Group Theory in Quantum Mechanics
Spring 2013 PHYS-5093

2013 Detailed List of Lecture Topics

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It contains known issues with symbols and is posted mainly for the purpose of quick reference.

Lecture 1. (1.15.13) Introduction to quantum amplitudes and analyzers

(Quantum Theory for Computer Age - Ch. 1 of Unit 1 )

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-2 of Ch. 1 )

Beam Sorters - Optical polarization sorting
Beam Sorters in Series and Transformation Matrices
Introducing Dirac bra-ket notation
"Abstraction" of bra and ket vectors from a Transformation Matrix
Introducing scalar and matrix products

Lecture 2. (1.17.13) Quantum amplitudes, analyzers, and axioms

(Quantum Theory for Computer Age - Ch. 1 of Unit 1 )

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-2 of Ch. 1 )

Review:"Abstraction" of bra and ket vectors from a Transformation Matrix

Introducing scalar and matrix products

Planck's energy and N-quanta (Cavity/Beam wave mode) Did Max Planck Goof? What's 1-photon worth? Feynman amplitude axiom 1
What comes out of a beam sorter channel or branch-b?
Sample calculations
Feynman amplitude axioms 2-3
Beam analyzers: Sorter-unsorters The "Do-Nothing" analyzer Feynman amplitude axiom 4
Some "Do-Something" analyzers
Sorter-counter, Filter, 1/2-wave plate, 1/4-wave plate

Monday, May 13, 2013 2

Lecture 3. (1.17.13) Analyzers, operators, and group axioms

(Quantum Theory for Computer Age - Ch. 1-2 of Unit 1 )

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 1 )Review: Axioms 1-4 and"Do-Nothing"vs" Do-Something" analyzers

Abstraction of Axiom-4 to define projection and unitary operators
Projection operators and resolution of identity
Unitary operators and matrices that do something (or "nothing") Diagonal unitary operators
Non-diagonal unitary operators and †-conjugation relations Non-diagonal projection operators and Kronecker -products Axiom-4 similarity transformation
Matrix representation of beam analyzers
Non-unitary "killer" devices: Sorter-counter, filter
Unitary "non-killer" devices: 1/2-wave plate, 1/4-wave plate

How analyzers "peek" and how that changes outcomes

Peeking polarizers and coherence loss
Classical Bayesian probability vs. Quantum probability

Lecture 4. (1.24.13) Matrix Eigensolutions and Spectral Decompositions

(Quantum Theory for Computer Age - Ch. 3 of Unit 1 )

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 1 )

Unitary operators and matrices that change state vectors
...and eigenstates ("ownstates) that are mostly immune
Geometric visualization of real symmetric matrices and eigenvectors
Circle-to-ellipse mapping (and I'm Ba-aaack!)
Ellipse-to-ellipse mapping (Normal space vs. tangent space)

Eigensolutions as stationary extreme-values (Lagrange λ-multipliers)

Matrix-algebraic eigensolutions with example M= {4,3;1,2}

Secular equation
Hamilton-Cayley equation and projectors
Idempotent projectors (how eigenvalueseigenvectors) Operator orthonormality and completeness

Spectral Decompositions

Functional spectral decomposition
Orthonormality vs. Completeness vis-a`-vis Operator vs. State
Lagrange functional interpolation formula
Proof that completeness relation is "Truer-than-true"

Diagonalizing Transformations (D-Ttran) from projectors

Eigensolutions for active analyzers

Monday, May 13, 2013 3

Lecture 5. (1.29.13) Spectral Decomposition with Repeated Eigenvalues

(Quantum Theory for Computer Age - Ch. 3 of Unit 1 ) Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 1 )

Review: matrix eigenstates ("ownstates) and Idempotent projectors ( Non-degeneracy case )

Operator orthonormality, completeness, and spectral decomposition(Non-degenerate e-values )

Eigensolutions with degenerate eigenvalues (Possible?... or not?) Secular Hamilton-CayleyMinimal equations Diagonalizability criterion

