 10−01 −3.4316 · 10−05

−1 · 10−04 1.0838 · 10−01 −4.3420 · 10−01 −3.4306 · 10−06

−1 · 10−05 1.0838 · 10−01 −4.3420 · 10−01 −3.4304 · 10−07

−1 · 10−06 1.0838 · 10−01 −4.3420 · 10−01 −3.4305 · 10−08

−1 · 10−07 1.0838 · 10−01 −4.3420 · 10−01 −3.4305 · 10−09

0 1.0838 · 10−01 −4.3420 · 10−01 01 · 10−07 1.0838 · 10−01 −4.3420 · 10−01 3.4305 · 10−09

1 · 10−06 1.0838 · 10−01 −4.3420 · 10−01 3.4305 · 10−08

1 · 10−05 1.0838 · 10−01 −4.3420 · 10−01 3.4304 · 10−07

1 · 10−04 1.0837 · 10−01 −4.3418 · 10−01 3.4304 · 10−06

1 · 10−03 1.0833 · 10−01 −4.3410 · 10−01 3.4292 · 10−05

1 · 10−02 1.0786 · 10−01 −4.3316 · 10−01 3.4188 · 10−04

1 · 10−01 1.0354 · 10−01 −4.2440 · 10−01 3.1908 · 10−03

3 · 10−01 9.5354 · 10−02 −4.0726 · 10−01 8.4054 · 10−03

1 · 10+00 7.5738 · 10−02 −3.6284 · 10−01 1.9781 · 10−02

1 · 10+01 2.5152 · 10−02 −2.0766 · 10−01 3.5588 · 10−02

3 · 10+01 1.1426 · 10−02 −1.3853 · 10−01 3.0880 · 10−02

5 · 10+01 7.6384 · 10−03 −1.1249 · 10−01 2.7266 · 10−02

8 · 10+01 5.1892 · 10−03 −9.2040 · 10−02 2.3720 · 10−02

1 · 10+02 4.3048 · 10−03 −8.3506 · 10−02 2.2060 · 10−02

3 · 10+02 1.6630 · 10−03 −5.0792 · 10−02 1.4691 · 10−02

5 · 10+02 1.0558 · 10−03 −4.0022 · 10−02 1.1914 · 10−02

8 · 10+02 6.9128 · 10−04 −3.2044 · 10−02 9.7416 · 10−03

1 · 10+03 5.6434 · 10−04 −2.8828 · 10−02 8.8328 · 10−03

3 · 10+03 2.0590 · 10−04 −1.6987 · 10−02 5.3642 · 10−03

5 · 10+03 1.2818 · 10−04 −1.3248 · 10−02 4.2228 · 10−03

8 · 10+03 8.2678 · 10−05 −1.0535 · 10−02 3.3784 · 10−03

1 · 10+04 6.7078 · 10−05 −9.4456 · 10−03 3.0362 · 10−03



N2 aau

kmin/N rrms√N E/N3

−1.1780 · 10+00 5.2713 · 10−01 2.3234 · 10+00 −1.4146 · 10−01

−1.0251 · 10+00 3.9101 · 10−01 3.1322 · 10+00 −1.3154 · 10−01

−1.0250 · 10+00 3.9097 · 10−01 3.1325 · 10+00 −1.3153 · 10−01

−1.0189 · 10+00 3.8894 · 10−01 3.1488 · 10+00 −1.3124 · 10−01

−1.0023 · 10+00 3.8370 · 10−01 3.1919 · 10+00 −1.3048 · 10−01

−9.7234 · 10−01 3.7514 · 10−01 3.2647 · 10+00 −1.2918 · 10−01

−9.2571 · 10−01 3.6362 · 10−01 3.3681 · 10+00 −1.2730 · 10−01

−8.5841 · 10−01 3.4973 · 10−01 3.5018 · 10+00 −1.2487 · 10−01

−7.6556 · 10−01 3.3417 · 10−01 3.6649 · 10+00 −1.2191 · 10−01

−6.4122 · 10−01 3.1755 · 10−01 3.8568 · 10+00 −1.1849 · 10−01

−4.7823 · 10−01 3.0039 · 10−01 4.0771 · 10+00 −1.1466 · 10−01

−3 · 10−01 2.8550 · 10−01 4.2897 · 10+00 −1.1112 · 10−01

−1 · 10−01 2.7187 · 10−01 4.5048 · 10+00 −1.0765 · 10−01

−1 · 10−02 2.6653 · 10−01 4.5950 · 10+00 −1.0625 · 10−01

−1 · 10−03 2.6601 · 10−01 4.6039 · 10+00 −1.0611 · 10−01

−1 · 10−04 2.6596 · 10−01 4.6048 · 10+00 −1.0610 · 10−01

−1 · 10−05 2.6596 · 10−01 4.6050 · 10+00 −1.0610 · 10−01

−1 · 10−06 2.6596 · 10−01 4.6049 · 10+00 −1.0610 · 10−01

−1 · 10−07 2.6596 · 10−01 4.6049 · 10+00 −1.0610 · 10−01

0 −−− −−− −−−1 · 10−07 2.6596 · 10−01 4.6049 · 10+00 −1.0610 · 10−01

1 · 10−06 2.6596 · 10−01 4.6049 · 10+00 −1.0610 · 10−01

1 · 10−05 2.6596 · 10−01 4.6050 · 10+00 −1.0610 · 10−01

1 · 10−04 2.6595 · 10−01 4.6050 · 10+00 −1.0610 · 10−01

1 · 10−03 2.6590 · 10−01 4.6059 · 10+00 −1.0608 · 10−01

1 · 10−02 2.6539 · 10−01 4.6147 · 10+00 −1.0595 · 10−01

1 · 10−01 2.6054 · 10−01 4.7007 · 10+00 −1.0464 · 10−01

3 · 10−01 2.5089 · 10−01 4.8815 · 10+00 −1.0198 · 10−01

1 · 10+00 2.2541 · 10−01 5.4332 · 10+00 −9.4498 · 10−02

1 · 10+01 1.3036 · 10−01 9.3948 · 10+00 −6.0846 · 10−02

3 · 10+01 8.6570 · 10−02 1.4147 · 10+01 −4.2300 · 10−02

5 · 10+01 7.0072 · 10−02 1.7478 · 10+01 −3.4818 · 10−02

8 · 10+01 5.7190 · 10−02 2.1415 · 10+01 −2.8784 · 10−02

1 · 10+02 5.1807 · 10−02 2.3640 · 10+01 −2.6214 · 10−02

3 · 10+02 3.1309 · 10−02 3.9116 · 10+01 −1.6164 · 10−02

5 · 10+02 2.4596 · 10−02 4.9792 · 10+01 −1.2781 · 10−02

8 · 10+02 1.9644 · 10−02 6.2346 · 10+01 −1.0256 · 10−02

1 · 10+03 1.7641 · 10−02 6.9424 · 10+01 −9.2282 · 10−03

3 · 10+03 1.0334 · 10−02 1.1851 · 10+02 −5.4434 · 10−03




N2 aau

kmin/N rrms√N E/N3

5 · 10+03 8.0405 · 10−03 1.5232 · 10+02 −4.2446 · 10−03

8 · 10+03 6.3771 · 10−03 1.9205 · 10+02 −3.3718 · 10−03

1 · 10+04 5.7111 · 10−03 2.1444 · 10+02 −3.0214 · 10−03

TAB. 5.31: Variational calculations with k+

N2 aau


4

−1.1780 · 10+00 −1.1165 · 10+00 −6.9972 · 10−01 2.6305 · 10−02

−1.0251 · 10+00 −7.2175 · 10−01 −4.9242 · 10−01 1.0736 · 10−02

−1.0250 · 10+00 −7.2166 · 10−01 −4.9236 · 10−01 1.0733 · 10−02

−1.0189 · 10+00 −7.1633 · 10−01 −4.8940 · 10−01 1.0566 · 10−02

−1.0023 · 10+00 −7.0263 · 10−01 −4.8180 · 10−01 1.0145 · 10−02

−9.7234 · 10−01 −6.8056 · 10−01 −4.6946 · 10−01 9.4813 · 10−03

−9.2571 · 10−01 −6.5126 · 10−01 −4.5294 · 10−01 8.6341 · 10−03

−8.5841 · 10−01 −6.1670 · 10−01 −4.3322 · 10−01 7.6824 · 10−03

−7.6556 · 10−01 −5.7885 · 10−01 −4.1134 · 10−01 6.7018 · 10−03

