Angular Momentum Techniques In Quantum Mechanics

255 Pages·2002·9.7 MB·english

by  Devanathan

Angular Momentum Techniques in Quantum Mechanics
Fundamental Theories of Physics
An International Book Series on The Fundamental Theories of Physics:
Their Clarification, Development and Application
Editor:
ALWYN VAN DER MERWE, University of Denver, U.S.A.
Editorial Advisory Board:
LAWRENCE P. HORWITZ, Tel-Aviv University, Israel
BRIAN D. JOSEPHSON, University of Cambridge, U. K.
CLIVE KILMISTER, University of London, U. K.
PEKKA J. LAHTI, University of Turku, Finland
GÜNTER LUDWIG, Philipps-Universität, Marburg, Germany
NATHAN ROSEN, Israel Institute of Technology, Israel
ASHER PERES, Israel Institute of Technology, Israel
EDUARD PRUGOVECKI, University of Toronto, Canada
MENDEL SACHS, State University of New York at Buffalo, U.S.A.
ABDUS SALAM, International Centre for Theoretical Physics, Trieste, Italy
HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der
Wissenschaften, Germany
Volume 108
Angular Momentum
Techniques in Quantum
Mechanics
by
V. Devanathan
Department of Nuclear Physics,
University of Madras
and
Crystal Growth Centre
Anna University,
Chennai, India
KLUWER ACADEMIC PUBLISHERS
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eBookISBN: 0-306-47123-X
Print ISBN: 0-792-35866-X
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To my teacher
Professor Alladi Ramakrishnan
who has inspired me to take to research and teaching
This page intentionally left blank.
CONTENTS
Preface xiii
1 ANGULAR MOMENTUM OPERATORS AND THEIR
MATRIX ELEMENTS 1
1.1 Quantum Mechanical Definition . . . . . . . . . . . . . . . . . 1
1.2 Physical Interpretation of Angular Momentum Vector ... 2
1.3 Raising and Lowering Operators . . . . . . . . . . . . . . . 3
1.4 Spectrum of Eigenvalues . . . . . . . . . . . . . . . . . . . . . 4
1.5 Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 Angular Momentum Matrices . . . . . . . . . . . . . . . . . . 6
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Solutions to Selected Problems . . . . . . . . . . . . . . . . 9
2 COUPLING OF TWO ANGULAR MOMENTA 10
2.1 The Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . 10
2.2 Some Simple Properties of C.G. Coefficients . . . . . . . . . 11
2.3 General Expressions for C.G. Coefficients . . . . . . . . . . 13
2.4 Symmetry Properties of C.G. Coefficients . . . . . . . . . . 14
2.5 Iso-Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . 16
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Solutions to Selected Problems . . . . . . . . . . . . . . . . . 19
3 VECTORS AND TENSORS IN SPHERICAL BASIS 24
3.1 The Spherical Basis . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Scalar and Vector Products in Spherical Basis . . . . . . . . 26
3.3 The Spherical Tensors . . . . . . . . . . . . . . . . . . . . . . 27
3.4 The Tensor Product . . . . . . . . . . . . . . . . . . . . . . . 29
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . 30
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Solutions to Selected Problems . . . . . . . . . . . . . . . . . 31
vii
viii
4 ROTATION MATRICES - I 34
4.1 Definition of Rotation Matrix . . . . . . . . . . . . . . . . . . 34
4.2 Rotation in terms of Euler Angles . . . . . . . . . . . . . . 34
4.3 Transformation of a Spherical Vector under Rotation of
Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 The Rotation Matrix D1 (α,β,γ) . . . . . . . . . . . . . . . . 38
4.5 Construction of other Rotation Matrices . . . . . . . . . . . . 38
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . 39
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Solutions to Selected Problems . . . . . . . . . . . . . . . . 39
5 ROTATION MATRICES - II 42
5.1 The Rotation Operator . . . . . . . . . . . . . . . . . . . . . 42
5.2 The (β) ) Matrix . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 The Rotation Matrix for Spinors . . . . . . . . . . . . 45