Nilpotents and "Bad degeneracy" examples: B= , and: N= Applications of Nilpotent operators later on

Idempotents and "Good degeneracy" example: G= Secular equation by minor expansion
Example of minimal equation projection
Orthonormalization of degenerate eigensolutions
Projection Pj-matrix anatomy (Gramian matrices)
Gram-Schmidt procedure
Orthonormalization of commuting eigensolutions. Examples: G= and: H=

The old "1=1.1 trick"-Spectral decomposition by projector splitting

Irreducible projectors and representations (Trace checks)

Lecture 6. (1.31.13) Spectral Decomposition of Bi-Cyclic (C2U(2)) Operators

(Quantum Theory for Computer Age - Ch. 7-9 of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 2 )

Review:How symmetry groups become eigen-solvers

C2 Symmetric two-dimensional harmonic oscillators (2DHO)
C2 (Bilateral σB reflection) symmetry conditions:
Minimal equation of σB and spectral decomposition of C2(σB)
C2 Symmetric 2DHO eigensolutions
C2 Mode phase character table
C2 Symmetric 2DHO uncoupling
2D-HO beats and mixed mode geometry

Monday, May 13, 2013 4

Lecture 7. (2.5.13) Spectral Analysis of U(2) Operators

(Quantum Theory for Computer Age - Ch. 10 of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 5 )

Review:How C2 (Bilateral σB reflection) symmetry is eigen-solver & three famous 2-state systems

U(2) vs R(3):2-State Schrodinger: it(t)=H(t)vs. Classical 2D-HO: 2 x=-K•x

Hamilton-Pauli spinor symmetry ( σ-expansion in ABCD-Types) H= σ

ωµ µ

Deriving σ-exponential time evolution (or revolution) operator U=e-iHt=e-iσµωµt

Spinor exponentials like complex exponentials ("Crazy-Thing"-Theorem) Geometry of U(2) evolution (or R(3) revolution) operator U=e-iHt=e-iσµωµt
The "mysterious" factors of 2 (or 1/2): 2D Spinor vs 3D Spin Vector space
2D Spinor vs 3D vector rotation
NMR Hamiltonian: 3D Spin Moment m in B field
Euler's state definition using rotations R(α,0,0), R(0,β,0),and R(0,0,γ) Spin-1 (3D-real vector) case
Spin-1/2 (2D-complex spinor) case
3D-real Stokes Vector defines 2D-HO polarization ellipses and spinor states

Asymmetry SA =SZ, Balance SB =SX, and Chirality SC =SY

Lecture 8. (2.7.13) Spinor and vector representations of U(2) and R(3) Operators

(Quantum Theory for Computer Age - Ch. 10A-B of Unit 3 & Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 5 )

Review:How "Crazy-Thing"-Theorem makes spinor and vector representation matrices

Half-angle Θ/2 =ϕ replacement and Darboux crank axis operators
Operator-on-Operator transformations

Product algebra for Pauli's σµ and Hamilton's qµ = -iσµ

Group product algebra

Jordan-Pauli identity and U(2) product R[Θ]R[Θ′]=R[Θ′′′]- formula

Transformation R[Θ]σµR[Θ] of spinor σµ-operators Transformation R[Θ]R[Θ′]R[Θ] of group-operators Operator-on-Operator transformations

Geometry of groups: Hamilton's turns and It's all done with mirrors!
Group product geometry

U(2) product R[Θ]R[Θ′]=R[Θ′′′]- geometry

Transformation R[Θ]R[Θ′]R[Θ] geometry

Euler R(αβγ) versus Darboux R[ϕϑΘ]

Euler R(αβγ) related to Darboux R[ϕϑΘ]

Euler R(αβγ) rotation Θ =0-4π-sequence [ϕϑ] fixed R(3)-U(2) slide rule for converting R(αβγ) R[ϕϑΘ] Euler R(αβγ) Sundial