−6.4122 · 10−01 −5.3951 · 10−01 −3.8824 · 10−01 5.7507 · 10−03

−4.7823 · 10−01 −5.0003 · 10−01 −3.6468 · 10−01 4.8679 · 10−03

−3 · 10−01 −4.6676 · 10−01 −3.4448 · 10−01 4.1794 · 10−03

−1 · 10−01 −4.3705 · 10−01 −3.2618 · 10−01 3.6087 · 10−03

−1 · 10−02 −4.2456 · 10−01 −3.1906 · 10−01 3.4004 · 10−03

−1 · 10−03 −4.2445 · 10−01 −3.1838 · 10−01 3.3807 · 10−03

−1 · 10−04 −4.2442 · 10−01 −3.1830 · 10−01 3.3787 · 10−03

−1 · 10−05 −4.2441 · 10−01 −3.1830 · 10−01 3.3786 · 10−03

−1 · 10−06 −4.2441 · 10−01 −3.1830 · 10−01 3.3785 · 10−03

−1 · 10−07 −4.2441 · 10−01 −3.1830 · 10−01 3.3785 · 10−03

0 −−− −−− −−−1 · 10−07 −4.2441 · 10−01 −3.1830 · 10−01 3.3785 · 10−03

1 · 10−06 −4.2441 · 10−01 −3.2554 · 10−01 3.3785 · 10−03

1 · 10−05 −4.2441 · 10−01 −3.1830 · 10−01 3.3785 · 10−03

1 · 10−04 −4.2440 · 10−01 −3.2554 · 10−01 3.3783 · 10−03

1 · 10−03 −4.2429 · 10−01 −3.1822 · 10−01 3.3764 · 10−03

1 · 10−02 −4.2320 · 10−01 −3.1756 · 10−01 3.3571 · 10−03

1 · 10−01 −4.1294 · 10−01 −3.1112 · 10−01 3.1763 · 10−03

3 · 10−01 −3.9280 · 10−01 −2.9838 · 10−01 2.8362 · 10−03

1 · 10+00 −3.4142 · 10−01 −2.6520 · 10−01 2.0570 · 10−03

1 · 10+01 −1.7267 · 10−01 −1.4718 · 10−01 3.9787 · 10−04




N2 aau


4

3 · 10+01 −1.0708 · 10−01 −9.5844 · 10−02 1.1651 · 10−04

5 · 10+01 −8.4368 · 10−02 −7.7000 · 10−02 6.1790 · 10−05

8 · 10+01 −6.7384 · 10−02 −6.2476 · 10−02 3.3592 · 10−05

1 · 10+02 −6.0484 · 10−02 −5.6456 · 10−02 2.4972 · 10−05

3 · 10+02 −3.5269 · 10−02 −3.3798 · 10−02 5.5121 · 10−06

5 · 10+02 −2.7377 · 10−02 −2.6468 · 10−02 2.6725 · 10−06

8 · 10+02 −2.1669 · 10−02 −2.1090 · 10−02 1.3613 · 10−06

1 · 10+03 −1.9389 · 10−02 −1.8923 · 10−02 9.8601 · 10−07

3 · 10+03 −1.1207 · 10−02 −1.1047 · 10−02 1.9819 · 10−07

5 · 10+03 −8.6824 · 10−03 −8.5862 · 10−03 9.3355 · 10−08

8 · 10+03 −6.8654 · 10−03 −6.8046 · 10−03 4.6574 · 10−08

1 · 10+04 −6.1410 · 10−03 −6.0920 · 10−03 3.3454 · 10−08


N2 aau

T/N2 Vu/N2 Vc/N

2

−1.1780 · 10+00 4.1681 · 10−01 −8.4118 · 10−01 −2.7535 · 10−01

−1.0251 · 10+00 2.2932 · 10−01 −6.2396 · 10−01 −9.7792 · 10−02

−1.0250 · 10+00 2.2932 · 10−01 −6.2390 · 10−01 −9.7758 · 10−02

−1.0189 · 10+00 2.2690 · 10−01 −6.2066 · 10−01 −9.5668 · 10−02

−1.0023 · 10+00 2.2084 · 10−01 −6.1228 · 10−01 −9.0354 · 10−02

−9.7234 · 10−01 2.1110 · 10−01 −5.9864 · 10−01 −8.1918 · 10−02

−9.2571 · 10−01 1.9832 · 10−01 −5.8024 · 10−01 −7.1020 · 10−02

−8.5841 · 10−01 1.8347 · 10−01 −5.5810 · 10−01 −5.8598 · 10−02

−7.6556 · 10−01 1.6750 · 10−01 −5.3326 · 10−01 −4.5588 · 10−02

−6.4122 · 10−01 1.5126 · 10−01 −5.0674 · 10−01 −3.2766 · 10−02

−4.7823 · 10−01 1.3535 · 10−01 −4.7934 · 10−01 −2.0686 · 10−02

−3 · 10−01 1.2227 · 10−01 −4.5560 · 10−01 −1.1159 · 10−02

−1 · 10−01 1.1087 · 10−01 −4.3384 · 10−01 −3.2130 · 10−03

−1 · 10−02 1.0656 · 10−01 −4.2532 · 10−01 −3.0516 · 10−04

−1 · 10−03 1.0614 · 10−01 −4.2450 · 10−01 −3.0040 · 10−05

−1 · 10−04 1.0610 · 10−01 −4.2442 · 10−01 −3.0022 · 10−06

−1 · 10−05 1.0610 · 10−01 −4.2441 · 10−01 −3.0020 · 10−07

−1 · 10−06 1.0610 · 10−01 −4.2441 · 10−01 −3.0021 · 10−08

−1 · 10−07 1.0610 · 10−01 −4.2441 · 10−01 −3.0021 · 10−09

0 −−− −−− −−−1 · 10−07 1.0610 · 10−01 −4.2441 · 10−01 3.0021 · 10−09

1 · 10−06 1.0610 · 10−01 −4.2441 · 10−01 3.0021 · 10−08




N2 aau

T/N2 Vu/N2 Vc/N

2

1 · 10−05 1.0610 · 10−01 −4.2441 · 10−01 3.0020 · 10−07

1 · 10−04 1.0609 · 10−01 −4.2440 · 10−01 3.0018 · 10−06

1 · 10−03 1.0605 · 10−01 −4.2432 · 10−01 3.0000 · 10−05

1 · 10−02 1.0565 · 10−01 −4.2350 · 10−01 2.9830 · 10−04

1 · 10−01 1.0182 · 10−01 −4.1576 · 10−01 2.8224 · 10−03

3 · 10−01 9.4420 · 10−02 −4.0036 · 10−01 7.5606 · 10−03

1 · 10+00 7.6220 · 10−02 −3.5970 · 10−01 1.8278 · 10−02

1 · 10+01 2.5492 · 10−02 −2.0802 · 10−01 3.5354 · 10−02

3 · 10+01 1.1241 · 10−02 −1.3814 · 10−01 3.1058 · 10−02

5 · 10+01 7.3652 · 10−03 −1.1182 · 10−01 2.7452 · 10−02

8 · 10+01 4.9060 · 10−03 −9.1262 · 10−02 2.3878 · 10−02

1 · 10+02 4.0260 · 10−03 −8.2672 · 10−02 2.2188 · 10−02

3 · 10+02 1.4704 · 10−03 −4.9962 · 10−02 1.4693 · 10−02

5 · 10+02 9.0750 · 10−04 −3.9250 · 10−02 1.1873 · 10−02

8 · 10+02 5.7882 · 10−04 −3.1346 · 10−02 9.6772 · 10−03

1 · 10+03 4.6682 · 10−04 −2.8150 · 10−02 8.7614 · 10−03

3 · 10+03 1.6019 · 10−04 −1.6490 · 10−02 5.2832 · 10−03

5 · 10+03 9.6976 · 10−05 −1.2830 · 10−02 4.1476 · 10−03

8 · 10+03 6.1000 · 10−05 −1.0176 · 10−02 3.3106 · 10−03

1 · 10+04 4.8926 · 10−05 −9.1136 · 10−03 2.9726 · 10−03


5.4.1 Relative error between the numerical and varia-tional calculations

We want to search for an error estimate of our results. Therefore we calculatethe relative error between the numerical and variational calculations by means ofthe formula

