# Lecture 1. 2-Wave Interference: Phase and Group Dynamics (1.14.14)

(Ch. 0-1 of Unit 8 CMwBang!; p.1-20 Relativity&QuantumTheory by Rule&Compass)

1. Review of basic formulas for waves in space-time (x,t) or per-space-time (ω,k)

1-Plane-wave phase velocity

2-Plane-wave phase velocity and group velocity (1/2-sum &1/2-diff.)

2-Plane-wave real zero grid in (x,t) or (ω,k)

Geometric analysis of Bohr-Schrodinger ″matter-wave″

Algebraic analysis of Bohr-Schrodinger ″matter-wave″

2. Geometric construction of wave-zero grids

Continuous Wave (CW) grid based on Kphase=(Ka+Kb)/2 and Kgroup=(Ka-Kb)/2 vectors

Pulse Wave (PW) grid based on primitive Ka=Kphase+Kgroup and Kb=Kphase-Kgroup vectors

When this doesn't work (When you don't need it!)

3. Beginning wave relativity

Dueling lasers make lab frame space-time grid

Einstein PW Axioms versus Evenson CW Axioms (Occam at Work)

Only CW light clearly shows shift

Dueling lasers make lab frame space-time grid

# Lecture 2. Relativity of wave-optics and Lorentz-Minkowski coordinates I. (1.16-23.14)

(Ch. 2 of Unit 8 CMwBang!; p.1-23 Relativity&QuantumTheory by Rule&Compass)

1. Optical wave coordinates and frames

Old-fashioned vs. New-fashioned spacetime frames

Dueling lasers make lab frame space-time grid (CW or PW)

Comparing Continuous-Wave (CW) vs. Pulse-Wave (PW) frames with Review of Light

2. Applying Occam’s razor to relativity axioms

Einstein PW Axioms versus Evenson CW Axioms (Traditional: The ′′Roadrunner′′ Axiom)

CW light clearly shows shifts

Check that red is red is red,...green is green is green,...blue is blue is blue,... etc.

Is dispersion linear?... does astronomy work?... how about spectroscopy?

Is Doppler a geometric factor or arithmetic sum?

Introducing rapidity ρ = ln b.

That old Time-Reversal meta-Axiom (that is so-oo-o neglected!)

3. theory of Einstein-Lorentz relativity

Applying Shifts to per-space-time (ck,ω) graph

CW Minkowski space-time coordinates (x,ct) and PW grids

Relating b or r=1/b to velocity u/c or rapidity ρ

Connection: Conventional approach to relativity and old-fashioned formulas

# Lecture 3. Relativity of wave-optics and Lorentz-Minkowski coordinates II. (1.28-30.14)

(Ch. 3 of Unit 8 CMwBang!; p.1-28 Relativity&QuantumTheory by Rule&Compass)

1. theory of Einstein-Lorentz relativity

Applying to per-space-time (ck,ω) graph

CW Minkowski space-time coordinates (x,ct) and PW grids

Relating b or r=1/b to velocity u/c or rapidity ρ

Connection: Conventional approach to relativity and old-fashioned formulas

Invariant hyperbolas and hyperbolic relations

2. Reciprocal dilation and contraction properties

The most old-fashioned form(ula) of all: Thales & Euclid means

Galileo wins one! (...in gauge space)

# Lecture 3b. Relativity of wave-optics and Lorentz-Minkowski coordinates III. (1.30.14)

(Ch. 0-3 of Unit 8 CMwBang!)

1. That “old-time” relativity (Circa 600BCE- 1905CE)

(“Bouncing-photons” in smoke & mirrors and Thales, again)

The Ship and Lighthouse saga

Light-conic-sections make invariants

A politically incorrect analogy of rotational transformation and Lorentz transformation

The straight scoop on “angle” and “rapidity” (They’re area!)

Introducing the “Sin-Tan Rosetta Stone” (Thanks, Thales!)