Monday, May 13, 2013 5

Lecture 9. (2.12.13) Applications of U(2) and R(3) representations

(Quantum Theory for Computer Age - Ch. 10A-B of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 5 and Ch. 7 )

Review: Fundamental Euler R(αβγ) and Darboux R[ϕϑΘ] representations of U(2) and R(3) Euler R(αβγ) derived from Darboux R[ϕϑΘ] and vice versa

Euler R(αβγ) rotation Θ =0-4π-sequence [ϕϑ] fixed

R(3)-U(2) slide rule for converting R(αβγ) R[ϕϑΘ] and Sundial

U(2) density operator approach to symmetry dynamics
Bloch equation for density operator
The ABC's of U(2) dynamics-Archetypes
Asymmetric-Diagonal A-Type motion
Bilateral-Balanced B-Type motion
Circular-Coriolis… C-Type motion
The ABC's of U(2) dynamics-Mixed modes

AB-Type motion and Wigner's Avoided-Symmetry-Crossings

ABC-Type elliptical polarized motion

Ellipsometry using U(2) symmetry coordinates
Conventional amp-phase ellipse coordinates
Euler Angle (αβγ) ellipse coordinates

Lecture 10. (2.26.13) Representations of cyclic groups C3 C6 C2

(Quantum Theory for Computer Age - Ch. 6-9 of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 3-7 of Ch. 2 )

C3 gg-product-table and basic group representation theory

C3 H-and-rp-matrix representations and conjugation symmetry

C3 Spectral resolution: 3rd roots of unity and ortho-completeness relations

C3 character table and modular labeling

Ortho-completeness inversion for operators and states

Comparing wave function operator algebra to bra-ket algebra

Modular quantum number arithmetic
C3-group jargon and structure of various tables
C3 Eigenvalues and wave dispersion functions
Standing waves vs Moving waves
C6 Spectral resolution: 6th roots of unity and higher
Complete sets of coupling parameters and Fourier dispersion
Gauge shifts due to complex coupling

Monday, May 13, 2013 6

Lecture 11. (2.19-28 to 3.5.13) Symmetry and Dynamics of CN cyclic systems

(Geometry of U(2) characters - Ch. 6-9 of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 3-7 of Ch. 2 )

Polygonal geometry of U(2)CN character spectral function
Algebra
Geometry
Introduction to wave dynamics of phase, mean phase, and group velocity
Expo-Cosine identity
Relating space-time and per-space-time
Wave coordinates
Pulse-waves (PW) vs Continuous -waves (CW)
Introduction to CN beat dynamics and "Revivals" due to Bohr-dispersion
∞-Square well PE versus Bohr rotor
SinNx/x wavepackets bandwidth and uncertainty
SinNx/x explosion and revivals
Bohr-rotor dynamics
Gaussian wave-packet bandwidth and uncertainty
Gaussian Bohr-rotor revivals Farey-Sums and Ford-products Phase dynamics

Lecture 12. (3.7.13) CN symmetry systems coupled, uncoupled, and re-coupled

(Geometry of U(2) characters - Ch. 6-12 of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-12 of Ch. 2 )

Breaking CN cyclic coupling into linear chains
Review of 1D-Bohr-ring related to infinite square well (and review of revival) Breaking C2N+2 to approximate linear N-chain
Band-It simulation: Intro to scattering approach to quantum symmetry
Breaking C2N cyclic coupling down to CN symmetry
Acoustical modes vs. Optical modes Intro to other examples of band theory Avoided crossing view of band-gaps
Finally! Symmetry groups that are not just CN
The "4-Group(s)" D2 and C2v
Spectral decomposition of D2
Some D2 modes
Outer product properties and the Group Zoo

Monday, May 13, 2013 7

Lecture 13. (3.12.13) Smallest non-Abelian isomorphic groups D3 ~C3v (Int.J.Mol.Sci p.755-774 , QTCA Unit 5 Ch. 15 , PSDS - Ch. 3)