∆x[%] = 100(xvar − xnum)

xnum

, (5.20)

where xvar and xnum are the variables for the variational and numerical values,respectively.

aau

rrms/% ρmax/% E/%

−1.0251 · 10+00 29.1058 73.0173 8.0464−1.0250 · 10+00 27.2446 71.6566 8.0407−1.0189 · 10+00 20.6237 65.9553 7.7990




rrms/% ρmax/% E/%

−1.0023 · 10+00 15.1900 59.9734 7.2934−9.7234 · 10−01 10.8791 54.0207 6.6379−9.2571 · 10−01 7.5448 48.3449 5.9095−8.5841 · 10−01 5.0107 43.0843 5.1733−7.6556 · 10−01 3.1084 38.3001 4.4628−6.4122 · 10−01 1.6932 33.9684 3.7964−4.7823 · 10−01 0.6552 30.0473 3.1896−3 · 10−01 0.0036 26.9646 2.7174−1 · 10−01 0.4622 24.3152 2.3213−1 · 10−02 0.6125 23.3102 2.1733−1 · 10−03 0.6239 23.2197 2.1608−1 · 10−04 0.6242 23.2096 2.1588−1 · 10−05 0.6245 23.2088 2.1583

0 −−− −−− −−−1 · 10−05 0.6240 23.2080 2.15851 · 10−04 0.6243 23.2072 2.15801 · 10−03 0.6268 23.1970 2.15781 · 10−02 0.6411 23.0973 2.14301 · 10−01 0.7662 22.1852 2.00933 · 10−01 0.9632 20.4504 1.76801 · 10+00 1.3918 15.7123 1.11911 · 10+01 0.6888 1.2399 0.13833 · 10+01 0.7935 10.0970 0.03875 · 10+01 1.5759 13.9127 0.26858 · 10+01 2.2114 17.4753 0.46551 · 10+02 2.5430 18.9258 0.60053 · 10+02 3.9403 25.3067 1.22285 · 10+02 4.5248 27.5116 1.54528 · 10+02 4.9854 29.2838 1.83561 · 10+03 5.1942 30.0431 2.06183 · 10+03 6.0724 32.7500 2.78675 · 10+03 6.3985 33.6475 3.09458 · 10+03 6.6606 34.3404 3.45871 · 10+04 6.7750 34.6466 3.6049

TAB. 5.34: Error between the numerical and variational calculations


aau

T/% Vu/% Vc/% µ/%

−1.0251 · 10+00 45.5309 26.6956 64.8452 30.4231−1.0250 · 10+00 43.8355 25.5794 63.1656 29.1466−1.0189 · 10+00 37.0929 21.3029 56.2197 24.2339−1.0023 · 10+00 30.5662 17.3862 49.0856 19.7057−9.7234 · 10−01 24.5746 13.9581 42.1720 15.7278−9.2571 · 10−01 19.3126 11.0640 35.7934 12.3624−8.5841 · 10−01 14.8294 8.6848 30.1193 9.5941−7.6556 · 10−01 11.1171 6.7620 25.1914 7.3669−6.4122 · 10−01 8.0764 5.2216 20.9765 5.5863−4.7823 · 10−01 5.6187 3.9939 17.4013 4.1852−3 · 10−01 3.9074 3.1588 14.9883 3.2220−1 · 10−01 2.5998 2.5028 12.8923 2.4973−1 · 10−02 2.1446 2.2752 12.2263 2.2452−1 · 10−03 2.1060 2.2557 12.4600 2.2219−1 · 10−04 2.1019 2.2519 12.4846 2.2186−1 · 10−05 2.1020 2.2537 12.4846 2.2207

0 −−− −−− −−− −−−1 · 10−05 2.1010 2.2541 12.4857 2.22111 · 10−04 2.1012 2.2515 12.4907 2.22321 · 10−03 2.0970 2.2522 12.5102 2.21991 · 10−02 2.0512 2.2263 12.7437 2.19211 · 10−01 1.6576 2.0334 11.5455 1.97793 · 10−01 0.9777 1.6920 10.0494 1.60051 · 10+00 0.6365 0.8611 7.5972 0.70491 · 10+01 1.3518 0.1782 0.6573 0.26033 · 10+01 1.6149 0.2782 0.5826 0.31115 · 10+01 3.5753 0.6026 0.6856 0.67878 · 10+01 5.4564 0.8451 0.6716 0.95311 · 10+02 6.4757 0.9976 0.5874 1.12213 · 10+02 11.5799 1.6315 0.0137 1.78695 · 10+02 14.0511 1.9262 0.3434 2.09388 · 10+02 16.2659 2.1736 0.6592 2.34621 · 10+03 17.2777 2.3457 0.8074 2.56083 · 10+03 22.1997 2.9218 1.5069 3.18895 · 10+03 24.3444 3.1510 1.7790 3.45118 · 10+03 26.2180 3.4111 2.0009 3.77011 · 10+04 27.0605 3.5135 2.0923 3.9006

TAB. 5.35: Error between the numerical and variational calculations

5.5. Comparison between ψTF−G and ψnum 99

The mean-field energy E has an error of a few per cent while for the wavefunction sensitive quantities, e.g. the peak density ρmax, error up to the value of70% may appear.

5.5 Comparison between ψTF−G and ψnum

For the “TF-G” region we have derived in section 3.8 an analytical wave functionof the form

ψTF−G (r) =1

2π√

aau

√√√√√sin

(r√

aau

)

r. (5.21)

This form is only valid if the constraint

sin

(r√aau

)≥ 0 (5.22)

is fulfilled. The argument of the sine must be less than or equal to π, viz.

r√aau

≤ π,

r ≤ π

√a

au. (5.23)

We compare the analytical wave function ψTF−G with the wave function ψnum de-termined numerically in the“TF-G”region, where the kinetic energy is negligible,viz. for large scattering lengths a

au. Therefore we assume three large scattering

lengths aau

= 5 · 10+03, aau

= 8 · 10+03 and aau

= 1 · 10+04. We calculate theappropriate constraints

r ≤ π√

5 · 10+03 . 222.1441, (5.24)

r ≤ π√

8 · 10+03 . 280.9926, (5.25)

r ≤ π√

1 · 10+04 . 314.1593. (5.26)

and present our results in table form.

r ψnum ψTF−G rel. error in %1 · 10−10 2.6429 · 10−04 2.6766 · 10−04 1.2772

20 2.6252 · 10−04 2.6721 · 10−04 1.280840 2.5723 · 10−04 2.6054 · 10−04 1.289350 2.5327 · 10−04 2.5655 · 10−04 1.2982




r ψnum ψTF−G rel. error in %70 2.4277 · 10−04 2.4597 · 10−04 1.319690 2.2884 · 10−04 2.3196 · 10−04 1.3663100 2.2059 · 10−04 2.2369 · 10−04 1.4089120 2.0151 · 10−04 2.0464 · 10−04 1.5576135 1.8485 · 10−04 1.8814 · 10−04 1.7820142 1.7633 · 10−04 1.7977 · 10−04 1.9519150 1.6597 · 10−04 1.6965 · 10−04 2.2218155 1.5914 · 10−04 1.6302 · 10−04 2.4419160 1.5197 · 10−04 1.5614 · 10−04 2.7458170 1.3692 · 10−04 1.4155 · 10−04 3.3840180 1.2073 · 10−04 1.2569 · 10−04 4.1113190 1.0370 · 10−04 1.0820 · 10−04 4.3414195 9.5008 · 10−05 9.8640 · 10−05 3.8230200 8.6265 · 10−05 8.8337 · 10−05 2.4027205 7.7675 · 10−05 7.7026 · 10−05 0.8344210 6.9241 · 10−05 6.4209 · 10−05 7.2669215 6.1138 · 10−05 4.8750 · 10−05 20.2615220 5.3445 · 10−05 2.6422 · 10−05 50.5611

