2017 Detailed List of Lecture Topics    

Lecture 1. Introduction to quantum amplitudes and analyzers (1.17.17)   From 2015

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

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

Beam Sorters

2-State Sorters: spin-1/2 vs. optical polarization

Geometry of optical polarization selection and Brewster’s angle

Feynman’s lever

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

Textbooks (Online!)

Link → Quantum Theory for the Computer Age URL is http://www.uark.edu/ua/modphys/markup/QTCAWeb.html
Link → Principles of Symmetry, Dynamics, and Spectroscopy URL is http://www.uark.edu/ua/modphys/markup/PSDSWeb.html

Lecture 2. Quantum amplitudes, analyzers, and axioms (1.19.17)  

(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

From the Quantum Theory in the Computer Age:

Link → http://www.uark.edu/ua/modphys/pdfs/QTCA_Pdfs/QTCA_Text_2013/QTCA_Unit_1_Ch._1_2013.pdf
Link → http://www.uark.edu/ua/modphys/pdfs/QTCA_Pdfs/QTCA_Text_2013/QTCA_Unit_7_Ch._22_2005.pdf

See also the 2005 Pirelli Challenge Relativity Site:

Link → http://www.uark.edu/ua/pirelli/html/light_energy_flux_1.html
Link → http://www.uark.edu/ua/pirelli/html/amplitude_probability_1.html

Lecture 3. Analyzers, operators, and group axioms (1.24.17)  

(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

Feynman 〈j⏐k〉-axioms compared to Group axioms

Lecture 4. Matrix Eigensolutions and Spectral Decompositions (1.26.17)  

(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 =

Secular equation

Hamilton-Cayley equation and projectors

Idempotent projectors (how eigenvalues⇒eigenvectors)

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

Lecture 5. Spectral Decomposition with Repeated Eigenvalues (1.31.17)  

(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)

(Preparing for: Degenerate eigenvalues)

Eigensolutions with degenerate eigenvalues (Possible?... or not?)

Secular → Hamilton-Cayley → Minimal 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 P_j-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)

Minimal equation for projector P = P²

How symmetry groups become eigen-solvers

Lecture 6. Spectral Decomposition of Bi-Cyclic (C₂ ⊂ U(2)) Operators (2.2.17)  

(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

How C₂ (Bilateral σ_B reflection) symmetry is eigen-solver

C₂ Symmetric two-dimensional harmonic oscillators (2DHO)

C₂ (Bilateral σ_B reflection) symmetry conditions:

Minimal equation of σ_B and spectral decomposition of C₂(σ_B)

C₂ Symmetric 2DHO eigensolutions

C₂ Mode phase character table

C₂ Symmetric 2DHO uncoupling and mixed mode projector algebra

2D-HO beats and mixed mode geometry

Three famous 2-state systems and two-complex-component coordinates

ANALOGY: U(2) vs R(3): 2-State Schrodinger: iħ∂_t|Ψ(t)〉= H|Ψ(t)〉 vs. Classical 2D-HO: ∂²_tx=-K•x

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

Lecture 7. Spectral Analysis of U(2) Operators (2.7.17)  Part 1 Part 2

(Quantum Theory for Computer Age - Ch. 10 of Unit 3)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-3 of Ch. 5)

Review of Lecture 6: ₂ symmetry is 2D oscillators and three famous 2-state systems

Review of Lecture 6: 2-State Schrodinger: iħ∂_t|Ψ(t)〉= H|Ψ(t)〉 vs. Classical 2D-HO: ∂²_tx=-K•x

Review of Lecture 6: Hamilton-Pauli spinor symmetry ( σ-expansion in ABCD-Types) H = ω_μσ_μ

Derive σ-exponential time evolution (or revolution) operator U = e^-iHt = e^-iσµ^ωµ^t

Spinor arithmetic like complex arithmetic

Spinor vector algebra like complex vector algebra

Spinor exponentials like complex exponentials (“Crazy-Thing”-Theorem)