3-Dihedral-axes group D3 vs. 3-Vertical-mirror-plane group C3v D3 and C3v are isomorphic (D3 ~ C3v share product table) Deriving D3 ~ C3v products:

By group definition g=g1 of position ket g

By nomograms based on U(2) Hamilton-turns

Deriving D3 ~ C3v equivalence transformations and classes

Non-commutative symmetry expansion and Global-Local solution

Global vs Local symmetry and Mock-Mach principle

Global vs Local matrix duality for D3

Global vs Local symmetry expansion of D3 Hamiltonian

1st-Stage spectral decomposition of global/local D3 Hamiltonian

All-commuting operators and D3-invariant class algebra All-commuting projectors and D3-invariant characters Group invariant numbers: Centrum, Rank, and Order

Spectral resolution to irreducible representations (or "irreps") foretold by characters or traces

Crystal-field splitting: O(3)D3 symmetry reduction and DD3 splitting

Lecture 14. (3.14.13) Spectral decomposition of groups D3 ~C3v (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 3 )

Review:Spectral resolution of D3 Center (Class algebra) Group theory of equivalence transformations and classes

Lagrange theorems

All-commuting class projectors and D3-invariant characters

Character ortho-completeness

Group invariant numbers: Centrum, Rank, and Order

2nd-Stage spectral decompositions of global/local D3

Splitting class projectors using subgroup chains D3C2 and D3C3

Splitting classes

3rd-stage spectral resolution to irreducible representations (irreps) and Hamiltonian eigensolutions

Tunneling modes and spectra for D3C2 and D3C3 local subgroup chains

Monday, May 13, 2013 8

Lecture 15. (3.26.13) Projector algebra and Hamiltonian local-symmetry eigensolution (Int.J.Mol.Sci,14,714(2013) p.755-774, QTCA Unit 5 Ch. 1, PSDS - Ch. 4 )

Review: Spectral resolution of D3 Center (Class algebra) and its subgroup splitting

General formulae for spectral decomposition (D3 examples)

Weyl g-expansion in irrep Dµ

(g) and projectors Pµ

jk transforms right-and-left

jk -expansion in g-operators

Dµ

jk(g) orthogonality relations

Class projector character formulae

Pµ in terms of κ

and κg

in terms of Pµ

Details of Mock-Mach relativity-duality for D3 groups and representations

Lab-fixed(Extrinsic-Global) vs. Body-fixed (Intrinsic-Local)

Compare Global vs Localg-basis and Global vs LocalP(µ)-basis

Hamiltonian and D3 group matrices in global and local P(µ)-basis

Hamiltonian local-symmetry eigensolution

Lecture 16. (3.28.13) Local-symmetry eigensolutions and vibrational modes (Int.J.Mol.Sci,14,714(2013) p.755-774, QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4)

Review: Projector formulae and subgroup splitting

Algebra and geometry of irreducible Dµ

(g) and projector Pµ

transformation

Example of D3 transformation by matrix DEjk(r1)

Details of Mock-Mach relativity-duality for D3 groups and representations

Lab-fixed(Extrinsic-Global) vs. Body-fixed (Intrinsic-Local) Compare Global vs Localg-basis and Global vs LocalP(µ)-basis

Hamiltonian and D3 group matrices in global and local P(µ)-basis

Hamiltonian local-symmetry eigensolution

Molecular vibrational mode eigensolution

Local symmetry limit

Global symmetry limit (free or "genuine" modes)

Monday, May 13, 2013 9

Lecture 17. (4.2.13) Vibrational modes and symmetry reciprocity: Induced reps (Int.J.Mol.Sci, 14, 714(2013) p.755-774, QTCA Unit 5 Ch. 15)(PSDS - Ch. 4)

Review: Hamiltonian local-symmetry eigensolution in global and localP(µ)-basis

Molecular vibrational modes vs. Hamiltonian eigenmodes

Molecular K-matrix construction

D3C2(i3) local-symmetry K-matrix eigensolutions D3-direct-connection K-matrix eigensolutions D3C3(r±1) local symmetry K-matrix eigensolutions