222.1441 5.0324 · 10−05 9.6857 · 10−06 80.7533

TAB. 5.36: Wave function as a function of the condensate radius for aau

= 5·10+03


20 1.8541 · 10−04 1.8736 · 10−04 1.054735 1.8381 · 10−04 1.8575 · 10−04 1.056250 1.8134 · 10−04 1.8326 · 10−04 1.059965 1.7800 · 10−04 1.7990 · 10−04 1.069685 1.7222 · 10−04 1.7408 · 10−04 1.0856100 1.6690 · 10−04 1.6873 · 10−04 1.1013115 1.6074 · 10−04 1.6254 · 10−04 1.1248135 1.5122 · 10−04 1.5300 · 10−04 1.1786150 1.4309 · 10−04 1.4487 · 10−04 1.2493175 1.2756 · 10−04 1.2947 · 10−04 1.5002187 1.1916 · 10−04 1.2122 · 10−04 1.5919195 1.1318 · 10−04 1.1538 · 10−04 1.9523200 1.0928 · 10−04 1.1160 · 10−04 2.1262210 1.0106 · 10−04 1.0368 · 10−04 2.5937


5.5. Comparison between ψTF−G and ψnum 101


r ψnum ψTF−G rel. error in %220 9.2324 · 10−05 9.5242 · 10−05 3.1613230 8.3042 · 10−05 8.6198 · 10−05 3.8008240 7.3291 · 10−05 7.6400 · 10−05 4.2421250 6.3294 · 10−05 6.5584 · 10−05 3.6182260 5.3323 · 10−05 5.3217 · 10−05 0.1984265 4.8447 · 10−05 4.6097 · 10−05 4.8488270 4.3681 · 10−05 3.7916 · 10−05 13.1976275 3.9044 · 10−05 2.7763 · 10−05 28.8906280 3.4756 · 10−05 1.1202 · 10−05 67.7689

280.992 3.3944 · 10−05 2.7245 · 10−07 99.1973


= 8·10+03


15 1.5733 · 10−04 1.5885 · 10−04 0.970335 1.5601 · 10−04 1.5753 · 10−04 0.975550 1.5433 · 10−04 1.5584 · 10−04 0.982365 1.5207 · 10−04 1.5357 · 10−04 0.986785 1.4815 · 10−04 1.4962 · 10−04 0.9973100 1.4455 · 10−04 1.4599 · 10−04 1.0000115 1.4036 · 10−04 1.4179 · 10−04 1.0199135 1.3390 · 10−04 1.3530 · 10−04 1.0499150 1.2839 · 10−04 1.2978 · 10−04 1.0877165 1.2232 · 10−04 1.2370 · 10−04 1.1343185 1.1330 · 10−04 1.1472 · 10−04 1.2577200 1.0581 · 10−04 1.0731 · 10−04 1.4218220 9.4756 · 10−05 9.6482 · 10−05 1.8218240 8.2275 · 10−05 8.4433 · 10−05 2.6236250 7.5461 · 10−05 7.7870 · 10−05 3.1928260 6.8285 · 10−05 7.0867 · 10−05 3.7822280 5.3103 · 10−05 5.5049 · 10−05 3.6661290 4.5410 · 10−05 4.5713 · 10−05 0.6688300 3.7906 · 10−05 3.4518 · 10−05 8.9361310 3.0896 · 10−05 1.8432 · 10−05 40.3400

314.159 2.8162 · 10−05 1.4627 · 10−07 99.4806


= 1·10+04


The results are shown in FIG. G.1 and FIG. G.2 in appendix G. The analyticalform of the wave function is in general in good agreement with the numericallysolved wave function but shows significant deviations at large r where it cannotreproduce the exponential decay.

5.6 Accuracy of the numerical energy values

The accuracy of the numerically calculated values can be determined by varyingthe accuracy of the Runge-Kutta routine. We calculate the mean-field energyof the system for different accuracies tabulated in TAB. 5.39 and compare theirvalues. From this it follows that the first three digits after the decimal point thatare different from zero remain identical.

accuracy E10−15 −0.00527746298544138710−14 −0.00527774922319977310−13 −0.00527924372072980310−12 −0.00527833910895737010−11 −0.00527015824907360810−10 −0.005276796851201327

TAB. 5.39: Comparison of the energies for different accuracies by b = 6

We see that the energies are correct to three to four significant digits. Hence theenergy eigenvalues “that make sense” are tabulated in TAB. 5.40. For this reasonwe have rounded all values tabulated in this thesis up to the first four digits afterthe decimal point.

accuracy E10−15 −0.00527(7)10−14 −0.00527(8)10−13 −0.00527(9)10−12 −0.00527(8)10−11 −0.00527(0)10−10 −0.00527(7)

TAB. 5.40: Rounded energy eigenvalues for different accuracies by b = 6

5.7. Tangent bifurcation 103

5.7 Tangent bifurcation

Both in the numerical and in the variational calculations we obtain two solutionsdown to a certain negative scattering length for one and the same scatteringlength. Results are listed in TAB. 5.41. The scattering length is given in red,the second solution in magenta, while the original solution is given in blue. Thecause for this behaviour can be seen from FIG. 5.4. If we insert (4.83) in (4.82)we obtain the variational GTO mean-field energy as a function of the condensateradius:

E(r) =9

4

1

r2+ γ2r2 +

3

2

√3

π

a

au

1

r3−

√3

π

1

r. (5.27)

Up to a certain value of the negative scattering length N2a/au = −0.25 thereexists no solution. At the value of N2a/au = −0.2 we obtain two solutionsbecause of the two stationary points of the energy. At the values of N2a/au = 0.0,N2a/au = 0.2 and N2a/au = 3.0 we get only one solution because the energy hasthere one stationary point.

FIG. 5.4: Mean-field energy as a function of the condensate radius [O’Dell et al.(2000)].

b ν N2 aau

E/N3 µ/N2

−3.0 2.2447 −0.5953 3.1564 · 10−01 −2.6477−2.0 1.5470 −0.8356 −6.5278 · 10−02 −1.3952




b ν N2 aau

E/N3 µ/N2

−1.4468 1.2198 −0.9723 −1.3119 · 10−01 −9.5589 · 10−01

−1.4 1.1941 −0.9817 −1.3382 · 10−01 −9.2503 · 10−01

−1.3 1.1403 −0.9996 −1.3823 · 10−01 −8.6199 · 10−01

−1.1 1.0371 −1.0226 −1.4268 · 10−01 −7.4713 · 10−01

−1.02516 1.0000 −1.0251 −1.4305 · 10−01 −7.0774 · 10−01

−0.7 0.8484 −0.9723 −1.3836 · 10−01 −5.5708 · 10−01

−0.3 0.6839 −0.6412 −1.2317 · 10−01 −4.1122 · 10−01

−3.35 · 10−08 0.5766 −1 · 10−07 −1.0844 · 10−01 −3.2554 · 10−01

TAB. 5.41: Numerical results for the energy and the chemical potential for dif-ferent negative scattering lengths

For the variational calculations with negative scattering lengths we use the neg-ative sign in (4.88) denoted by k−, so that we obtain

kmin ≡ k− =1

2

√π

2

1

N aau

(−

√1 +

8

3πN2

a

au− 1

). (5.28)

We present some values for the chemical potential µ and the mean-field energyE in TAB. 5.42. For the critical value N2 a

au≈ −1.1780 · 10+00 the parameters k−

and k+ as well as the other quantities become equivalent. We compare the valuesof TAB. 5.42 with TAB. 5.31 and TAB. 5.32 and represent them in FIG. 5.5.

N2 aau

µ/N2 E/N3

−1.1780 · 10+00 −7.1514 · 10−01 −1.4145 · 10−01

−1.16 · 10+00 −8.3027 · 10−01 −1.3863 · 10−01

−1.1 · 10+00 −1.0186 · 10+00 −1.2446 · 10−02

−1.0251 · 10+00 −1.2304 · 10+00 −9.6563 · 10−02

−9.7234 · 10−01 −1.3897 · 10+00 −6.8547 · 10−02

−8.5841 · 10−01 −1.7975 · 10+00 2.5791 · 10−02

−7.6556 · 10−01 −2.2349 · 10+00 1.5576 · 10−01

−4.7823 · 10−01 −5.1604 · 10+00 1.4577 · 10+00

−1 · 10−01 −8.8213 · 10+01 6.8647 · 10+01

−1 · 10−03 −7.8639 · 10+05 7.8439 · 10+05

TAB. 5.42: Variational results with k− for the energy and the chemical potentialfor different negative scattering lengths

From FIG. 5.5 (a) it is evident that at the critical value of N2a/au = −1.0251two solutions are born “out of nowhere” in a tangent bifurcation.

5.7. Tangent bifurcation 105

-1.2 -0.9 -0.6 -0.3 0N

2a/a

u

-6

-5

-4

-3

-2

-1

0µ/

N3

-1.0251

-1.1780

(a) Chemical potential as a function of N2a/au.