Geometry of U(2) evolution (or R(3) revolution) operator U = e^-iHt = e^{-iσ_μω_μt}

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 S_A = S_Z, Balance S_B = S_X, and Chirality S_C = S_Y

Polarization ellipse and spinor state dynamics

Lecture 8. Spinor and vector representations of U(2) and R(3) Operators (2.9.17)  Part 1 Part 2

(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 the “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 q_μ = -iσ_μ

Group product algebra

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

Transformation R[Θ]σ_μR[Θ]^† of spinor σ_μ-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′] - 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

Lecture 9. Applications of U(2) and R(3) representations (2.14.17)  Part 1 Part 2

(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 (and “real-world” applications)

U(2) density operator approach to symmetry dynamics

Bloch equation for density operator

Quick U(2) way to find eigen-solutions for 2-by-2 H

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. Applications of U(2) and R(3) representations II (2.16.17)  

(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 (and “real-world” applications)

U(2) density operator approach to symmetry dynamics

Bloch equation for density operator

Quick U(2) way to find eigen-solutions for 2-by-2 H

The ABC’s of U(2) dynamics-Archetypes

Asymmetric-Diagonal A-Type motion

Bilateral-Balanced B-Type motion

Circular-Coriolis… C-Type motion

Lecture 10 Thur. 2.16.17 starts here ⤵

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 11. Representations of cyclic groups C₃ ⊂ C₆ ⊃ C₂ (2.21.17)  

(Quantum Theory for Computer Age - Ch. 6-9 of Unit 3)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 3-7 of Ch. 2)

Review of C₂ spectral resolution for 2D oscillator (Lecture 6 : p. 11, p. 17, and p. 11)

C₃ g^†g-product-table and basic group representation theory

C₃ H-and-r^p-matrix representations and conjugation symmetry

C₃ Spectral resolution: 3^rd 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: 6^th roots of unity and higher

Complete sets of coupling parameters and Fourier dispersion

Gauge shifts due to complex coupling

Introduction to C_N beat dynamics and “Revivals” in Lecture 12

Web apps used:

Link → WaveIt
Link → MolVibes

Lecture 12. Symmetry and Dynamics of C_N cyclic systems I. (2.23.17)  

(Geometry of U(2) characters - Ch. 6-9 of Unit 3)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 3-7 of Ch. 2)

C₆ Spectral resolution: 6^th roots of unity and higher

Complete sets of coupling parameters and Fourier dispersion

Gauge shifts due to complex coupling

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)

Wave coordinates for Linear Dispersion

Wave coordinates for Bohr-Schrodinger Dispersion

Einstein-Lorentz-Minkowski laser coordinates

Introduction to C_N 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

Web apps used:

Link → WaveIt
Link → BohrIt

Lecture 13. Symmetry and Dynamics of C_N cyclic systems II. (2.28.17)  Part 1 Part 2

(Geometry of U(2) characters - Ch. 6-9 of Unit 3)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 3-7 of Ch. 2)

Two wave archetypes: Pulse-Wave (PW) versus Continuous-Wave (CW)

Comparing spacetime uncertainty (Δx or Δt) with per-spacetime bandwidth (Δκ or Δυ)

Introduction to beat dynamics and “Revivals” due to Bohr-dispersion

Relating ∞-Square-well waves to Bohr rotor waves

∞-Square-well wave dynamics

SinNx/x wavepacket bandwidth and uncertainty

∞-Square-well revivals: SinNx/x packet explodes! (and then UNexplodes!)

Bohr-rotor wave dynamics

Gaussian wave-packet bandwidth and uncertainty

Gaussian Bohr-rotor revivals and quantum fractals

Understanding fractals using geometry of fractions (Rationalizing rationals)

Farey-Sums and Ford-products

Discrete CN beat phase dynamics (Characters gone wild!)