Introducing the stellar aberration angle σ vs. rapidity ρ

How Minkowski’s space-time graphs help visualize relativity

Group vs. phase velocity and tangent contacts

# Lecture 4. Relativity of lightwaves and Lorentz-Minkowski coordinates IV. (1.30.14)

(Ch. 0-3 of Unit 8 CMwBang!)

More connections to conventional approach to relativity and old-fashioned formulas

Catching up to light (Coyote finally triumphs! Rest-frame at last.)

The most old-fashioned form(ula) of all: Thales & Euclid means

Galileo wins one! (...in gauge space) That “old-time” relativity (Circa 600BCE- 1905CE)

“Bouncing-photons” in smoke & mirrors

The Ship and Lighthouse saga

Light-conic-sections make invariants

A politically incorrect analogy of rotational transformation and Lorentz transformation

The straight scoop on “angle” and “rapidity” (They both are area!)

Introducing the “Sin-Tan Rosetta Stone” (Thanks, Thales!)

Introducing the stellar aberration angle σ vs. rapidity ρ

How Minkowski’s space-time graphs help visualize relativity

Group vs. phase velocity and tangent contacts

# Lecture 4b. Relativity of wave-optics and Lorentz-Minkowski coordinates IV-b. (2.6.14)

(Ch. 0-3 of Unit 8 CMwBang!)

More connections to conventional approach to relativity and old-fashioned formulas

Catching up to light (Coyote finally triumphs! Rest-frame at last.)

The most old-fashioned form(ula) of all: Thales & Euclid means

Galileo wins one! (...in gauge space) That “old-time” relativity (Circa 600BCE- 1905CE)

“Bouncing-photons” in smoke & mirrors

The Ship and Lighthouse saga

Light-conic-sections make invariants

A politically incorrect analogy of rotational transformation and Lorentz transformation

The straight scoop on “angle” and “rapidity” (They both are area!)

Introducing the “Sin-Tan Rosetta Stone” (Thanks, Thales!)

Introducing the stellar aberration angle σ vs. rapidity ρ

How Minkowski’s space-time graphs help visualize relativity

Group vs. phase velocity and tangent contacts

# Lecture 5. Relativity of wave-optics and Lorentz-Minkowski coordinates V. (2.11.14)

(Ch. 0-4 of Unit 8 CMwBang!)

Review of space-time (x,ct) and per-space-time (ω,ck) geometry

Space-time (x,ct) and per-space-time (ω,ck) geometry and its physics

All of those contraction and expansion coefficients

Detailed views Einstein time dilation

The old “smoke and mirrors” trick

Detailed views Lorentz contraction

Heighway’s paradox 1 and 2

Phase invariance used to derive (x,ct)↔(x′,ct′) Einstein Lorentz Transformations (ELT)

Introducing the stellar aberration angle σ vs. rapidity ρ

Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)†Lewis Carroll Epstein, Relativity Visualized
Trigonometry: From circular to hyperbolic and backInsight Press, San Francisco, CA 94107
Group vs. phase velocity and tangent contacts
See also: L. C. Epstein, Thinking Physics
Insight Press, San Francisco, CA 94107

# Lecture 5b. Relativity of wave-optics and Lorentz-Minkowski coordinates V-b. (2.13.14)

(Ch. 0-4 of Unit 8 CMwBang!)

Review of space-time (x,ct) and per-space-time (ω,ck) geometry

Space-time (x,ct) and per-space-time (ω,ck) geometry and its physics

All of those contraction and expansion coefficients

Detailed views Einstein time dilation

The old “smoke and mirrors” trick

Detailed views Lorentz contraction

⇒ Heighway’s paradox 1 and 2

Phase invariance used to derive (x,ct)↔(x′,ct′) Einstein Lorentz Transformations (ELT)

Introducing the stellar aberration angle σ vs. rapidity ρ

Finish “Sin-Tan” blackboard construction

Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)†Lewis Carroll Epstein, Relativity Visualized
Trigonometry: From circular to hyperbolic and backInsight Press, San Francisco, CA 94107
Group vs. phase velocity and tangent contacts
See also: L. C. Epstein, Thinking Physics
Insight Press, San Francisco, CA 94107

# Lecture 6. Introduction to Relativistic Classical and Quantum Mechanics I. (2.18.14)

(Ch. 2-5 of Unit 8 CMwBang!)