Applied symmetry reduction and splitting

Subduced irrep Dα(D3)C2 =d02d12⊕.. correlation

Subduced irrep Dα(D3)C3 =d03d13⊕.. correlation

Spontaneous symmetry breaking and clustering: Frobenius Reciprocity , band structure

Induced rep da(C2)D3 =DαDβ⊕.. correlation

Induced rep da(C3)D3 =DαDβ⊕.. correlation

D6 symmetry and Hexagonal Bands

Cross product of the C2 and D3 characters gives all D6 =D3 ×C2 characters and irreps

Lecture 18. (4.4.13) Hexagonal D6D6h and octahedral-tetrahedral O~Td symmetry (Int.J.Mol.Sci, 14, 714(2013) p.755 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4)

Review: Symmetry reduction and splitting: Subduced irrep Dα(D3)C2 =d02d12⊕.. correlation

Symmetry induction and clustering: Induced rep da(C2)D3 =DαDβ⊕.. correlation

D3-C2 Coset structure of dm2(C2)D3 induced representation basis

D3-Projection of dm2(C2)D3 induced representation basis

Derivation of Frobenius reciprocity

D6D2C2 =D3 ×C2 symmetry and outer product geometry

Irreducible characters Irreducible representations Correlations with D6 characters:

...and C2(i3) characters......and C6(1,h1,h2,...) characters

D6 symmetry and induced representation band structure
Introduction to octahedral tetrahedral symmetryOhO~TdT

Monday, May 13, 2013 10

Lecture 19. (4.9.13) Octahedral-tetrahedral O~Td symmetries (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4)

Introduction to octahedral/ tetrahedral symmetry OhO~TdT Octahedral-cubic O symmetry and group operations Tetrahedral symmetry becomes Icosahedral

Octahedral groups OhO~TdT Octahedral O and spin-OU(2)

Tetrahedral T class algebra

Tetrahedral T class minimal equations

Tetrahedral T class projectors and characters

Octahedral O class algebra

Octahedral O class minimal equations

Octahedral O class projectors and characters

Octahedral OhO subgroup correlations Octahedral OhO subgroup correlations OhOD4 subgroup correlations OhOD4C4 subgroup correlations Applications

Lecture 20. (4.11.13) Octahedral-tetrahedral O~Td representations and spectra (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4 ) (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4)

Review Octahedral Oh⊃O group operator structure

Review Octahedral Oh⊃O⊃D4C4 subgroup chain correlations

Comparison of O⊃D4C4 and O⊃D4D2 correlations and level/projector splitting

O⊃D4C4 subgroup chain splitting

O⊃D4D2 subgroup chain splitting (nOrmal D2 vs. unOrmal D2)

Oh⊃O⊃D4C4v and Oh⊃O⊃D4C4vC2v subgroup splitting

Simplest Oh⊃O⊃D4C4 spectral analysis problems

Elementary induced representation 04(C4)↑O

Projection reduction of induced representation 04(C4)↑O

Introduction to ortho-complete eigenvalue calculations

Monday, May 13, 2013 11

Lecture 21. (4.16.13) Octahedral OhOD4C4 eiegensolution in coset spaces (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4 ) (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4 )

Review Octahedral O⊃D4C4 subgroup chain and coset bases

Coset factored splitting of O⊃D4C4 projectors and levels

Coset spaces based on m4(C4)↑O

Splitting class projectors into C4 cosets and m4(C4)↑O bases

General development of irrep projectors Pµm4m4

Calculating PE0404

Calculating PE2424

Calculating PT10404

Calculating PT11414

Calculating PT22424

Structure and applications of various subgroup chain irreps

OhD4hC4v OhD3hC3v OhC2v

Lecture 22. (4.18.13) Octahedral OhOD4C4 eiegensolution in coset spaces II (Int.J.Mol.Sci, 14,755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4)