The classical bouncing-ball Monster-Mash

Polygonal geometry of U(2) ⊃ C_N character spectral function

Algebra

Geometry

Lecture 14. C_N symmetry systems coupled, uncoupled, and re-coupled I. (3.2.17)  

(Geometry of U(2) characters - Ch. 6-12 of Unit 3)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-12 of Ch. 2)

Bohr-rotor wave dynamics

Gaussian wave-packet bandwidth and uncertainty

Gaussian Bohr-rotor revivals and quantum fractals

Understanding fractals using geometry of fractions (Rationalizing rationals)

Farey-Sums and Ford-products

Discrete C_N beat phase dynamics (Characters gone wild!)

The classical bouncing-ball Monster-Mash

Breaking C_N cyclic coupling into linear chains

Breaking C_2N+2 to approximate linear N-chain

Band-It simulation: Intro to scattering approach to quantum symmetry

Breaking C_2N cyclic coupling down to C_N symmetry

Acoustical modes vs. Optical modes

Intro to other examples of band theory

Type-AB avoided crossing view of band-gaps

Lecture 14.5 C_N symmetry systems coupled, uncoupled, and re-coupled II. (3.7.17)  Part 1 Part 2

(Quantum Theory for the Computer Age - Unit 3-5)

(Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 1-12 of Ch. 2)

Breaking C_N cyclic coupling into linear chains

Review of 1D-Bohr-ring related to infinite square well

Breaking C_2N+2 to approximate linear N-chain (Examples C₂ C₆ C₁₄)

Band-It simulation: Intro to scattering approach to quantum symmetry

     How Band-It works: Match each Ψ and DΨ, Let L=0 at Right end

Breaking C_2N cyclic coupling down to C_N 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 C_N

The “4-Group(s)” D₂ and C_2v     The CPT subgroup of Lorentz Group

Spectral decomposition of D₂

Some D₂ modes

Outer product properties and the Crystal-Point Group Zoo

Polygonal geometry of U(2)⊃C_N character spectral function.

Algebra

     Geometry

Lecture 15. Smallest non-Abelian group D₃ (and isomorphic C_3v ~ D₃) (3.9.17)  Part 1 Part 2

(QTCA Unit 5 Ch. 15; Int.J.Mol.Sci, 14, 714(2013) p.755-774)

(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 D ~ C products: By group definition ⏐g〉=g⏐1〉 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 E operators) Group theory of equivalence transformations and classes Lagrange theorems All-commuting operators and D3-invariant class algebra (center) All-commuting projectors and D3-invariant characters Group invariant numbers: Centrum, Rank, and Order

Lecture 16. Spectral decomposition of groups D₃ ~C_3v (3.14.17)  Part 1 Part 2

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 5 Ch. 15)

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

Review: 1st-Stage Spectral resolution of D3 Center (Class algebra) Group theory of equivalence transformations and classes Lagrange theorems All-commuting class projectors D3-invariant character ortho-completeness Spectral resolution to irreducible representations (“irreps”) foretold by characters or traces Subgroup splitting or correlation frequency formula: f(α)(D(α)(G)↓H) Atomic ℓ-level or 2ℓ+1-multiplet splitting      D3 examples for ℓ=1-6 Group invariant numbers: Centrum, Rank, and Order 2nd-Stage spectral decompositions of global/local D3 Splitting class projectors using subgroup chains D3 ⊃ C2 and D3 ⊃ C3 Splitting classes 3rd-stage spectral resolution to irreducible representations (irreps) and Hamiltonian eigensolutions Tunneling modes and spectra for D3 ⊃ C2 and D3 ⊃ C3 local subgroup chains

(Review of Lectures 15-16 with more detailed and rigorous derivations)

Lecture 18. Projector algebra and Hamiltonian local-symmetry eigensolution (3.28.17)  Part 1 Part 2

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 4)

