(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
(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
(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
(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 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
Monday, May 13, 2013 3
(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-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 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)
(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
(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: i∂t|Ψ(t)〉=H|Ψ(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
(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
(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
(Quantum Theory for Computer Age - Ch. 6-9 of Unit 3 Principles of Symmetry, Dynamics, and Spectroscopy - Sec. 3-7 of Ch. 2 )
C3 g†g-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
(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
(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
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〉=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
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 D↓D3 splitting
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 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
Monday, May 13, 2013 8
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 Local ⏐g〉-basis and Global vs Local ⏐P(µ)〉-basis
Hamiltonian and D3 group matrices in global and local ⏐P(µ)〉-basis
Hamiltonian local-symmetry eigensolution
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 Local ⏐g〉-basis and Global vs Local ⏐P(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
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 irrep Dα(D3)↓C2 =d02⊕d12⊕.. correlation
Subduced irrep Dα(D3)↓C3 =d03⊕d13⊕.. correlation
Spontaneous symmetry breaking and clustering: Frobenius Reciprocity , band structure
Induced rep da(C2)↑D3 =Dα⊕Dβ⊕.. correlation
Induced rep da(C3)↑D3 =Dα⊕Dβ⊕.. correlation
D6 symmetry and Hexagonal Bands
Cross product of the C2 and D3 characters gives all D6 =D3 ×C2 characters and irreps
Review: Symmetry reduction and splitting: Subduced irrep Dα(D3)↓C2 =d02⊕d12⊕.. correlation
Symmetry induction and clustering: Induced rep da(C2)↑D3 =Dα⊕Dβ⊕.. 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
D6⊃D2⊃C2 =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 symmetryOh⊃O~Td⊃T
Monday, May 13, 2013 10
Introduction to octahedral/ tetrahedral symmetry Oh⊃O~Td⊃T Octahedral-cubic O symmetry and group operations Tetrahedral symmetry becomes Icosahedral
Octahedral groups Oh⊃O~Td⊃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 Oh⊃O subgroup correlations Octahedral Oh⊃O subgroup correlations Oh⊃O⊃D4 subgroup correlations Oh⊃O⊃D4⊃C4 subgroup correlations Applications
Review Octahedral Oh⊃O group operator structure
Review Octahedral Oh⊃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)
Oh⊃O⊃D4⊃C4v and Oh⊃O⊃D4⊃C4v⊃C2v subgroup splitting
Simplest Oh⊃O⊃D4⊃C4 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
Review Octahedral O⊃D4⊃C4 subgroup chain and coset bases
Coset factored splitting of O⊃D4⊃C4 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
Oh⊃D4h⊃C4v Oh⊃D3h⊃C3v Oh⊃C2v
Review Coset factored splitting of O⊃D4⊃C4 projectors and levels
Irreducible idempotent projectors Pµ
of O⊃C ~T ⊃C
m,m
4 d 4i
Calculating PE0404 PE2424 PT10404 PT11414 PT22424
Factoring out O⊃C4 subgroup cosets:
Factoring PE0404 PE2424 PT10404 PT11414 PT22424
Irreducible nilpotent projectors Pµ
Fundamental Pµ
definitions:
Review of D3⊃C2 ~ C3v⊃Cv
Calculating and Factoring PT11404
Structure and applications of various subgroup chain irreps
Oh⊃D4h⊃C4v
Oh⊃D3h⊃C3v
Oh⊃C2v
Monday, May 13, 2013 12
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 a†a algebra of U(1) representations Creation-Destruction a†a 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
Review : 1-D a†a 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 a†a 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
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
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
Monday, May 13, 2013 14