-1.2 -0.9 -0.6 -0.3 0N

2a/a

u

-0.2

0

0.2

0.4

0.6

0.8

1

E/N

3

-1.0251-1.1780

(b) Mean-field energy as a function of N2a/au.

FIG. 5.5: Solid violet lines for variational computations with k− and dashed withk+; green lines for numerical computations.

Chapter 6

Beyond mean-field approximation

In this chapter we present some results of the theory beyond the mean-fieldapproximation. The basics are discussed in detail in [Braaten and Nieto (1997);McKinney et al. (2004)].

6.1 Quantum corrections to the ground state of

self-trapped BEC

The correction to the contact potential can be derived by means of a many-body perturbation-theory [McKinney et al. (2004)] or using a gradient expansioncarrying out a self-consistent one-loop calculation. The correction term takes intoaccount the dominant effects of quantum fluctuations around the mean-field.

The modified GP equation (MGP) reads

[T (r) + Vext(r) +N

∫d3r′ |ψ(r′)|2W (r, r′)

]ψ(r) = µψ(r) (6.1)

with the two-body potential

W (r, r′) =4π~2a

mδ(3)(r − r′)

(1 +

32

3√πa

32N

12ψ

)− u

|r − r′| , (6.2)

including the correction term. In “atomic” units (6.2) takes the form

W (r, r′) = Wc(r, r′) +Wc,qc(r, r

′) +Wu(r, r′), (6.3)

W (r, r′) = 8πa

auδ(3)(r − r′)

(1 +

32

3√π

(a

au

) 32

N12ψ

)− 2

|r − r′| (6.4)

107

108 Chapter 6. Beyond mean-field approximation

with

Vc(r) = 8πa

au|ψ(r)|2, (6.5)

Vc,qc(r) = 8πa

au|ψ(r)|2 32

3√π

(a

au

) 32

N12ψ = Vc(r)

32

3√π

(a

au

) 32

N12ψ, (6.6)

Vu(r, r′) = −2

∫d3r′

|ψ(r′)|2|r − r′| . (6.7)

We obtain the modified energy functional

EMGP [ψ] =N

⟨−∆r + γ2r2 + 4Nπ

a

au|ψ(r)|2

(1 +

128

15√π

(a

au

) 32

N12ψ

)

−N∫d3r′

|ψ(r′)|2|r − r′|

⟩

=N

⟨µ− N

2Vc − N

2Vu − N

2Vc

64

5√π

(a

au

) 32

N12ψ

⟩, (6.8)

with the corresponding chemical potential

µ =∂EMGP

∂N

= −∆r + γ2r2 + 8Nπa

au|ψ(r)|2

(1 +

32

3√π

(a

au

) 32

N12ψ

)

− 2N

∫d3r′

|ψ(r′)|2|r − r′| . (6.9)

We compare the results of the numerical calculation of the contact potential Vcwith the corresponding correction term Vc,qc in TAB. 6.1 for different scatteringlengths.

N2 aau

Vc/N2 Vc,qc/N

2

1 · 10−07 3.4305 · 10−09 3.3111 · 10−23

1 · 10−06 3.4305 · 10−08 1.0470 · 10−20

1 · 10−05 3.4304 · 10−07 3.3111 · 10−18

1 · 10−04 3.4304 · 10−06 1.0469 · 10−15

1 · 10−03 3.4292 · 10−05 3.3088 · 10−13

1 · 10−02 3.4188 · 10−04 1.0397 · 10−10

1 · 10−01 3.1908 · 10−03 2.8038 · 10−08

3 · 10−01 8.4054 · 10−03 3.2429 · 10−07

1 · 10+00 1.9781 · 10−02 2.9676 · 10−06


6.1. Quantum corrections to the ground state of self-trapped BEC 109


N2 aau

Vc/N2 Vc,qc/N

2

1 · 10+01 3.5588 · 10−02 1.8418 · 10−05

3 · 10+01 3.0880 · 10−02 1.6781 · 10−05

5 · 10+01 2.7266 · 10−02 1.4094 · 10−05

8 · 10+01 2.3720 · 10−02 1.1373 · 10−05

1 · 10+02 2.2060 · 10−02 1.0126 · 10−05

3 · 10+02 1.4691 · 10−02 5.1552 · 10−06

5 · 10+02 1.1914 · 10−02 3.6092 · 10−06

8 · 10+02 9.7416 · 10−03 2.5552 · 10−06

1 · 10+03 8.8328 · 10−03 2.1584 · 10−06

3 · 10+03 5.3642 · 10−03 9.1030 · 10−07

5 · 10+03 4.2228 · 10−03 6.0060 · 10−07

8 · 10+03 3.3784 · 10−03 4.0726 · 10−07

1 · 10+04 3.0362 · 10−03 3.3805 · 10−07

TAB. 6.1: Quantum corrections for the contact potential

Obviously the quantum corrections are smaller by factors of 10−3–10−4 com-pared with the values of the contact interaction of the ground state of self-trappedBEC1.

1We do not include the correction term of the gravity-like potential.

110 Chapter 6. Beyond mean-field approximation

Chapter 7

Summary

In this chapter we give a brief summary of the objectives and the results of thepresent thesis. At the end follows an outlook on future issues beyond this thesis.

7.1 Purpose of the study

• Firstly we solved the radial Schrodinger equation of self-trapped Bose-Einstein condensates with electromagnetically induced 1/r-interaction sothat we were able to obtain solutions for the wave functions and poten-tials. To that aim we extended the numerics that was used for the solutionsof the Newton-Schrodinger equation [Greiner (2005)]. Thereafter we cal-culated energies, sizes and peak densities of self-trapped BEC with 1/r-interaction for different particle numbers.

• As an alternative to the numerical calculations we used an approximationmethod based on the variational principle. We “guessed” a wave functionfor the ground state of the BEC and calculated the energy functional of thesystem. It has been found that the “Gaussian Type Orbital” (GTO) rep-resents a more adapted wave function for our nonlinear Schrodinger equa-tion, because it better minimizes the energy functional of the system thanan STO wave function. We performed calculations for energies, sizes andpeak densities and compared the results of the numerical and variationalcalculations.

7.2 The results

• This thesis presents for the first time accurate numerical solutions for ener-gies, sizes and peak densities of self-trapped Bose-Einstein condensates withelectromagnetically induced 1/r-interaction. The results are calculated via

111

112 Chapter 7. Summary

a fourth-order Runge-Kutta routine with adaptive step-size control andcompared with analytical results of a variational ansatz.

• We introduce “atomic” units and show that these are the most adequateunits to understand the physics of self-trapped BEC, with the trappingpotential switched off.

• The physics of self-trapping depends only on the parameter N2 aau

, where Nis the particle number, a the scattering length, and au the “Bohr” radius.

• We can also make some qualitative conclusions

– As the number of particles is increased, the energy eigenvalues de-crease, at fixed scattering length. This is due to the fact that anincreased particle number induces a more strongly binding potential.

– With increasing number of particles the solutions are stronger local-ized. This is also caused by a more strongly binding potential.

– For increasing a/au the potential grows shallower and the wave func-tion becomes more extended.

– For negative scattering lengths, the potential becomes ever more bind-ing, until the condensate becomes unstable with respect to collapseleading to bifurcation at a certain critical value of N2 a

au= −1.0251.

• Two new physical regions are characterized by the parameter N2 aau

. The“TF-G” region appears due to the balance of the contact potential and thegravity-like attraction for N2 a

augreater than 1, where the kinetic energy is

negligible. The “G” region, which corresponds to the Newton-Schrodingerscheme and has been investigated in detail in the literature, appears due tothe balance of the kinetic energy and the gravity-like attraction for N2 a

au

less than 1, where the scattering is negligible.

7.3 Outlook

• In the future the algorithms should be improved so that one can increasethe accuracy of the numerical results. That can probably be achieved bythe finite element method. The solutions can be evolved in terms of simplerbasis functions such as B-splines. Details can be found in [Zienkiewicz etal. (2006)]. A further improvement is to start the numerical calculations byr = 0 with the Taylor-expansions (3.109a) and (3.109b).

• For the numerical integration we make use of the trapezoidal rule. An exam-ple is given in appendix H. Instead of this one can use better discretizationmethods such as the Simpson rule.

7.3. Outlook 113

• The dynamical behavior of the system can be analyzed.