Review: Spectral resolution of D3 Center (Class algebra) and its subgroup splitting Review: General formulae for spectral decomposition (D3 examples) Weyl g-expansion in irreps Dμjk(g) and projectors Pμjk Pμjk transforms right-and-left Pμjk -expansion in g-operators Dμjk(g) orthogonality relations Details omitted from Lecture 15-16 Class projector character formulae Pμ in terms of and in terms of Pμ Review: Details of Mock-Mach relativity-duality for D3 groups and representations Lab-fixed(Extrinsic-Global) vs. Body-fixed (Intrinsic-Local) Compare Global vs Local ⏐g〉-basis and Global vs Local ⏐P(μ)〉>-basis Review: Hamiltonian and D3 group matrices in global and local ⏐P(μ)>-basis Hamiltonian local-symmetry eigensolution

Lecture 19. Octahedral-tetrahedral O~T_d symmetries (3.30.17)  Part 1 Part 2

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 4)

Introduction to Octahedral/ Tetrahedral symmetry O_h ⊃ O ~ T_d ⊃ T

Octahedral-cubic O symmetry and group operations

Tetrahedral symmetry becomes Icosahedral

Octahedral groups O_h ⊃ O ~ T_d ⊃ T

Octahedral O and spin-O ⊂ U(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 O_h ⊃ O: Inversion (g&u) parity

Octahedral O_h ⊃ O ⊃ C_I subgroup correlations

O_h ⊃ O ⊃ D₄ subgroup correlations

O_h ⊃ O ⊃ D₄ ⊃ C₄ subgroup correlations

Preview of applications to high resolution spectroscopy

Lecture 18. Vibrational modes and symmetry reciprocity: Induced reps (4.4.17)  Part 1 Part 2

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 5 Ch. 15)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 4)

Review: Hamiltonian local-symmetry eigensolution in global and local ⏐P(μ)〉-basis

Molecular vibrational modes vs. Hamiltonian eigenmodes

Molecular K-matrix construction

D₃ ⊃ C₂(i₃) local-symmetry K-matrix eigensolutions

D₃-direct-connection K-matrix eigensolutions

D₃ ⊃ C₃(r^±1) local symmetry K-matrix eigensolutions

Applied symmetry reduction and splitting

Subduced irep D^α(D₃)↓C₂ = d^0₂⊕d^1₂⊕.. correlation

Subduced irep D^α(D₃)↓C₃ = d^0₃⊕d^1₃⊕.. correlation

Spontaneous symmetry breaking and clustering: Frobenius Reciprocity , band structure

Induced rep d^a(C₂)↑D₃ = D^α⊕D^β⊕.. correlation

Induced rep d^a(C₃)↑D₃ = D^α⊕D^β⊕.. correlation

D₆ symmetry and Hexagonal Bands

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

Induced rep d^a(C₂)↑D₆ = D^α⊕D^β⊕.. correlation

Induced rep d^a(C₆)↑D₆ = D^α⊕D^β⊕.. correlation

Lecture 19. Octahedral-tetrahedral O~T_d symmetries (4.6.17)  

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 4)

Introduction to Octahedral/ Tetrahedral symmetry O_h ⊃ O ~ T_d ⊃ T

Octahedral-cubic O symmetry and group operations

Tetrahedral symmetry becomes Icosahedral

Octahedral groups O_h ⊃ O ~ T_d ⊃ T

Octahedral O and spin-O ⊂ U(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 O_h ⊃ O: Inversion (g&u) parity

Octahedral O_h ⊃ O ⊃ C_I subgroup correlations

O_h ⊃ O ⊃ D₄ subgroup correlations

O_h ⊃ O ⊃ D₄ ⊃ C₄ subgroup correlations

Preview of applications to high resolution spectroscopy

Lecture 20. Octahedral-tetrahedral O~T_d representations and spectra (4.12.17)  Part 1 Part 2

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 4)