Review Coset factored splitting of O⊃D4C4 projectors and levels

Irreducible idempotent projectors Pµ

of OC ~T C

m,m

4 d 4i

Calculating PE0404 PE2424 PT10404 PT11414 PT22424

Factoring out OC4 subgroup cosets:

Factoring PE0404 PE2424 PT10404 PT11414 PT22424

Irreducible nilpotent projectors Pµ

Fundamental Pµ

definitions:

Review of D3C2 ~ C3vCv

Calculating and Factoring PT11404

Structure and applications of various subgroup chain irreps

OhD4hC4v

OhD3hC3v

OhC2v

Monday, May 13, 2013 12

Lecture 19. (4.9.13) Octahedral-tetrahedral O~Td symmetries (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15 )(PSDS - Ch. 4)

Lecture 23 (4.23.13) Harmonic oscillator symmetry U(1)U(2)U(3)...(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 20-22 )(PSDS - Ch. 8 )

1-D aa algebra of U(1) representations Creation-Destruction aa algebra Eigenstate creationism (and destruction)

Vacuum state
1st excited state
Normal ordering for matrix calculation Commutator derivative identities Binomial expansion identities

Matrix ana†n calculations

Number operator and Hamiltonian operator
Expectation values of position, momentum, and uncertainty for eigenstate n
Harmonic oscillator beat dynamics of mixed states

Lecture 24. (4.25.13) Harmonic oscillator symmetry U(1)U(2)U(3)... (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-22 )(PSDS - Ch. 8)

Review : 1-D aa algebra of U(1) representations

U(1) Oscillator coherent states ("Shoved" and "kicked" states) Translation operators vs. boost operators

Applying boost-translation combinations
Time evolution of coherent state
Properties of coherent state and "squeezed" states

2-D aa algebra of U(2) representations and R(3) angular momentum operators

2D-Oscillator basics
Commutation relations
Bose-Einstein symmetry vs Pauli-Fermi-Dirac (anti)symmetry

Anti-commutation relations

Two-dimensional (or 2-particle) base states: ket-kets and bra-bras
Outer product arrays
Entangled 2-particle states
Two-particle (or 2-dimensional) matrix operators
U(2) Hamiltonian and irreducible representations
2D-Oscillator eigensolutions

Monday, May 13, 2013 13

Lecture 25. (4.30.13) Rotational symmetry U(2)U(3) and R(3) (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-22 )(PSDS - Ch. 5, 7)

Review : 2-D aa algebra of U(2) representations

Angular momentum generators by U(2) analysis

Angular momentum raise-n-lower operators s+ and s-

SU(2)U(2) oscillators vs. R(3)O(3) rotors
Angular momentum commutation relations
Key Lie theorems
Angular momentum magnitude and uncertainty
Angular momentum uncertainty angle
Generating R(3) rotation and U(2) representations
Applications of R(3) rotation and U(2) representations
Molecular and nuclear wavefunctions
Molecular and nuclear eigenlevels
Generalized Stern-Gerlach and transformation matrices
Angular momentum cones and high J properties

Lecture 26. (5.2.13) Symmetry product analysis U(m)*Sn tensors (Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 23-26 )(PSDS - Ch. 5, 7)

Review : 2-D aa algebra of U(2) representations

Spin-spin (1/2)2 product states: Hydrogen hyperfine structure
Kronecker product states and operators
Spin-spin interaction reduces symmetry U(2)proton×U(2)electron to U(2)e+p
Clebsch-Gordan Coefficients
Hydrogen hyperfine structure: Fermi-contact interaction plus B-field gives avoided crossing
Higher-J product states (J=1)(J=1)=2⊕1⊕0 case General U(2) case
Multi-spin (1/2)N product states
Magic squares - Intro to Young Tableaus
Tensor operators for spin-1/2 states: Outer products give Hamilton-Pauli-spinors
Tensor operators for spin-1 states: U(3) generalization of Pauli spinors

Monday, May 13, 2013 14