Quantum Mechanics for Nuclear Structure, Volume 2

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Professor Kris Heyde. Quantum Mechanics for Nuclear Structure, Volume 2

IOP Series in Nuclear Spectroscopy and Nuclear Structure

Contents

Preface

Author biographies

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 1. Representation of rotations, angular momentum and spin

1.1 Rotations in (3, R)

1.2 Matrix representations of spin and angular momentum operators

1.3 The Pauli spin matrices

1.4 Matrix representations of rotations in ket space

1.5 Tensor representations for SU(2)

1.6 Tensor representations for SO(3)

1.7 The Schwinger representations for SU(2)

1.8 A spinor function basis for SU(2)

1.9 A spherical harmonic basis for SO(3)

1.10 Spherical harmonics and wave functions

1.11 Spherical harmonics and rotation matrices

1.12 Properties of the rotation matrices

1.13 The rotation of 〈jm∣

1.14 The rotation of the Ylm(θ,ϕ)

1.15 Exercises

1.16 Spin-12 particles; neutron interferometry

1.17 The Bargmann representation

1.17.1 Representation of operators

1.18 Coherent states for SU(2)

Comments

1.19 Properties of SU(2) from coherent states

1.20 Exercises

References

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 2. Addition of angular momenta and spins

2.1 The coupling of two spin-12 particles

2.2 The general coupling of two particles with spin or angular momentum

2.3 Spin–orbit coupling

2.4 Vector spherical harmonics

2.5 Clebsch–Gordan coefficients and rotation matrices

Exercises

2.6 The coupling of many spins and angular momenta and their recoupling

2.6.1 6-j coefficients

2.6.2 9-j coefficients

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 3. Vector and tensor operators

3.1 Vector operators

3.2 Tensor operators

3.3 Matrix elements of spherical tensor operators and the Wigner–Eckart theorem

Exercises

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 4. Identical particles

4.1 Slater determinants

4.2 The occupation number representation for bosons

4.3 The occupation number representation for fermions

4.4 Hamiltonians and other operators in the occupation number representation

4.4.1 Exercises

4.5 Condensed states (superconductors and superfluids)

4.5.1 Two fermions in a degenerate set of levels with a pairing force

4.5.2 Many fermions in a degenerate set of levels with a pairing force: the quasispin formalism

4.5.3 BCS theory

4.6 The Lipkin model

Reference

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 5. Group theory and quantum mechanics

5.1 Definition of a group

Definition of an Abelian group

5.2 Groups and transformation

5.2.1 Translations

5.2.2 Rotations

5.2.3 Space–time transformation

5.3 Transformation on physical systems

5.4 Quantum mechanics: a synoptic view

5.5 Symmetry transformations in quantum mechanics

5.5.1 The unitary transformations for translations, rotations, and time evolution in quantum mechanics

5.5.2 Consequences of symmetry in quantum mechanics

5.6 Models with symmetry in quantum mechanics

5.7 Groups and algebras

5.8 Dynamical or spectrum generating algebras

5.9 Matrix groups

5.9.1 Discrete matrix groups

Example 5-1. Matrix representations for the symmetric group, S3

5.9.2 Continuous matrix groups

5.9.3 Compact and non-compact groups

5.9.4 Polynomial representation of groups

5.10 Generators of continuous groups and Lie algebras

5.10.1 The matrix group SO(3) and its generators

5.10.2 Unitary groups and SU(2)

5.11 The unitary and orthogonal groups in n dimensions, U(n) and SO(n)

Example 5-2. Derivation of the Lie algebra u(2) from the Lie group U(2)

5.12 Casimir invariants and commuting operators

5.12.1 The Casimir invariants of u(n)

Example 5-3. The Casimir invariants of u(2) and su(2)

5.12.2 The Casimir invariants of so(n)

Example 5-4. The Casimir invariant of SO(3)

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 6. Algebraic structure of quantum mechanics

6.1 Angular momentum theory as an application of a Lie algebra

6.2 The Lie algebra su(1,1) ∼ sp(1,R)

6.3 Rank-2 Lie algebras

6.3.1 su(3) and the isotropic harmonic oscillator in three dimensions

6.3.2 so(4) and the hydrogen atom (Kepler problem)

6.4 so(5) and models with ‘quadrupole’ degrees of freedom (Bohr model)

6.5 The Lie algebra sp(3,R) and microscopic models of nuclear collectivity

6.6 Young tableaux

6.6.1 SU(3) tensor tableau calculus

6.6.2 Multiplicity of a weight state in an SU(3) irrep

6.6.3 Dimension of an SU(3) irrep: Robinson ‘hook-length’ method (figure 6.8)

6.6.4 SU(2) irreps contained in an SU(3) irrep. Example

6.6.5 Kronecker products. Examples (figure 6.11)

Rules (SU(n))

6.7 Introduction to Cartan theory of Lie algebras

6.7.1 Cartan structure of the so(4) Lie algebra

6.7.2 Cartan structure of the su(3) Lie algebra

6.7.3 The generic Lie algebra

6.7.4 Irrep quantum numbers: Cartan subalgebras and Casimir operators

Examples

Irrep quantum numbers

Example: irreps for SU(3)

Reference

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 7. Perturbation theory and the variational method

7.1 Time-independent perturbation theory

7.1.1 Exercises

7.2 Time-independent perturbation theory for systems with degeneracy

7.3 An example of (second-order) degenerate perturbation theory

7.4 Perturbation theory and symmetry

7.4.1 Example

7.4.2 Inversion symmetry

7.4.3 Example

7.4.4 Exercises

7.5 The variational method

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 8. Time-dependent perturbation theory

8.1 The interaction picture

8.2 Time-dependent perturbation theory

8.3 Constant perturbations and Fermi’s golden rule

Exercise

Reference

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 9. Electromagnetic fields in quantum mechanics

9.1 The quantization of the electromagnetic field

9.2 The interaction of the electromagnetic field with matter

9.3 The emission and absorption of photons by atoms

References

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Chapter 10. Epilogue

Reference

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Appendix A. Clebsch–Gordan coefficients and 3-j symbols. A.1 Clebsch–Gordan coefficients (tables A.1–A.4)

A.2 3-j symbols (table A.5)

A.3 Tables of 3-j symbol numerical values

A.4 A worked example using 3-j symbols

References

IOP Publishing. Quantum Mechanics for Nuclear Structure, Volume 2. An intermediate level view. Kris Heyde and John L Wood. Appendix B. The mode density for the electromagnetic field

Отрывок из книги

Series Editors

John Wood, Georgia Institute of Technology, USA

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8.2 Time-dependent perturbation theory

8.3 Constant perturbations and Fermi’s golden rule

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