Review: Octahedral O_h ⊃ O group operator structure

Review: Octahedral O_h ⊃ O ⊃ D₄ ⊃ C₄ subgroup chain correlations

Comparison of O ⊃ D₄ ⊃ C₄ and O ⊃ D₄ ⊃ D₂ correlations and level/projector splitting

O ⊃ D₄ ⊃ C₄ subgroup chain splitting

O ⊃ D₄ ⊃ D₂ subgroup chain splitting (nOrmal D₂ vs. unOrmal D₂)

O_h ⊃ O ⊃ D₄ ⊃ C₄ and O_h ⊃ O ⊃ D₄ ⊃ C_4v ⊃ C_2v subgroup splitting

Splitting O class projectors P^μ into irreducible projectors P^μ_m4m4 for O ⊃ C₄

Development of irreducible projectors P^μ_m4m4 and representations D^μ_m4m4

Calculating PE0404 , PE2424, PT10404, PT11414, PT22424, PT21414

O ⊃ C₄ induced representation 0₄(C₄)↑O ~ A₁ ⊕ T₁ ⊕ E and spectral analysis examples

Elementary induced representation 0₄(C₄)↑O

Projection reduction of induced representation 0₄(C₄)↑O

Introduction to ortho-complete eigenvalue-parameter relations

Examples in SF₆ spectroscopy

Lecture 21. Octahedral O_h ⊃ subgroup tunneling parameter modeling (4.13.17)  Part 1 Part 2

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 5 Ch. 15)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 4)

Review: Calculating idempotent projectors P^μ_m,m of O ⊃ C₄:

      PE0404 , PE2424, PT10404, PT11414, PT22424, PT21414

Review: Coset factored splitting of O ⊃ D₄ ⊃ C₄ projectors

Review: Broken-class-ordered splitting of O ⊃ D₄ ⊃ C₄ projectors and levels

Subgroup-defined tunneling parameter modeling

Comparing two diagonal O ⊃ C₄ parameter sets to SF₆ spectra

Comparing two diagonal O ⊃ C₃ parameter sets to SF₆ spectra

Why O ⊃ C₂ parameter sets require off-diagonal nilpotent P^μ_m,n (m≠n)

Irreducible nilpotent projectors P^μ_m,n (m≠n)

Using fundamental g⇄P^μ_m,n relations: ( from Lecture 16):

      (a) P^μ_m,mgP^μ_n,n=D^μ_m,n(g)P^μ_m,n

      (b) g=Σ_μΣ_m,nD^μ_m,n(g)P^μ_m,n

      (c) P^μ_m,n=(ℓ^μ/°G)Σ_gD^μ*_m,n(g)g

Review of D₃ ⊃ C₂ ~ C_3v ⊃ C_v

Calculating and Factoring PT11404 and PT11434

Structure and applications of various subgroup chain irreducible representations

O_h ⊃ D_4h ⊃ C_4v , O_h ⊃ D_3h ⊃ C_3v , O_h ⊃ C_2v

Comparing O_h ⊃ D_4h ⊃ D_2h and O_h ⊃ D_3d ⊃ C₂ representations (T₁ vector-type)

Examples of off-diagonal tunneling coefficients D^E₀₄₂₄

Comparing Local C₄, C₃, and C₂ symmetric spectra

When Local C₂ symmetry dominates

Comparing off-diagonal O ⊃ C₂ parameter sets to CH₄ models with “cluster-crossings”

Lecture 22. Harmonic oscillator symmetry U(1) ⊂ U(2) ⊂ U(3) ... (4.18.17)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 20-22)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 8)

1-D a^†a algebra of U(1) representations

Creation-Destruction a^†a algebra

Eigenstate creationism (and destruction)

Vacuum state

1^st 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

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 a^†a algebra of U(2) representations and R(3) angular momentum operators

Lecture 23. Harmonic oscillator symmetry U(1) ⊂ U(2) ⊂ U(3) ... (4.20.17)  

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-22)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 8)