• The research on excitations of BEC with attractive 1/r-interaction can alsobe analyzed [Wagner (2007)].

• The biggest efforts will be to go beyond the mean-field approximation andto calculate the correction term of the gravity-like potential.

• A further challenge is the solution of the unsymmetrical extended Gross-Pitaevskii equation including the angular components.

114 Chapter 7. Summary

Appendix A

Integral representation of ψ(r)and U(r)

Our main goal in chapter 3.6 is to derive an integral formulation of the equations(3.52). These equations are integrated twice and we obtain

ψ(r)− ψ(0) = −r∫

0

1

y2

y∫

0

x2U(x)ψ(x) dx dy + C

r∫

0

1

y2

y∫

0

x2ψ(x)3 dx dy, (A.1a)

U(r)− U(0) = −8π

r∫

0

1

y2dy

r∫

0

x2ψ(x)2 dx. (A.1b)

An integration by parts with the formula∫uv dx = (

∫u dx)v − ∫

(∫u dx) dv

dxdx

[Abramowitz and Stegun (1970)] lead us finally to

U(r)− U(0) = 8π

−

r∫

0

1

y2dy

r∫

0

x2ψ(x)2 dx

+

r∫

0

y∫

0

1

y2dy

d

dy

y∫

0

x2ψ(x)2 dx

dy

= 8π

−

(−1

y

∣∣∣∣r

0

) r∫

0

x2ψ(x)2 dx+

r∫

0

(−1

y

∣∣∣∣y

0

)y2ψ(y)2 dy

=1

r

r∫

0

x2ψ(x)2 dx−r∫

0

1

yy2ψ(y)2 dy

, (A.2)

U(r) = U(0) + 8π

r∫

0

x(xr− 1

)ψ(x)2 dx. (A.3)

115

116 Appendix A. Integral representation of ψ(r) and U(r)

ψ(r)− ψ(0) = −

r∫

0

1

y2dy

r∫

0

x2U(x)ψ(x) dx

+

r∫

0

y∫

0

1

y2dy

d

dy

y∫

0

x2U(x)ψ(x) dx

dy

+ C

r∫

0

1

y2dy

r∫

0

x2ψ(x)3 dx

− C

r∫

0

y∫

0

1

y2dy

d

dy

y∫

0

x2ψ(x)3 dx

dy

= −(−1

y

∣∣∣∣r

0

) r∫

0

x2U(x)ψ(x) dx

+

r∫

0

(−1

y

∣∣∣∣y

0

)y2U(y)ψ(y) dy

+ C

(−1

y

∣∣∣∣r

0

) r∫

0

x2ψ(x)3 dx

− C

r∫

0

(−1

y

∣∣∣∣y

0

)y2ψ(y)3 dy

=1

r

r∫

0

x2U(x)ψ(x) dx−r∫

0

1

yy2U(y)ψ(y) dy

− C

r

r∫

0

x2ψ(x)3 dx+ C

r∫

0

1

yy2ψ(y)3 dy

=

r∫

0

x(xr− 1

)U(x)ψ(x) dx− C

r∫

0

x(xr− 1

)ψ(x)3 dx

=

r∫

0

x(xr− 1

)ψ(x) dx

[U(x)− Cψ(x)2

], (A.4)

ψ(r) = ψ(0) +

r∫

0

x(xr− 1

)ψ(x) dx

[U(x)− Cψ(x)2

]. (A.5)

Appendix B

Derivatives for the Taylorexpansion

We calculate the derivatives at r = 0 of ψ and U , where the even derivativeshave to vanish because of the symmetry. This fact takes also into account thatthe first derivatives of ψ and U vanish, too. Now we begin to calculate the firstderivative of the two functions

∂rψ0 = ∂r (rψ0) = ψ0 (∂rr) + r(∂rψ0) = ψ0 + r(∂rψ0)︸︷︷︸=0

= ψ0, (B.1)

∂rU0 = ∂r(rU0) = U0, (B.2)

followed by the second derivative

∂2r ψ0 = −1

rU0ψ0 +

C

r2ψ3

0 = −rψ0U0 + Crψ0 = 0, (B.3)

∂2r U0 = −8π

rψ2

0 = −8πrψ20 = 0. (B.4)

To keep track of we calculate the third derivative of the two functions individual.Thus the third derivative of ψ reads

∂3r ψ0 = ∂r

[−1

rU0ψ0 +

C

r2ψ3

0

]

=

[(−∂r 1

r

)U0ψ0 − 1

r

(∂rU0ψ0

)+

(∂rC

r2

)ψ3

0 +C

r2

(∂rψ

30

)]

=

[1

r2U0ψ0 − 1

r(ψ0∂rU0 + U0∂rψ0)− 2C

r3ψ3

0 +3C

r2ψ2

0∂rψ0

]

= U0ψ0 − ψ0U0 + U0ψ0 − 2Cψ30 + 3Cψ3

0

= −U0ψ0 + Cψ30 (B.5)

117

118 Appendix B. Derivatives for the Taylor expansion

following from the third derivative of U

∂3r U0 = ∂r∂

2r U0 = −8π∂r

(1

rψ2

0

)= −8π

(− 1

r2ψ2

0 +2

rψ0∂rψ0

)

= −8π

(− 1

r2r2ψ2

0 +2

rrψ2

0

)= −8π(−ψ2

0 + 2ψ20)

= −8πψ20. (B.6)

Now we need the fourth derivative of ψ

∂4r ψ0 = ∂r

[1

r2U0ψ0 − 1

r(ψ0∂rU0 + U0∂rψ0)− 2C

r3ψ3

0 +3C

r2ψ2

0∂rψ0

]

=1

r2

(U0∂rψ0 + ψ0∂rU0

)− 2

r3U0ψ0 +

1

r2

(ψ0∂rU0 + U0∂rψ0

)

− 1

r

(ψ0∂

2r U0 + 2∂rU0∂rψ0 + U0∂

2r ψ0

)+

6C

r4ψ3

0 −6C

r3ψ2

0∂rψ0

− 6C

r3ψ2

0∂rψ0 +6C

r2ψ0∂rψ0∂rψ0 +

3C

r2ψ2

0∂2r ψ0

=1

rU0ψ0 +

1

rU0ψ0 − 2

rU0ψ0 +

1

rU0ψ0 +

1

rU0ψ0

− 2

rU0ψ0 +

6C

rψ3

0 −12C

rψ3

0 +6C

rψ3

0

=2

r(U0ψ0 − U0ψ0 + U0ψ0 − U0ψ0)

= 0 (B.7)

and the fourth derivative of U

∂4r U0 = ∂r

[−8π

(− 1

r2ψ2

0 +2

rψ0∂rψ0

)]

= −8π

[2

r3ψ2

0 −2

r2ψ0∂rψ0 − 2

r2ψ0∂rψ0 +

2

r

((∂rψ0

)2+ ψ0∂

2r ψ0

)]

= −8π

[2

rψ2

0 −2

rψ2

0 −2

rψ2

0 +2

rψ2

0

]

= 0. (B.8)

Finally we calculate the fifth derivative of ψ

∂5r ψ0 = ∂r

[− 2

r3U0ψ0 +

2

r2

(U0∂rψ0 + ψ0∂rU0

)

− 1

r

(ψ0∂

2r U0 + 2∂rU0∂rψ0 + U0∂

2r ψ0

)+

6C

r4ψ3

0

−12C

r3ψ2

0∂rψ0 +6C

r2ψ0∂rψ0∂rψ0 +

3C

r2ψ2

0∂2r ψ0

]

119

=6

r4U0ψ0 − 6

r3

(ψ0∂rU0 + U0∂rψ0

)+

3

r2

(ψ0∂

2r U0 + 2ψ0∂rU0 + U0∂

2r ψ0

)

− 1

r

(ψ0∂

3r U0 + 3∂2

r U0∂rψ0 + 3∂2r ψ0∂rU0 + U0∂

3r ψ0

)

− 24C

r5ψ3

0 +54C

r4ψ2

0∂rψ0 − 36C

r3ψ0

(∂rψ0

)2 − 18C

r3ψ2

0∂2r ψ0

+6C

r2

(∂rψ0

)3+

18C

r2ψ0∂rψ0∂

2r ψ0 +

3C

r2ψ2

0∂3r ψ0

=6

r2U0ψ0 − 12

r2U0ψ0 +

6

r2U0ψ0

− 1

r

(rψ0

(−8πψ20

)+ rU0

(−U0ψ0 + Cψ30

))