Review of group velocity vs. phase velocity and tangent contacts

“Sin-Tan” geometry in “Relawavity”  (https://hosted.uark.edu/~modphys/markup/RelaWavityWeb.html)

“Sinh-Tanh” geometry

“Per-Space-Time” geometry

“The rare case where group velocity is both Δω/Δk and dω/dk

How optical CW group and phase properties give relativistic mechanics

What’s the Matter with Mass?

Brief look at Higgs

Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)

What’s the matter with light?

Bohr-Schrodinger (BS) approximation throws out Mc2

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# Lecture 7. Relativistic Lagrangian and Hamiltonian Mechanics (2.20.14)

(Ch. 2-5 of Unit 8 CMwBang!)

Reviewing geometry of relativistic mechanics

Bohr-Schrodinger (BS) approximation throws out Mc2 (Is frequency really relative?)

Effect on group velocity (None) and phase velocity (Absurd)

Defining phase variables Lagrangian L, Hamiltonian H, and Action S

Poincare Invariants and Lagrangian-Hamiltonian relations

Poincare-Legendre contact transformation geometry

Feynman’s flying clock and phase minimization

Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)

# Lecture 8. Relativity of transverse waves and 4-vectors (2.25.14)

(Ch. 2-5 of CMwBang-Unit 8 Ch. 6 of QTforCA Unit 2)

Reviewing “Relawavity” geometry

Reviewing the stellar aberration angle σ vs. rapidity ρ

Pattern recognition: “Occam’s Sword”

Introducing per-spacetime 4-vector (ω0xyz) =(ω,ckx,cky,ckz) transformation

More details of Lorentz boost of North-South-East-West plane-wave 4-vectors ω0xyz)

Thales-like construction of Lorentz boost in 2D and 3D

The ellipsoid

Combination and interference of 4-vector plane waves (Idealized polarization case)

Combination group and phase waves define 4D Minkowski coordinates

Combination group and phase waves define wave guide dynamics

Waveguide dispersion and geometry

1st-quantized cavity modes

(And introducing 2nd-quantized cavity modes)

Lorentz symmetry effects

How it makes momentum and energy be conserved

# Lecture 9. Relativity of interfering and galloping waves: Amplitude and SWR. (3.04.14)-(3.06.14)

(Ch. 2-5 of CMwBang-Unit 8 Ch. 6 of QTforCA Unit 2)

Unmatched amplitudes giving galloping waves

Standing Wave Ratio (SWR) and Standing Wave Quotient (SWQ)

Analogy with group and phase

Galloping waves

Galloping dynamics algebra

Waves that go back in time - The Feynman-Wheeler Switchback

The Ship-Barn-and-Butler saga of confused causality

1st Quantization: Quantizing phase variables ω and k

Understanding how quantum transitions require “mixed-up” states

Closed cavity vs ring cavity

Relativistic effects on charge, current, and Maxwell Fields

Current density changes by Lorentz asynchrony

Magnetic B-field is relativistic sinhρ 1st order-effect

# Lecture 10. Relativity of 1st Quantization and electromagnetic fields (3.12.14)

(Ch. 2-5 of CMwBang-Unit 8 Ch. 6 of QTforCA Unit 2)

1st Quantization: Quantizing phase variables ω and k

Understanding how quantum transitions require “mixed-up” states

Closed cavity vs ring cavity

2nd Quantization: Quantizing amplitudes (“photons”,“vibrons”, and “what-ever-ons”)

Analogy with molecular Born-Oppenheimer-Approximate energy levels

Introducing coherent states (What lasers use)

Analogy with (ω,k) wave packets

Wave coordinates need coherence

Relativistic effects on charge, current, and magnetic fields

Current density changes by Lorentz asynchrony

Magnetic B-field is relativistic sinhρ 1st order-effect

# Lecture 11. Quantum theory of harmonic oscillators U(1)⊂U(2)⊂U(3)... (3.13.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 20-22; PSDS - Ch.8)

EM Waves are made of (relativistic) oscillators?