Review: 1-D a^†a algebra of U(1) representations

Review: Translate T(a) and/or Boost B(b) to construct coherent state

Review: Time evolution of coherent state (and “squeezed” states)

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

R(3) 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

Lecture 24. Rotational symmetry U(2) ⊂ U(3) and O(3) (4.25.17)

(Int.J.Mol.Sci, 14, 714 (2013) p.755-774 , QTCA Unit 7 Ch. 21-22)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 5, 7)

Review: 2-D a^†a algebra of U(2) representations

Angular momentum generators by U(2) analysis

Review: 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 25. Introduction to Rotational Eigenstates and Spectra I. (4.27.17)  

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-25 )

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 5, 7)

Three (3) applications of R(3) rotation and U(2) unitary representations D^J_mn(α,β,γ)

1. Atomic and molecular D^J*_mn(α,β,γ)-wavefunctions

“Mock-Mach” lab-vs-body-defined states |^J_mn〉=P^J_mn|(0,0,0)〉=∫ d(α,β,γ)D^J*_mn(α,β,γ)R(α,β,γ)|(0,0,0)〉

2. R(3) rotation and U(2) unitary D^J_mn)-transformation matrices

General Stern-Gerlach and polarization transformations R(α,β,γ)|^J_mn〉=Σ_m′ D^J_m'n(α,β,γ)|^J_m'n〉

Angular momentum cones and high J properties

3. Atomic and molecular multipole Hamiltonian tensor operators T_q^k and eigenvalues

Multipole T_q^k expansion of asymmetric-rotor Hamiltonians H=AJ_x²+BJ_y²+CJ_z²

Multipole T_q^k expansion of symmetric-rotor Hamiltonians H=BJ_x²+BJ_y²+CJ_z²

Rotational Energy Surfaces (RE or RES) of symmetric rotor and eigensolutions

Rotational Energy Surfaces (RE or RES) of asymmetric rotor and energy levels

Sketch of modern molecular electronic, vibrational, and rotational spectroscopy

Example of CO₂ rovibrational (υ=0) ⇔ (υ=1) bands

Introduction to RE symmetry and RES analysis of rovibrational Hamiltonians

Asymmetric Top eigensolutions for J=1-2

Lecture 26. Introduction to Rotational Eigenstates and Spectra II. (5.2.17)  

Based on AMOP Lectures 14-20

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-25)

(Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 5, 7)

Review: Symmetric rigid quantum rotor analysis of R(2) Hamiltonian H = BJ_x²+BJ_y²+CJ_z²

Review: Rotational Energy Surfaces (RE or RES) and R(3) ~ U(2) representations

Asymmetric rigid quantum rotor analysis of D₂ Hamiltonian H = AJ_x²+BJ_y²+CJ_z²

RES and Multipole T_q^k tensor expansions

Atomic or molecular R(3) ℓ-level or 2ℓ+1-multiplet splitting

R(3) ⊃ D₂ character analysis of ℓ-level or 2ℓ+1-multiplet splitting in D₂

Detailed angular momentum operator analysis for J = 1-2 for D₂ symmetry

      Asymmetric rotor levels and RES plots for high-J

Octahedral semi-rigid quantum rotor analysis of O_h Hamiltonian H = BJ•J + t₀₄₄T^[4]

Spherical rotor levels and RES plots of O_h tensor eigenvalues

R(3) ⊂ O(3) ⊃ O_h ⊃ O character analysis of ℓ-level or 2ℓ+1-multiplet splitting in O

SF₆ spectral fine structure P(88)

CF₄ spectral fine structure P(54)

Lecture 27. Introduction to Rotational Eigenstates and Spectra III. (5.3.17)  

Based on: QTCA Lectures 24-25 (2013); AMOP Lectures 17-18 (2014)

Int.J.Mol.Sci, 14, 714(2013) p.755-774; Computer Phys. Reports 8,319-394 (1988)

(QTCA Unit 7 Ch. 21-25; Principles of Symmetry, Dynamics, and Spectroscopy - Ch. 5, 7)

Review: Asymmetric rotor levels of H = AJ_x²+BJ_y²+CJ_z² and RES plots

D₂ ⊃ C₂ symmetry correlation

Review: Spherical rotor levels and RES plots

Spectral fine structure of SF₆, SiF₄, C₈H₈, ...

