(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
(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
(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)
(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!)
Galilean velocity addition becomes rapidity addition
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
(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!)
Galilean velocity addition becomes rapidity addition
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
(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!)
Galilean velocity addition becomes rapidity addition
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
(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
(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
(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
<(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)
Twin paradox resolved
(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 (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation
More details of Lorentz boost of North-South-East-West plane-wave 4-vectors ω0,ωx,ωy,ωz)
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
(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
(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
(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 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
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 a†a algebra of U(2) representations and R(3) angular momentum operators
(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) H=ωµσµ
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}
(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 a†a 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) H=ωµσµ
Spinor-complex variable analogies: arithmetic, vector algebra, operator calculus
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
(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 a†a 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
(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
(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
(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)