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

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

NEXT Lect 12:2-D aa algebra of U(2) representations and R(3) angular momentum operators

# Lecture 12. Spectral Analysis of U(2) Operators (3.22.14)-(3.23.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 21-22; PSDS - Ch. 8)

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

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

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

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

Spinor arithmetic like complex arithmetic

Spinor vector algebra like complex vector algebra

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

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

How probability ψ-waves and flux ψ-waves evolved

Properties of amplitude ψ*ψ-squares

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

Polarization ellipse and spinor state dynamics

Fundamental Euler R(αβγ) and Darboux R[ϕϑΘ] representations of U(2) and R(3) {From QTCALectures 8-9}

# Lecture 13. Quantum theory of harmonic oscillators U(1)⊂U(2)⊂U(3)... II. (4.1.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 21-22; PSDS - Ch. 8)

Based on QTCA Lectures 7, 23-25 from Group Theory in Quantum Mechanics

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

Review : 2-D Classical and semi-classical harmonic oscillator ABCD-analysis

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

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

Spinor-complex variable analogies: arithmetic, vector algebra, operator calculus

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

# Lecture 14. Rotational symmetry U(2)⊂U(3) and O(3) (4.1.14)-(4.3.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 21-22; (PSDS - Ch. 5, 7)

Based on QTCA Lectures 24-25 from Group Theory in Quantum Mechanics

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 15. Introduction to Rotational Eigenstates and Spectra I (4.9.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 21-25; PSDS - Ch. 5, 7 )

Based on QTCA Lectures 24-25 from Group Theory in Quantum Mechanics

Review : Applications of R(3) rotation and U(2) representations

Molecular and nuclear wavefunctions

Molecular and nuclear eigenlevels

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

Generalized Stern-Gerlach and transformation matrices

Angular momentum cones and high J properties

Asymmetric Top eigensolutions for J=1-2

New geometric approach to rotational eigenstates and spectra

# Lecture 16. Introduction to Rotational Eigenstates and Spectra II. (4.10.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 21-25; PSDS - Ch. 5, 7 )

Based on QTCA Lectures 24-25 from Group Theory in Quantum Mechanics

Review : Asymmetric Top eigensolutions for J=1-2 and D2 symmetry

New geometric approach to rotational eigenstates and spectra

Introduction to Rotational Energy Surfaces (RES) and multipole tensor expansion

Rank-2 tensors from D2-matrix

Building Hamiltonian H = AJx2 + BJy2 + CJz2 out of scalar and tensor operators

Comparing quantum and semi-classical calculations

Symmetric rotor levels and RES plots

Asymmetry rotor levels and RES plots

Spherical rotor levels and RES plots

SF6 spectral fine structure

CF4 spectral fine structure

# Lecture 17. Introduction to Rotational Eigenstates and Spectra III. (4.15.14)

(Int.J.Mol.Sci, 14, 714(2013) p.755-774; QTCA Unit 7 Ch. 21-25; PSDS - Ch. 5, 7 )

Based on QTCA Lectures 24-25 from Group Theory in Quantum Mechanics

Review : Building Hamiltonian H = AJx2 + BJy2 + CJz2 out of scalar and tensor operators

Review : Symmetric rotor levels and RES plots

Asymmetric rotor levels and RES plots

D2⊃C2 symmetry correlation

Spherical rotor levels and RES plots

Spectral fine structure of SF6, SiF4, C8H8, CF4,...

O⊃C4 and O⊃C3 symmetry correlation

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

Beginning theory

Rovibronic nomograms and PQR structure

Rovibronic energy surfaces (RES) and cone geometry

Spin symmetry correlation, tunneling, and entanglement

Analogy between PE surface dynamics and RES

Rotational Energy Eigenvalue Surfaces (REES)