O ⊃ C₄ and O ⊃ C₃ symmetry correlation

Some more examples of J=30 levels (including T^[6] vs T^[4] effects)

Details of P(88) v₄ SF₆ and P(54) v₄ CF₄ spectral structure and implications

Beginning theory

Rovibronic nomograms and PQR structure

Rovibronic energy surfaces (RES) and cone geometry

Spin symmetry correlation, tunneling, and entanglement

Hyperfine vs. superfine structure (Case 1. vs Case 2.)

Spin-0 nuclei give Bose Exclusion

      The spin-symmetry species mixing problem

Analogy between PE surface dynamics and RES

Rotational Energy Eigenvalue Surfaces (REES)

Lecture 28. Rotational energy and eigenstate surfaces for Coriolis dynamics (Draft lecture for 5.4.17)

Based on AMOP Lectures 19-20

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-25)

(PSDS - Ch. 5-8, Rev. Mod. Phys. 50,1,37-83(1978) , Computer Phys. Reports 8, 319-394 (1988))

Some ways to picture Atomic Molecular and Optical (AMO) eigenstates

J-power-law energy eigenvalue spectra and tensor operators

Introducing U(2), U(3),... tensor 2^k-multipole expansions and Wigner Eckart forms

Born-Oppenheimer Approximations

(BOA) for PES

(BOA) for RES and LAB-BOD “hook-up” frame transformation

Semiclassical Rotor-“Gyro”-Spin coupling *

Semiclassical Rotor-“Gyro” RES

Semiclassical Rotor analogy of Anharmonic Vibrator

Analogies between energy surfaces of potential (PES) and rotation (RES)

Jahn-Teller-Renner analogies

Rotational energy eigenvalue surfaces (REES)

Introducing “Sherman the Shark” ZIPPed* and unZIPPed*

REES for high-J Coriolis spectra in υ₃ CF₄ (with Review: SF₆ Coriolis PQR structure)

REES for high-J and high-υ ro-vibrational polyads

CF₄ -υ₄/2υ₃ dyad

          *ZIPP (Zero-Interaction-Potential-`Proximation

Lecture 29. Rotational eigenstates and spin-permutation symmetry (5.5.17)  

Based on AMOP Lectures 18-19

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 21-25)

(PSDS - Ch. 5, 7 )

Review: SF₆ levels and nomograms for Coriolis PQR structure

Review: SF₆ spectral clusters of symmetry species O ⊃ C₄ and O⊃C₃ symmetry correlation

Conservation (or not!) of rovibronic spin-symmetry-species

Entanglement and related issues

Diatomic or linear molecule symmetry O(3) ⊃ D_∞h ⊃ C_∞v

Labeling by symmetry O(3) ⊃ D_∞h

Coriolis and λ (or Λ)-doubling levels

Dipole-allowed transitions

S_n Young Tableaus and spin-symmetry for X_n and XY_n molecules

S₄ and spin-symmetry for XY₄ molecules (Introducing hook-length formulae)

S₆ and spin-symmetry for XY₆ molecules

Entanglement and Disentanglement

Resulting hyperfine spectra

Superhyperfine spectra

Spin-0 nuclei give Bose Exclusion

C₆₀ Buckminster Fullerene (“Buckyball”) structure and spectra

Lecture 30. Symmetry product analysis U(m)*S_n tensors (Draft lecture for 5.10.17)  

(Int.J.Mol.Sci, 14, 714(2013) p.755-774 , QTCA Unit 7 Ch. 23-26)

(PSDS - Ch. 5, 7)

Review: 2-D a^†a 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