Hyperspherical Harmonics Expansion Techniques: Application to Problems in Physics

170 Pages·2015·2.387 MB·English

by  Tapan Kumar Das

Theoretical and Mathematical Physics
Tapan Kumar Das
Hyperspherical
Harmonics
Expansion
Techniques
Application to Problems in Physics
Hyperspherical Harmonics Expansion Techniques
Theoretical and Mathematical Physics
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More information about this series at http://www.springer.com/series/720
Tapan Kumar Das
Hyperspherical Harmonics
Expansion Techniques
Application to Problems in Physics
123
Tapan KumarDas
Department ofPhysics
University of Calcutta
Kolkata, West Bengal
India
ISSN 1864-5879 ISSN 1864-5887 (electronic)
Theoretical andMathematicalPhysics
ISBN978-81-322-2360-3 ISBN978-81-322-2361-0 (eBook)
DOI 10.1007/978-81-322-2361-0
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Preface
A major part of research in physics involves solving the Schrödinger equation.
While one-body motion in a potential field and a two-body system with mutual
interactionaresubjectmattersofstandardtextsofQuantumMechanics,anabinitio
formal solution of themany-bodySchrödinger equation forinteractingmany-body
systems is not commonly encountered. The reason for this is that mathematical
complexity increases enormously as the number of particles increases from two to
three. It is not just the difficulty arising from the increasing number of position
coordinates,butalsothedifficultyinimposingthedesiredsymmetryofthesystem,
identificationofappropriateconservedquantumnumbers,etc.Naturally,physicists
tend to depend on approximate many-body techniques e.g. Born-Oppenheimer
approximation, variational and perturbation techniques, mean-field theories like
Hartree-FockandHartree-Fock-Bogoliubovmethods,etc. orontheuseofmodels,
e.g.,shell,collectiveorliquiddropmodelsinNuclearPhysics.However,anumber
of problems involving systems containing a few particles demand description in
terms of coordinates of individual particles. In such cases it is necessary to handle
the few-body Schrödinger equation in an exact manner. Hyperspherical harmonics
is the appropriate basis for this. With developments in mathematical and compu-
tational tools, it is becoming increasingly easy to handle the hyperspherical har-
monics basis. Hence it is becoming popular as an effective tool in theoretical
research.Thehypesphericaltechniqueisquitehandyforuseintheessentiallyexact
Monte Carlo methods for a fairly large number of interacting particles.
Unfortunately, there is a dearth of monographs dealing with the hyperspherical
technique.Thismonographisaimedatfulfillingthisnecessity.Besidesintroducing
the hyperspherical variables (which are many-body generalization of ordinary
spherical polar coordinates) and hyperspherical harmonics basis for the expansion
ofamany-bodywavefunction,methodstointroducedesiredsymmetryofthewave
function has been discussed. Approximation methods, which simplify the calcu-
lations, without loosing sight of the interesting physics sought after, have also
been included. Finally, discussion of a number of current topics in physics like
v
vi Preface
Bose–Einsteincondensation, wherethistechnique hasbeenveryuseful,havebeen
incorporated.
I take this opportunity to thank all my colleagues and friends at the Physics
DepartmentoftheUniversityofCalcuttafortheirhelpandcooperation.Ispecially
thank Prof. A. Raychaudhuri and Prof. P. Sen for their continuous encourage-
ment and support. I also thank all my collaborators, and in particular
Prof.B.Chakrabarti,fortheirhelpinresearch.ThanksareduetotheDepartmentof
Science and Technology, Government of India, for financial assistance for this
project(No.HR/UR/08/2011).Last,butnottheleast,Iexpressmyappreciationand
thanks tomywife Rita anddaughterArunima,fortheirunderstandingandbearing
with me, as I could not spare much time for them due to this project.
Catalysed and supported by the Science & Engineering Research Board,
DepartmentofScience&TechnologyunderitsUtilisationofScientificExpertiseof
RetiredScientistsScheme.
Kolkata Tapan Kumar Das
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Systems of One or More Particles. . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 One-Body System: A Particle in a Potential Field. . . . . . . . . . . 5
2.2 Two-Body System with Mutual Interaction . . . . . . . . . . . . . . . 7
2.2.1 Two Distinct Particles. . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Two Identical Particles: Symmetry of Wave
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Inclusion of Spin Degrees of Freedom . . . . . . . . . . . . 10
2.2.4 Introduction of Isospin Degrees of Freedom
for Nucleons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 System of Several Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Independent Particle Model: Mean-Field
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Many-Body Description . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Three-Body System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Jacobi Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Hyperspherical Harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Schrödinger Equation for Relative Motion. . . . . . . . . . . . . . . . 24
3.4 Calculation of Potential Matrix Element . . . . . . . . . . . . . . . . . 25
3.4.1 Expansion of Potential in Hyperspherical Multipoles . . 26
3.4.2 Calculation of Potential Multipole . . . . . . . . . . . . . . . 27