− 24C

r2ψ3

0 +54C

r2ψ3

0 −36C

r2ψ3

0 +6C

r2ψ3

0 + 3Cψ20

(−U0ψ0 + Cψ30

)

= −1

r

(−8πrψ30 − rψ0U

20 + rCU0ψ

30

)− 3Cψ30U0 + 3C2ψ5

0

= 8πψ30 + ψ0U

20 − 4CU0ψ

30 + 3C2ψ5

0 (B.9)

and the fifth derivative of U

∂5r U0 = −8π∂r

[2

r3ψ2

0 −4

r2ψ0∂rψ0 +

2

r

(∂rψ0

)2+

2

rψ0∂

2r ψ0

]

= −8π

[− 6

r4ψ2

0 +4

r3ψ0∂rψ0 +

8

r3ψ0∂rψ0 − 4

r2

((∂rψ0

)2+ ψ0∂

2r ψ0

)

− 2

r2

(∂rψ0

)2+

2

r

(2∂2

r ψ0∂rψ0

)− 2

r2ψ0∂

2r ψ0

+2

r

(∂rψ0∂

2r ψ0 + ψ0∂

3r ψ0

)]

= −8π

[− 6

r4ψ2

0 +12

r3ψ0∂rψ0 − 6

r2

(∂rψ0

)2

− 6

r2ψ0∂

2r ψ0 +

6

r∂rψ0∂

2r ψ0 +

2

rψ0∂

3r ψ0

]

= −8π

[− 6

r2ψ2

0 +12

r2ψ2

0 −6

r2ψ2

0 + 2ψ0

(−U0ψ0 + Cψ30

)]

= 16π(U0ψ

20 − Cψ4

0

). (B.10)

120 Appendix B. Derivatives for the Taylor expansion

Appendix C

Integrals

All integrals are from [Bronstein et al. (2001)].

∞∫

0

dx xne−ax =n!

an+1for a > 0, n = N. (C.1)

∞∫

0

dr xne−2ax =n!

(2a)n+1for a > 0, n = N. (C.2)

∞∫

0

dx x2e−a2x2

=

√π

4a3for a > 0. (C.3)

∞∫

0

dx x2e−2a2x2

=

√π

4(√

2a)3 for a > 0. (C.4)

∞∫

0

dx x4e−a2x2

=3√π

8a5for a > 0. (C.5)

∫dx x2eax = eax

(x2

a− 2x

a2+

2

a3

). (C.6)

121

122 Appendix C. Integrals

Appendix D

Non-normalized solutions

0 5 10 15 20rN

-3

-2

-1

0

1

2

3SU

(a) S0 = 2 and U0 = 1/100

0 0.1 0.2 0.3 0.4rN

1.8

1.84

1.88

1.92

1.96

2SU

(b) magnification near r = 0

0 5 10 15 20rN

-3

-2

-1

0

1

2

3SU

(c) S0 = 1 and U0 = 2

0 0.1 0.2 0.3 0.4rN

0.9

0.92

0.94

0.96

0.98

1

SU

(d) magnification near r = 0

123

124 Appendix D. Non-normalized solutions

0 5 10 15 20rN

-3

-2

-1

0

1

2

3SU

(e) S0 = 1/2 and U0 = 100

0 0.1 0.2 0.3 0.4rN

0.45

0.46

0.47

0.48

0.49

0.5

SU

(f) magnification near r = 0

FIG. D.1: Non-normalized wave functions S and potentials U for b = 0; Sarbitrary but constant, as U(r = 0) is varied.

Appendix E

Wave functions and potentials

0 2 4 6 8 10 12 14rN

0

0.03

0.06

0.09

0.12

0.15

0.18

ψ/Ν

3/2

(a) wave function

0 5 10 15 20 25 30 35rN

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0V

/N3

(b) blue line: gravity-like potential; green line:effective self-binding potential

FIG. E.1: Wave function and corresponding potentials for N2 aau

= −1.

0 2 4 6 8 10 12 14rN

0

0.02

0.04

0.06

0.08

0.1

ψ/Ν

3/2

(a) wave function

0 5 10 15 20 25 30 35rN

-0.5

-0.4

-0.3

-0.2

-0.1

0

V/N

3



= −6 · 10−01.

125

126 Appendix E. Wave functions and potentials

0 2 4 6 8 10 12 14rN

0

0.02

0.04

0.06

0.08ψ

/Ν3/

2

(a) wave function

0 5 10 15 20 25 30 35rN

-0.4

-0.3

-0.2

-0.1

0

V/N

3



= −3 · 10−01.

0 5 10 15 20 25 30rN

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

ψ/Ν

3/2

(a) wave function

0 5 10 15 20 25 30rN

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

V/N

3

(b) self-trapping gravity-like potential


= 0.

0 5 10 15 20 25 30 35rN

0

0.004

0.008

0.012

0.016

0.02

0.024

ψ/Ν

3/2

(a) wave function

0 20 40 60 80 100rN

-0.18

-0.15

-0.12

-0.09

-0.06

-0.03

0

V/N

3



= 1 · 10+01.

127

0 10 20 30 40 50 60 70rN

0

0.001

0.002

0.003

0.004

0.005

ψ/Ν

3/2

(a) wave function

0 20 40 60 80 100rN

-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01

0

V/N

3



= 1 · 10+02.

0 20 40 60 80 100 120rN

0

0.0003

0.0006

0.0009

0.0012

0.0015

ψ/Ν

3/2

(a) wave function

0 50 100 150 200 250rN

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

V/N

3



= 5 · 10+02.

0 50 100 150 200rN

0

0.0002

0.0004

0.0006

0.0008

0.001

ψ/Ν

3/2

(a) wave function

0 60 120 180 240 300 360rN

-0.02

-0.016

-0.012

-0.008

-0.004

0

V/N

3



= 1 · 10+03.

128 Appendix E. Wave functions and potentials

0 40 80 120 160 200 240 280rN

0

0.0001

0.0002

0.0003

0.0004

ψ/Ν

3/2

(a) wave function

0 70 140 210 280 350 420rN

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

V/N

3



= 3 · 10+03.

0 50 100 150 200 250 300 350rN

0.0

6.0×10-5

1.2×10-4

1.8×10-4

2.4×10-4

3.0×10-4

ψ/Ν

3/2

(a) wave function

0 80 160 240 320 400 480rN

-0.01

-0.008

-0.006

-0.004

-0.002

0

V/N

3



= 5 · 10+03.

0 100 200 300 400 500rN

0.02.5×10

-55.0×10

-57.5×10

-51.0×10

-41.2×10

-41.5×10

-41.8×10

-4

ψ/Ν

3/2

(a) wave function

0 100 200 300 400 500 600rN

-7.0×10-3

-6.0×10-3

-5.0×10-3

-4.0×10-3

-3.0×10-3

-2.0×10-3

-1.0×10-3

0.0

V/N

3



= 1 · 10+04.

Appendix F

Graphs

We plot all energies, sizes and peak densities of section 5.4 for which tabularvalues were given in.

-1 0 1 2 3 4 5N

2a/a

u

0

2

4

6

8

10

r r.m

.s√

G TF-G

N

(a) radius of the condensate for −1 / N2 aau≤

5

10-3

10-2

10-1

100

101

102

103

104

105

N2a/a

u

100

101

102

103

r r.m

.s√

~ √N

2a

N

TF-GG

(b) radius of the condensate for 1 ≤ N2 aau≤

105

-1 -0.5 0 0.5 1N

2a/a

u

0

5

10

15

20

25

30

r r.m

.s√

in

%N

(c) relative error for −1 / N2 aau≤ 1

100

101

102

103

104

N2a/a

u

0

1

2

3

4

5

6

7

r r.m

.s√

in

[%]

N

(d) relative error for 1 ≤ N2 aau≤ 104

FIG. F.1: Green lines: numerical computations; blue lines: variational compu-tations; red lines: relative error between the numerical and variational computa-tions.