5.4 The Clebsch-Gordan Series . . . . . . . . . . . . . . . . . . 48
5.5 The Inverse Clebsch-Gordan Series . . . . . . . . . . . . . . 49
5.6 Unitarity and Symmetry Properties of the Rotation Matrices 50
5.7 The Spherical Harmonic Addition Theorem . . . . . . . . . 52
5.8 The Coupling Rule for the Spherical Harmonics . . . . . . . 54
5.9 Orthogonality and Normalization of the Rotation Matrices . 56
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Solutions to Selected Problems . . . . . . . . . . . . . . . . . . 58
6 TENSOR OPERATORS AND REDUCED MATRIX
ELEMENTS 61
6.1 Irreducible Tensor Operators . . . . . . . . . . . . . . . . . . 61
6.2 Racah’s Definition . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.3 The Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . 63
6.4 Proofs of the Wigner-Eckart Theorem . . . . . . . . . . . . . 65
6.4.1 Method I . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4.2 Method II . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4.3 Method III . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.5 Tensors and Tensor Operators . . . . . . . . . . . . . . . . . . . 72
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Solutions to Selected Problems . . . . . . . . . . . . . . . . . . 75
7 COUPLING OF THREE ANGULAR MOMENTA 77
7.1 Definition of the U-Coefficient . . . . . . . . . . . . . . . . . . 77
7.2 The U-Coefficient in terms of C.G. Coefficients . . . . . . . 78
ix
7.3 Independence of U-Coefficient from Magnetic Quantum
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.4 Orthonormality of the U-Coefficients . . . . . . . . . . . . . 80
7.5 The Racah Coefficient and its Symmetry Properties . . . . 81
7.6 Evaluation of Matrix Elements . . . . . . . . . . . . . . . . . 82
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Solutions to Selected Problems . . . . . . . . . . . . . . . . . 85
8 COUPLING OF FOUR ANGULAR MOMENTA 88
8.1 Definition of LS-jj Coupling Coefficient . . . . . . . . . . . . 88
8.2 LS-jj Coupling Coefficient in terms of C.G. Coefficients . . . 89
8.3 Independence of the LS-jj Coupling Coefficients from the
Magnetic Quantum Numbers . . . . . . . . . . . . . . . . . . . 91
8.4 Simple Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.5 Expansion of 9-j Symbol into Racah Coefficients . . . . . . 93
8.6 Evaluation of Matrix Elements . . . . . . . . . . . . . . . . . 95
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . 96
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Solutions to Selected Problems . . . . . . . . . . . . . . . . . 97
9 PARTIAL WAVES AND THE GRADIENT FORMULA 100
9.1 Partial Wave Expansion for a Plane Wave . . . . . . . . . . 100
9.2 Distorted Waves . . . . . . . . . . . . . . . . . . . . . . . . . 102
9.3 The Gradient Formula . . . . . . . . . . . . . . . . . . . . . . 103
9.4 Derivation of the Gradient Formula . . . . . . . . . . . . . . 105
9.5 Matrix Elements Involving the ∇ Operator . . . . . . . . . 108
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . 112
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Solutions to Selected Problems . . . . . . . . . . . . . . . . 114
10 IDENTICAL PARTICLES 119
10.1 Fermions and Bosons . . . . . . . . . . . . . . . . . . . . . . . 119
10.2 Two Identical Fermions in j-j Coupling . . . . . . . . . . . . 119
10.3 Construction of Three-Fermion Wave Function . . . . . . . 120
10.4 Calculation of Fractional Parentage Coefficients . . . . . . . 123
10.5 The Iso-Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.6 The Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.7 The m-scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Review Questions . . . . . . . . . . . . . . . . . . . . . . . . . 128
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Solutions to Selected Problems . . . . . . . . . . . . . . . . 129