3.5 Symmetrization of HH Basis. . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6 Calculation of GSC for Central Potentials. . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 General Many-Body Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Jacobi Coordinates and Hyperspherical Variables. . . . . . . . . . . 34
4.2 Generalized Hyperspherical Harmonics. . . . . . . . . . . . . . . . . . 36
vii
viii Contents
4.3 Symmetrization of Wave Function . . . . . . . . . . . . . . . . . . . . . 39
4.3.1 Kinematic Rotation Vector (KRV) . . . . . . . . . . . . . . . 39
4.3.2 Symmetrization of Wave Function . . . . . . . . . . . . . . . 42
4.4 Schrödinger Equation: Coupled Differential Equations . . . . . . . 43
4.5 Approximation by Truncation of Basis . . . . . . . . . . . . . . . . . . 44
4.5.1 Restriction of Symmetry Components. . . . . . . . . . . . . 45
4.5.2 L Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
m
4.5.3 Optimal Subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5.4 Potential Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Truncation of Basis: Application to Particles and Nuclei. . . . . . 47
4.6.1 Baryons as Three-Quark Systems. . . . . . . . . . . . . . . . 48
4.6.2 Nuclear Few-Body Systems. . . . . . . . . . . . . . . . . . . . 49
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 The Trinucleon System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1 Symmetrization of Spin–Isospin Wave Function. . . . . . . . . . . . 56
5.1.1 States Having Total Isospin t¼1 and Spin s¼1 . . . . . 57
2 2
5.1.2 States Having Total Isospin t¼3 and Spin s¼1 . . . . . 60
2 2
5.1.3 States Having Total Isospin t¼1 and Spin s¼3 . . . . . 60
2 2
5.1.4 States Having Total Isospin t¼3 and Spin s¼3 . . . . . 61
2 2
5.2 Symmetrization of Total Wave Function. . . . . . . . . . . . . . . . . 61
5.2.1 General Expression for Fully Antisymmetric
Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.2 Construction of HH for Different Partitions . . . . . . . . . 63
5.3 Optimal Subset for the Trinucleon . . . . . . . . . . . . . . . . . . . . . 64
5.4 Calculation of Potential Matrix Element: GSC. . . . . . . . . . . . . 68
5.4.1 Coupling Among S and S0 States Through Central
Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.4.2 Coupling Between S and D State Through Tensor
Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4.3 Coupling Between S0and D States Through Tensor
Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4.4 Coupling Between Two D States Through Central
Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4.5 Coupling Between Two D States Through Tensor
Interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.6 Numerical Computation of GSCs. . . . . . . . . . . . . . . . 76
5.5 Results of Numerical Calculations for 3H and 3He. . . . . . . . . . 77
5.6 Addition of Three-Nucleon Forces . . . . . . . . . . . . . . . . . . . . . 79
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Contents ix
6 Application to Coulomb Systems. . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1 Two-Electron Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1.1 Exact Non-adiabatic Treatment of Two-Electron
Atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.2 Convergence of HH Expansion: Extrapolation
and Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 General Three-Body Coulomb Bound Systems. . . . . . . . . . . . . 90
6.2.1 Adiabatic Approximation in Coulomb Systems . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Potential Harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.1 Potential Harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.2 Potential Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.3 Overlap of PHs of Different Pairs. . . . . . . . . . . . . . . . . . . . . . 99
7.4 Potential Basis as Optimal Subset. . . . . . . . . . . . . . . . . . . . . . 100
7.4.1 Symmetrical PH Basis . . . . . . . . . . . . . . . . . . . . . . . 100
7.5 Potential Matrix in Unsymmetrized PH Basis. . . . . . . . . . . . . . 101
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8 Application to Bose–Einstein Condensates. . . . . . . . . . . . . . . . . . . 105
8.1 General Properties of BEC . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2 GP Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.2.1 Simplifying Assumptions and Their Limitations. . . . . . 110
8.2.2 Rigorous Proof of Existence of BEC
and Derivation of the GP Equation. . . . . . . . . . . . . . . 110
8.3 Many-Body Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.4 Need for a Short-Range Correlation Function. . . . . . . . . . . . . . 116
8.5 Results for Repulsive and Attractive Condensates. . . . . . . . . . . 119
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9 Integro-Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.1 Derivation of IDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.2 Applications of IDE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.2.1 Nuclear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.2.2 Bose–Einstein Condensates . . . . . . . . . . . . . . . . . . . . 136
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10 Computational Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.1 Solution of a Single Differential Equation. . . . . . . . . . . . . . . . 142
10.2 Solution of Coupled Differential Equations . . . . . . . . . . . . . . . 144
10.2.1 Exact Solution of the CDE . . . . . . . . . . . . . . . . . . . . 145
10.2.2 Introduction of Hypercentral Average. . . . . . . . . . . . . 149
10.2.3 Hyperspherical Adiabatic Approximation. . . . . . . . . . . 150