129

130 Appendix F. Graphs

-1 0 1 2 3 4 5N

2a/a

u

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

E/N

3

G TF-G

(a) mean-field energy for −1 / N2 aau≤ 5

10-3

10-2

10-1

100

101

102

103

104

105

N2a/a

u

-0.1

-0.08

-0.06

-0.04

-0.02

0

E/N

3 ~ - 1

N2a√

TF-GG

(b) mean-field energy for 1 ≤ N2 aau≤ 105

-1 -0.5 0 0.5 1N

2a/a

u

0

2

4

6

8

E/N

3 in [

%]


100

101

102

103

104

N2a/a

u

0

0.5

1

1.5

2

2.5

3

3.5E

/N3 in

[%

]


10 100rr.m.s

√

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

E/N

3

N

(e) mean-field energy as a function of the meanradius of the condensate


131

-1 0 1 2 3 4 5N

2a/a

u

0

0.1

0.2

0.3

0.4

T/N

3 TF-GG

(a) kinetic energy for −1 / N2 aau≤ 5

10-3

10-2

10-1

100

101

102

103

104

105

N2a/a

u

0

0.02

0.04

0.06

0.08

0.1

T/N

3 TF-GG

(b) kinetic energy for 1 ≤ N2 aau≤ 105

-1 -0.5 0 0.5 1N

2a/a

u

0

5

10

15

20

25

30

T/N

3 in [

%]


100

101

102

103

104

N2a/a

u

0

5

10

15

20

25

30T

/N3 in

[%

]

(d) relative error for 1 ≤ N2 aau≤ 1 · 104

10 100rr.m.s

√

0

0.1

0.2

0.3

0.4

T/N

3

N

(e) kinetic energy as a function of the meanradius of the condensate



-1 0 1 2 3 4 5N

2a/a

u

-0.24

-0.2

-0.16

-0.12

-0.08

-0.04

0

0.04

Vc/N

3

TF-GG

(a) contact potential for −1 / N2 aau≤ 5

10-2

10-1

100

101

102

103

104

105

N2a/a

u

0

0.01

0.02

0.03

0.04

Vc/N

3

√N

2a

~ 1

TF-GG

(b) contact potential for 1 ≤ N2 aau≤ 105

-1.0 -0.5 0.0 0.5 1.0N

2a/a

u

0

10

20

30

40

50

60

70

Vc/N

3 in [

%]


101

102

103

104

N2a/a

u

0

0.5

1

1.5

2V

c/N3 in

[%

]


10 100rr.m.s

√

-0.3

-0.24

-0.18

-0.12

-0.06

0

0.06

Vc/N

3

N

(e) contact potential as a function of the meanradius of the condensate


133

-1 0 1 2 3 4 5N

2a/a

u

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

Vu/N

3

G TF-G

(a) gravity-like potential for −1 / N2 aau≤ 5

10-3

10-2

10-1

100

101

102

103

104

105

N2a/a

u

-0.5

-0.4

-0.3

-0.2

-0.1

0

Vu/N

3

~ 1

N2a√

TF-GG

(b) gravity-like potential for 1 ≤ N2 aau≤ 105

-1 -0.5 0 0.5 1N

2a/a

u

0

4

8

12

16

20

24

28

Vu/N

3 in [

%]


100

101

102

103

104

N2a/a

u

0

0.5

1

1.5

2

2.5

3

3.5V

u/N3 in

[%

]


10 100rr.m.s

√

-0.9

-0.75

-0.6

-0.45

-0.3

-0.15

0

Vu/N

3

N

(e) gravity-like potential as a function of themean radius of the condensate



-1 0 1 2 3 4 5N

2a/a

u

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

µ/N

3

G TF-G

(a) chemical potential for −1 / N2 aau≤ 5

10-3

10-2

10-1

100

101

102

103

104

105

N2a/a

u

-0.4

-0.3

-0.2

-0.1

0

µ/N

3 √N

2a

~ - 1

TF-GG

(b) chemical potential for 1 ≤ N2 aau≤ 105

-1 -0.5 0 0.5 1N

2a/a

u

0

5

10

15

20

25

30

µ/N

3 in [

%]


100

101

102

103

104

N2a/a

u

0

1

2

3

4

µ/N

3 in [

%]


10 100rr.m.s

√

-0.4

-0.3

-0.2

-0.1

0

µ/N

3

N

(e) chemical potential as a function of themean radius of the condensate


135

-1 0 1 2 3 4 5N

2a/a

u

0

0.01

0.02

0.03

ρ max

/N4

TF-GG

(a) peak density for −1 / N2 aau≤ 5

10-3

10-2

10-1

100

101

102

103

104

105

N2a/a

u

10-7

10-6

10-5

10-4

10-3

10-2

ρ max

/N4

~ 1

N2

)3/2

a(TF-GG

(b) peak density for 1 ≤ N2 aau≤ 105

-1 -0.5 0 0.5 1N

2a/a

u

0

20

40

60

80

ρ max

/N4 in

[%

]


100

101

102

103

104

N2a/a

u

0

5

10

15

20

25

30

35ρ m

ax/N

4 in [

%]


5 10 15rr.m.s

√

0

1×10-2

2×10-2

3×10-2

4×10-2

ρ max

/N4

N

(e) peak density as a function of the radius ofthe condensate

FIG. F.7: Green lines: numerical computations; blue lines: variational com-putations for; red lines: relative error between the numerical and variationalcomputations.

Appendix G

Comparison between ψTF−G andψnum

0 50 100 150 200 250 300 350 400 450r

0.0

5.0×10-5

1.0×10-4

1.5×10-4

2.0×10-4

2.5×10-4

3.0×10-4

ψ

a = 5·103

a = 5·103

a = 8·103

a = 8·103

a = 1·104

a = 1·104

FIG. G.1: Wave functions as a function of the condensate radius; dashed linesfor analytical form (5.21); solid lines for numerical computations.

137

138 Appendix G. Comparison between ψTF−G and ψnum

0 40 80 120 160 200 240 280r

5

10

15

20

25

30

∆ψ in

%a = 5·10

3

a = 8·103

a = 1·104

FIG. G.2: Relative error between the numerical and the analytical wave function(5.21) as a function of r.

Appendix H

Numerical integration

The trapezoidal rule reads [Bronstein et al. (2001)]

r0+∆∫

r0

dr f(r) ≈ ∆

2(f(r0) + f(r0 + ∆)) , (H.1)

where ∆ is the step size. The function f(r) is replaced in the interval [r0, r0 + ∆]by a first degree polynomial, that interpolate f(r) at the nodes r0 and r0 + ∆.Now we discretise for instance the integral

r0+∆∫

r0

dr ψ2r2 (H.2)

by means of (H.1), so that we obtain

r0+∆∫

r0

dr ψ2r2 =∆

2

(ψ2r2

0 + ψ (r0 + ∆)2 (r0 + ∆)2) . (H.3)

Using the Taylor expansion up to first order

f (r0 + ∆) = f(r0) +∆

1!f ′(r0) + . . . (H.4)

we obtain

ψ (r0 + ∆)2 = ψ2 +∆

1!ψ′2 = ψ2 + 2ψψ′∆.. (H.5)

Inserting (H.5) in (H.3) lead us to

r0+∆∫

r0

dr ψ2r2 =∆

2

(ψ2r2

0 +(ψ2 + 2ψψ′∆

) (r20 + 2r0∆ +O

(∆2

)))

=∆

2

(ψ2r2

0 + ψ2r20 + 2ψ2r0∆ + 2ψψ′r2

0∆ + 4ψψ′r0∆2)

= ∆(ψ2r2

0 + ψ2r0∆ + ψψ′r20∆ + 2ψψ′r0∆2

). (H.6)

139

140 Appendix H. Numerical integration

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Acknowledgements

Many people contribute to the present diploma thesis. Therefore I would liketo thank several people that have been instrumental in allowing this work to becompleted. Without their support I would not have been able to make this thesis.

• Firstly special thanks go to my first supervisor Prof. Dr. Gunter Wunnerfor giving me the opportunity to work on this subject and for sharing hisknowledge with me.

• Secondly I would like to thank my second supervisor Prof. Dr. ManfredFahnle for his comments on this thesis.

• Thirdly I would like to thank secretary Karin Fairgrieve for her alwaysfriendly manner.

• Fourthly I would like to thank Prof. Dr. Jorg Main for his attentivelyproofreading of my thesis and for the official journey to Marburg.

• Next special thanks to my dear friend Farshid Karim Pour for his numeroushelp during my diploma thesis and his understanding.

• Also I would like to thank my friend Patrick Wagner for his motivating andhelpful discussions on the area of BEC.

• Last but not least I would like to thank Steffen Bucheler, Dirk Engel andHolger Cartarius for their kind offer to help me in any problems I had duringthe last year.

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