The first volume of Quantum Mechanics for Nuclear Structure introduced the reader to the basic elements that underpin the one-body formulation of quantum mechanics. Volume two follows on from its predecessor by examining topics essential for understanding the many-body formulation. The algebraic structure of quantum theory is emphasised throughout as an essential aspect of the mathematical formulation of many-body quantum systems.
The authors begin with a thorough treatment of angular momentum theory, covering representation and coupling of spin-angular momentum states and associated operators with a focus on tensor structure. Identical particles and the representation of many-body states and operators are then covered using second quantization, followed by an introduction to the role of group theory and algebraic structures in quantum mechanics. The final chapters cover perturbation theory and the variational method, as well as a brief treatment of electromagnetic fields.
Author(s): Kris Heyde, John Wood
Series: IOP SEries in Nuclear Spectroscopy and Nuclear Structure
Publisher: IOP Publishing
Year: 2020
Language: English
Pages: 220
City: Bristol
PRELIMS.pdf
Preface
Author biographies
Kris Heyde
John L Wood
CH001.pdf
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
CH002.pdf
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
CH003.pdf
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
CH004.pdf
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
CH005.pdf
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
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)
5.12 Casimir invariants and commuting operators
5.12.1 The Casimir invariants of u(n)
5.12.2 The Casimir invariants of so(n)
CH006.pdf
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
6.6.5 Kronecker products
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
Reference
CH007.pdf
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
CH008.pdf
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
CH009.pdf
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
CH010.pdf
Chapter 10 Epilogue
Reference
APP1.pdf
Chapter
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
APP2.pdf
Chapter
